Physics Book - User contributions [en]

Vectors

2018-11-26T04:35:53Z

Bradak3:

Written by Elizabeth Robelo

Improved by Aparajita Satapathy

Improved by Lichao Tang

Improved by Jimin Yoon

Improved by Sabrina Seibel

Claimed: Sanjana Kumar Fall 2017

Improved by Audrey Suh (Spring 2018)

'''Improved by Heeva Taghian (Fall 2018)'''

Improved by Ben Radak (Fall 2018)

In physics, a vector is a quantity with a magnitude and a direction. It helps determine the position of one point relative to another.

==The Main Idea==

A vector <math> \vec A </math> is a quantity with a magnitude <math> |\vec A| </math> and a direction <math> \hat A </math>. The magnitude of a vector is a scalar quantity which represents the length of the vector but does not have a direction. A vector is represented by an arrow. The orientation of the vector represents its direction. The length of the vector represents its magnitude. When a vector is drawn, the starting point is the tail and the ending point is called the head, or the 'tip', of the vector. In physics, a vector always starts at the source and directs to the observation location. In other words, if you are drawing a vector, the tail of the vector will be located at the original location and the tip of the vector will be located at the observation location. Refer to the image below for a visual representation:
[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors, place the head of one vector at the tail of the other. The sum vector will be the arrow starting from the tail of the first vector to the head of the second vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors, reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar, we can stretch, compress, or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compressed. If the scalar is greater than 1, the vector will get stretched. If the scalar has a negative sign, then the vector reverses its direction, even if it has also been compressed or stretched.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by ''x'', ''y'', and ''z'' coordinates. They are written in the form <math> < X,\ Y,\ Z > </math> or <math> (X\ i + Y\ j + Z\ k) </math>.

The following calculations can be performed on vectors:

'''Finding the Magnitude''': <math> |\vec A| = \sqrt{x^2 + y^2 + z^2} </math>

'''Finding the Unit Vector''': :<math> \hat{A} = \frac{\vec A}{|\vec A|} </math>

'''Dot Product''': <math> |\vec A \cdot \vec B| = |\vec A| \ast |\vec B| \ast \cos\Theta </math>

'''Cross Product''': <math> |\vec A \times \vec B| = |\vec A| \ast |\vec B| \ast \sin\Theta </math>

'''Adding Two Vectors''': <math> < A_1,\ A_2,\ A_3 > + < B_1,\ B_2,\ B_3 >\ =\ < A_1 + B_1,\ A_2 + B_2,\ A_3 + B_3 > </math>

'''Multiplying Two Vectors''': <math> \vec A \times \vec B = < A_1,\ A_2,\ A_3 > \times < B_1,\ B_2,\ B_3 > = < A_2 \ast B_3 - A_3 \ast B_2 >\ i\ +\ < A_1 \ast B_3 - A_3 \ast B_1 >\ j\ +\ < A_1 \ast B_2 - A_2 \ast B_1 >\ k </math>

'''Multiplying a vector and a scalar''': <math> C\ \ast\ < A_1,\ A_2,\ A_3 >\ =\ < C \ast A_1,\ C \ast A_2,\ C \ast A_3 > </math>

<math>\vec A \cdot \vec B\ =\ \sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

===A Computational Model===

[[File:glowscript_ide11.jpg|200px|thumb|left|Vectors in GlowScript]]

VIDLE is an interactive editor for VPython, a programming language that is commonly used in Physics to create 3D displays and animations. It is also used to perform iterative calculations using fundamental principles. Codes on VIDLE are instructions for a computer to follow to make these calculations.

In VIDLE code, arrow objects usually represent vector components. Arrows can be defined by their position, axis, and color. Each part can be manipulated to achieve different results. The position and axis of arrows are vectors, so they can be scaled by multiplying by a scalar. Arrows are often used to represent relative position vectors, starting at position A and ending at position B or by finding the "final minus initial" (B-A) position vector. In the image, the relative position vector is of the tennis ball with respect to the baseball, so the arrow points from the baseball to the tennis ball. If a relative position vector is in the 3D-plane, three further arrows can be used to denote the x, y, and z components. The ''z'' component vector is referenced using the formula ''vectorname.z''.

In Physics 2, you will learn that a vector always starts at a source and points to the observation location at which physical quantities such as the electric or magnetic fields need to be found. Furthermore, it is important to be able to calculate the magnitude and direction of a vector in 3D space. [https://trinket.io/embed/glowscript/e17d933a59?outputOnly=true Here] is an example of a VPython model that computationally calculates such values. The green arrow represents the position vector which starts from a proton (red ball) to the arbitrary observation location. Click 'Run' on the upper-left corner in order to display the model.

--[[User:Htaghian3|Htaghian3]] ([[User talk:Htaghian3|talk]]) 01:54, 16 September 2018 (EDT)

==Examples==

===Simple===
Which of the following statements is correct? (Can be more than one)

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2 and option number 4.

===Intermediate===
1. What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

2. What is the cross product of A = <1,2,3> and B = <9,4,5>?

Use the equation for cross product: a x b= <a1, a2, a3> x <b1, b2, b3> = <a2*b3 - a3*b2> i - <a1*b3 - a3*b1> j + <a1*b2 - a2*b1> k

A x B<math>=<2*5 - 3*4, 1*5 - 3*9, 1*4 - 2*9> = <-2, -22, -14></math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
1.Vectors will be used in many applications in most calculation based fields when movement and position are involved. Vectors can be two dimensional or three dimensional. Vectors are used to represent forces, fields, and momentum.

2.Vectors has been used in many application problems in engineering majors. In engineering applications, vectors are used to a lot of quantities which have both magnitude and direction. Dividing a magnitude into vector quantities in the x,y, and z directions clarify which components of a vector have quantity. For example, in Biomedical Engineering applications, vectors are used to represent the velocity of a flow to further calculate the flow rate and some other related quantities.

3. Vectors play a huge part in industry. For example, in process flow, vectors play a huge part in most calculations. For example, many calculations in different fields of science and math use vector components and direction. Our car's GPS uses vectors even if we don't realize it!

==History==
The discovery and use of vectors can date back to the ancient philosophers, Aristotle and Heron. The theory can also be found in the first article of Newtons Principia Mathematica. In the early 19th century Caspar Wessel, Jean Robert Argand, Carl Friedrich Gauss, and a few more depicted and worked with complex numbers as points on a 2D plane. in 1827, August Ferdinand published a book introducing line segments labelled with letters. he wrote about vectors without the name "vector".

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors.
Also in 1835 Hamilton founded "quaternions", which were 4D planes and equations with vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimensional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

VPython was released by David Scherer in the year 2000. He came up with the idea after taking a physics class at Carnegie Mellon University. Previous programs only allowed for 2D modeling, so he took it upon himself to make something better. VPython, also known as Visual Python, allows for 3D modeling.
== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

Here is a link on more computational work with vectors:
[http://vpython.org/contents/docs/vector.html]

Here is a link detailing the basics of vectors:
[https://www.physics.uoguelph.ca/tutorials/vectors/vectors.html]

===Further Reading===

Vector Analysis by Josiah Willard Gibbs

Introduction to Matrices and Vectors by Jacob T. Schwartz

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

[[Category:Which Category did you place this in?]]

[https://hijabersea.com/posts/intro-to-glowscript.html]

Specific Heat

2016-11-28T03:57:18Z

Bradak3: /* References */

'''Improved by Ben Radak Fall 2016'''

Specific heat, also known as the specific heat capacity, is defined as the amount of energy required to raise the temperature of a unit mass by one degree Celsius. This unit mass is one gram. The idea of heat capacity was first speculated by Albert Einstein is 1907 with his specific heat of solids lattice vibrations model, and later expanded by Peter Debye. The units for specific heat are Joules per gram-degree Celsius (J / g °C). Specific heat is important as it can determine the thermal interaction a material has with other materials. We can test the validity of models with specific heat since it is experimentally measurable. The study of thermodynamics was sparked by the research done on specific heat. Thermodynamics is the study of the conversion of energy involving heat and temperature change of a system.

==The Main Idea==

The most common definition is that specific heat is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius. The specific heat of a substance depends on its phase (solid, liquid, or gas) and its molecular structure. The relationship between heat and temperature change is best defined by the constant "c" in the equation below. Specific Heat is an intensive property, meaning that the amount of substance does not affect this property, only the composition of the subtance.

[[File:Specific Heat Equation.gif]]

where:

m = the mass of the system (grams, g)

Q = the amount of heat added to the system (Joules, J)

T = the temperature change (in Celsius ºC or Kelvin ºK)

c = the specific heat of the system ([Joules/(gram x degree celsius)], J/gºC)

The formula below is a reconfigured version of the relationship above. It shows how to calculate "c" specifically from the basic principle.

[[File:specific heat formula.png|300px|thumb|left|]]

This equation does not apply if a phase change occurs (say from a liquid state to a gaseous state). This is because the amount of heat added or removed during a phase change does not change the overall temperature of the substance. So we disregard this relationship when phase changes take place.

The specific heat for solid can be calculated by the change in energy of the atoms over the change in temperature. The change in energy of the atoms is calculated by dividing the change in the energy of the system by the number of atoms in the substance.

The specific heat most commonly known is the specific heat for water, which is 4.186 J/g°C or 1 calorie/g°C. The specific heat of water is higher than any other common substance. Water has a very large specific heat on a per-gram basis, meaning that it takes a lot more added heat to cause a change in its temperature. Since the specific heat of water is so high, water can be used for temperature regulation. Due to the difference in atomic structures, the specific heat per gram of water is much higher than that of a metal substance. It is possible to predict the specific heat of any material, as long as you know about its atomic structure, as a rise in temperature is the increase in energy at the atomic level of substances. Generally, it is more more useful to compare molar specific heats of substances.

There are two models to determine the specific heats of substances at an atomic level. These are the Dulong-Petit Law and the Einstein-Debye model. The Dulong-Petit Law states that the molar specific heats of most solids (at room temperature or above) are almost constant. The Einstein-Debye model of specific heat states that specific heats drop at lower temperatures, as atomic processes become more relevant.

Here is an example of how to calculate specific heat.

'''Example: Find the specific heat of 350 g of an unknown substance when 34,700 Joules of heat are applied, and the temperature rises from 22ºC to 173ºC with no phase change.'''

'''We know that'''

m = 350 g

Q = 34,700 J

T initial = 22ºC

T final = 173ºC

c = ?

'''Using the formula above, c = Q / (mΔT)'''

c = 34,700 / (350*(173-22))

c = 34,700 / (350*151)

c = 34,700 / 52,850

'''c = 0.657 J/(gºC)'''

== Common Specific Heats and Noticeable Trends ==

Air
[[File:Airspecific.PNG]]

Although with small temperature changes, the change in specific heat with temperature is ignored, it is worth noting that it does change with temperature as evident with air.
The temperatures represented in the graph above are accessible. Specific heat typically decreases with a decrease in temperature and increases with an increase in temperature.
Fun Fact: Specific heat approaches 0 near absolute zero. This is explained by the Debye model in a later section.

[[File:Specific-heat-capacity.PNG]]

These are various specific heats of materials. Notice the difference of specific heat in the different phases of water.

http://physics.tutorcircle.com/heat/specific-heat.html

== Using Specific Heat (Problem Solving) ==

More often, the specific heat of a substance can be looked up in a table or online, and it is unnecessary to find the specific heat. Instead, you will be asked to find the final temperature of a system. This can be done using the principle of the conservation of Energy

'''Example: Coffee at 93ºC with a specific heat of 4.2 J/gºC is mixed with 55.3 grams of creams at 5ºC (specific heat of 3.8). What is the final temperature?'''

[[File:Specifc_heat_in_equation.gif]]

[[File:Eq3.gif]]

[[File:Eq4.gif]]

[[File:Eq5.gif]]

[[File:Eq6.gif]]

The above solution was completed using the Principle of the Conservation of Energy. This principle states that the change of energy in a system should be equal to zero because energy cannot be created nor destroyed, which gives us the 0 on the right hand side of the equation. Assuming their is no heat transfer from the surroundings, this principle applies. For both substances, the masses are multiplied by the specific heats and the difference in temperature, and this is done to find their change in internal energy. Through the concept of convection we recognize that the substance with the higher temperature will lose energy, and this energy will be absorbed by the other substance, giving us a net energy change of 0. To convince yourself of this you can look at the units, the mass multiplied by the specific heat multiplied by the change in temperature.

g * J/gºC * ºC = J

leaves us with the energy unit Joules. Since the net change must equal zero we recognize that the energy lost by one substance is gained by the other, and the final temperature of the combined substances can be found.

In all cases, the two objects at different temperatures will reach the same final temperature. This is Thermal Equilibrium.

== Law of Dulong and Petit ==

The Law of Dulong and Petit is a thermodynamic law discovered in 1819 by the French physicists Pierre Louis Dulong and Alexis Thérèse Petit. It yields the expression for the molar specific heat capacity of certain chemical elements. They found, through experiments, that the mass specific heat capacity for many elements was close to a constant value, after it had been adjusted to reflect the relative atomic weight of the element.

Basically, Dulong and Petit found that the heat capacity of a mole of numerous solid elements is about 3R, where R is the universal gas constant. Dulong and Petit were unaware of the relationship to R, since it had not yet been defined. The value of 3R is about 25 joules per kelvin, and Dulong and Petit found that this was the approximate heat capacity of some solid elements per mole of atoms they contained.

'''Example''' The specific heat of copper is 0.389 J/gm K. The specific heat of lead is 0.128 J/gm K. Why are the values so different in these two metals? Did you notice that they are expressed as energy per unit mass? If you express each as energy per mole, they are actually very similar. The Law of Dulong and Petit addresses this similarity in molar specific heats. It can be accounted for by applying equipartition of energy to the atoms of solids.

[[File:Dulong.gif]]

When observed on a molar basis, the specific heats of copper and lead are as follows:

Copper: 0.386 J/gm K * 63.6 gm/mol = 24.6 J/mol K

Lead: 0.128 J/gm K * 207 gm/mol = 26.5 J/mol K

Other molar specific heats of metals are shown below:

Aluminum: 24.3 J/mol K

Gold: 25.6 J/mol K

Silver: 24.9 J/mol K

Zinc: 25.2 J/mol K

As you can see, molar specific heats of many metals are around 25 J/mol K and are really very similar.

== Einstein Debye Model ==

Einstein and Debye had developed models for specific heat separately. Einstein's model stated that low energy excitation of a solid material was caused by oscillation of a single atom, whereas Debye's model stated that phonons or collective modes iterating through a material caused excitations. However, these two models are able to be put together to find the specific heat given by the following formula.

[[File:einstein debye.png]]

For low temperatures, Einstein and Debye found that the Law of Dulong and Petit was not applicable. At lower temperatures, it was found that atomic interactions were deemed significant in calculating the molar specific heat of an object.

[[File: Einstein Debye Graphs.gif]]

According to the Einstein Debye Model for Copper and Aluminum, specific heat varies a lot at lower temperatures and goes much below the Dulong-Petit Model. This is due to increased effects on specific heat by interatomic forces. However, for very high temperatures, the Einstein-Debye Model cannot be used. In fact, at high temperatures, Einstein's expression of specific heat reduces to the Dulong-Petit mathematical expression.

The Einstein Debye Equation is below.

[[File:Einstein Debye Equation.gif]]

For high temperatures it may be reduced like this:

[[File:Einstein Debye for High Temperatures.gif]]

This actually reduces to the Dulong-Petit Formula for Specific Heat:

[[File:Dulong Petit.gif]]

===Specific Heats of Gases===

The specific heats of gases are usually expressed as molar specific heats. For an ideal gas, the internal energy is all in the form of kinetic energy. The expression for the internal energy is

[[File:U.png]]

There are two specific heats for gases, one for gases at a constant volume and one gases at a constant pressure. In the formula below, the gas has a constant volume:

[[File:constant volume.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For an ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant volume.gif]]

For a constant pressure, specific heat can be derived as:

[[File:Constant Pressure.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For and ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant pressure.gif]]

The molar specific heats of gases all gravitate towards these ranges depending on the conditions the gas is kept in.

== Specific Heat in Thermodynamics ==

While working on thermodynamic processes specific heat is differentiated between specific heat at a constant temperate <Cp> and specific heat at a constant volume <Cv>. For the scope of Physics 2211 it is not necessary to note this distinction. However, <Cp> and <Cv> can be indicative to the value of the other because

<Cp> = <Cv> + R

where R is the gas constant.

While dealing with in Thermodynamic processes it is also important to recognize that <Cp> values change through out different temperature ranges. To account for this the follow equation is utilized:

[[File:cpr.png]]

Where R is the gas constant, and A, B, C, and D are the heat capacity constants for the specific substance of interest which can be found in the back of thermodynamic textbooks or online.

'''Example of Heat Capacities Table'''

[[File:cpcon.png]]

You may notice is in the particular table there are no D values listed. That is because D values are equal to 0 for gases, and C values are equal to 0 in liquids and solids.

It is easiest to find the Cp value by plugging in the constants into a program.

If working with vPython or glowscript you would simply have to write down the constants. Below is an example of what the program may look like.

[[File:glow.png]]

The units which you should use for your temperature will be specified in your table where you get you A, B, C, and D values. From the table you will also note the units to determine which gas constant R is appropriate for the equation.

==Connectedness==

Specific heat and thermodynamics are used often in chemistry, nuclear engineering, aerodynamics, and mechanical engineering. It is also used in everyday life in the radiator and cooling system of a car.

Specific heat can have a lot to do with prosthetic manufacturing, which is a focus in Biomedical Engineering. Prosthetics materials must be durable and easy to manipulate in a normal range of temperatures. In order to created medical devices, specific heats must be known, especially for welding or molding things, which require a specific temperature to be effective. At higher temperatures, the Dulong-Petit law must be used to calculate the specific heat of an object. Especially for solid metal objects, which would be used in prosthetics, Dulong-Petit is very useful.

Cooking materials such as pots and pans are made to have a low specific heat so that they need less heat to raise their temperature. This allows for faster cooking processes. The handles of these cooking utensils are made of substances with high specific heats so that their temperature won’t rise too much if a large amount of heat is absorbed.

Have you ever noticed that sand on the beach can burn your feet but the ocean water is cool and refreshing? Sand has a lower specific heat than ocean water. So when the sun is beating down, the temperature of the land increases faster than that of the sea.

Insulation is made of materials with high specific heat so that they won't change temperature easily. For example, wood has a high specific heat. A wooden house helps keep the inside cooler during summer because it requires lots of heat to change its temperature. Builders can choose certain materials which allows us to build houses for specific locations and altitudes.

== See also ==

'''Further reading'''

[http://www.physicsbook.gatech.edu/Kinds_of_Matter Kinds of Matter]

[http://www.physicsbook.gatech.edu/Boiling_Point Boiling Point]

[http://www.physicsbook.gatech.edu/Melting_Point Melting Point]

[http://www.physicsbook.gatech.edu/Thermal_Energy Thermal Energy]

[http://www.physicsbook.gatech.edu/Heat_Capacity Heat Capacity]

[http://www.physicsbook.gatech.edu/Specific_Heat_Capacity Specific Heat Capacity]

[http://www.physicsbook.gatech.edu/First_Law_of_Thermodynamics Thermodynamics]

'''External Links'''

[http://www.wisegeek.org/what-is-specific-heat.htm WiseGeek.org]

[https://www.khanacademy.org/science/biology/water-acids-and-bases/water-as-a-solid-liquid-and-gas/v/specific-heat-of-water KhanAcademy.org]

==References==

This section contains the references I used while writing this page:

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html

http://scienceworld.wolfram.com/physics/SpecificHeat.html

http://www.wikihow.com/Calculate-Specific-Heat

https://www.khanacademy.org/

http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/heat/heatcapacityapplication.html

http://brainly.in/question/40990

http://www.tutorvista.com/content/physics/physics-iii/heat-and-thermodynamics/dulong-and-petit-law.php

Matter & Interactions Vol I. Chabay Sherwood

http://www.personal.utulsa.edu/~geoffrey-price/Courses/ChE7023/HeatCapacity-HeatOfFormation.pdf

https://engineering.ucsb.edu/~shell/che110a/heatcapacitycalculations.pdf

http://faculty.uca.edu/saddison/ThermalPhysics/Heat%20Capacity.pdf

https://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm

http://physics.tutorcircle.com/heat/specific-heat.html

Introduction to Chemical Engineering Thermodynamics Seventh Edition. J. M. Smith, H. C. Van Ness, Michael M. Abbott

This page was last modified on 27 November 2016, by Panna Rasania.

Specific Heat

2016-11-28T03:51:53Z

Bradak3: /* Common Specific Heats and Noticeable Trends */

'''Improved by Ben Radak Fall 2016'''

Specific heat, also known as the specific heat capacity, is defined as the amount of energy required to raise the temperature of a unit mass by one degree Celsius. This unit mass is one gram. The idea of heat capacity was first speculated by Albert Einstein is 1907 with his specific heat of solids lattice vibrations model, and later expanded by Peter Debye. The units for specific heat are Joules per gram-degree Celsius (J / g °C). Specific heat is important as it can determine the thermal interaction a material has with other materials. We can test the validity of models with specific heat since it is experimentally measurable. The study of thermodynamics was sparked by the research done on specific heat. Thermodynamics is the study of the conversion of energy involving heat and temperature change of a system.

==The Main Idea==

The most common definition is that specific heat is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius. The specific heat of a substance depends on its phase (solid, liquid, or gas) and its molecular structure. The relationship between heat and temperature change is best defined by the constant "c" in the equation below. Specific Heat is an intensive property, meaning that the amount of substance does not affect this property, only the composition of the subtance.

[[File:Specific Heat Equation.gif]]

where:

m = the mass of the system (grams, g)

Q = the amount of heat added to the system (Joules, J)

T = the temperature change (in Celsius ºC or Kelvin ºK)

c = the specific heat of the system ([Joules/(gram x degree celsius)], J/gºC)

The formula below is a reconfigured version of the relationship above. It shows how to calculate "c" specifically from the basic principle.

[[File:specific heat formula.png|300px|thumb|left|]]

This equation does not apply if a phase change occurs (say from a liquid state to a gaseous state). This is because the amount of heat added or removed during a phase change does not change the overall temperature of the substance. So we disregard this relationship when phase changes take place.

The specific heat for solid can be calculated by the change in energy of the atoms over the change in temperature. The change in energy of the atoms is calculated by dividing the change in the energy of the system by the number of atoms in the substance.

The specific heat most commonly known is the specific heat for water, which is 4.186 J/g°C or 1 calorie/g°C. The specific heat of water is higher than any other common substance. Water has a very large specific heat on a per-gram basis, meaning that it takes a lot more added heat to cause a change in its temperature. Since the specific heat of water is so high, water can be used for temperature regulation. Due to the difference in atomic structures, the specific heat per gram of water is much higher than that of a metal substance. It is possible to predict the specific heat of any material, as long as you know about its atomic structure, as a rise in temperature is the increase in energy at the atomic level of substances. Generally, it is more more useful to compare molar specific heats of substances.

There are two models to determine the specific heats of substances at an atomic level. These are the Dulong-Petit Law and the Einstein-Debye model. The Dulong-Petit Law states that the molar specific heats of most solids (at room temperature or above) are almost constant. The Einstein-Debye model of specific heat states that specific heats drop at lower temperatures, as atomic processes become more relevant.

Here is an example of how to calculate specific heat.

'''Example: Find the specific heat of 350 g of an unknown substance when 34,700 Joules of heat are applied, and the temperature rises from 22ºC to 173ºC with no phase change.'''

'''We know that'''

m = 350 g

Q = 34,700 J

T initial = 22ºC

T final = 173ºC

c = ?

'''Using the formula above, c = Q / (mΔT)'''

c = 34,700 / (350*(173-22))

c = 34,700 / (350*151)

c = 34,700 / 52,850

'''c = 0.657 J/(gºC)'''

== Common Specific Heats and Noticeable Trends ==

Air
[[File:Airspecific.PNG]]

Although with small temperature changes, the change in specific heat with temperature is ignored, it is worth noting that it does change with temperature as evident with air.
The temperatures represented in the graph above are accessible. Specific heat typically decreases with a decrease in temperature and increases with an increase in temperature.
Fun Fact: Specific heat approaches 0 near absolute zero. This is explained by the Debye model in a later section.

[[File:Specific-heat-capacity.PNG]]

These are various specific heats of materials. Notice the difference of specific heat in the different phases of water.

http://physics.tutorcircle.com/heat/specific-heat.html

== Using Specific Heat (Problem Solving) ==

More often, the specific heat of a substance can be looked up in a table or online, and it is unnecessary to find the specific heat. Instead, you will be asked to find the final temperature of a system. This can be done using the principle of the conservation of Energy

'''Example: Coffee at 93ºC with a specific heat of 4.2 J/gºC is mixed with 55.3 grams of creams at 5ºC (specific heat of 3.8). What is the final temperature?'''

[[File:Specifc_heat_in_equation.gif]]

[[File:Eq3.gif]]

[[File:Eq4.gif]]

[[File:Eq5.gif]]

[[File:Eq6.gif]]

The above solution was completed using the Principle of the Conservation of Energy. This principle states that the change of energy in a system should be equal to zero because energy cannot be created nor destroyed, which gives us the 0 on the right hand side of the equation. Assuming their is no heat transfer from the surroundings, this principle applies. For both substances, the masses are multiplied by the specific heats and the difference in temperature, and this is done to find their change in internal energy. Through the concept of convection we recognize that the substance with the higher temperature will lose energy, and this energy will be absorbed by the other substance, giving us a net energy change of 0. To convince yourself of this you can look at the units, the mass multiplied by the specific heat multiplied by the change in temperature.

g * J/gºC * ºC = J

leaves us with the energy unit Joules. Since the net change must equal zero we recognize that the energy lost by one substance is gained by the other, and the final temperature of the combined substances can be found.

In all cases, the two objects at different temperatures will reach the same final temperature. This is Thermal Equilibrium.

== Law of Dulong and Petit ==

The Law of Dulong and Petit is a thermodynamic law discovered in 1819 by the French physicists Pierre Louis Dulong and Alexis Thérèse Petit. It yields the expression for the molar specific heat capacity of certain chemical elements. They found, through experiments, that the mass specific heat capacity for many elements was close to a constant value, after it had been adjusted to reflect the relative atomic weight of the element.

Basically, Dulong and Petit found that the heat capacity of a mole of numerous solid elements is about 3R, where R is the universal gas constant. Dulong and Petit were unaware of the relationship to R, since it had not yet been defined. The value of 3R is about 25 joules per kelvin, and Dulong and Petit found that this was the approximate heat capacity of some solid elements per mole of atoms they contained.

'''Example''' The specific heat of copper is 0.389 J/gm K. The specific heat of lead is 0.128 J/gm K. Why are the values so different in these two metals? Did you notice that they are expressed as energy per unit mass? If you express each as energy per mole, they are actually very similar. The Law of Dulong and Petit addresses this similarity in molar specific heats. It can be accounted for by applying equipartition of energy to the atoms of solids.

[[File:Dulong.gif]]

When observed on a molar basis, the specific heats of copper and lead are as follows:

Copper: 0.386 J/gm K * 63.6 gm/mol = 24.6 J/mol K

Lead: 0.128 J/gm K * 207 gm/mol = 26.5 J/mol K

Other molar specific heats of metals are shown below:

Aluminum: 24.3 J/mol K

Gold: 25.6 J/mol K

Silver: 24.9 J/mol K

Zinc: 25.2 J/mol K

As you can see, molar specific heats of many metals are around 25 J/mol K and are really very similar.

== Einstein Debye Model ==

Einstein and Debye had developed models for specific heat separately. Einstein's model stated that low energy excitation of a solid material was caused by oscillation of a single atom, whereas Debye's model stated that phonons or collective modes iterating through a material caused excitations. However, these two models are able to be put together to find the specific heat given by the following formula.

[[File:einstein debye.png]]

For low temperatures, Einstein and Debye found that the Law of Dulong and Petit was not applicable. At lower temperatures, it was found that atomic interactions were deemed significant in calculating the molar specific heat of an object.

[[File: Einstein Debye Graphs.gif]]

According to the Einstein Debye Model for Copper and Aluminum, specific heat varies a lot at lower temperatures and goes much below the Dulong-Petit Model. This is due to increased effects on specific heat by interatomic forces. However, for very high temperatures, the Einstein-Debye Model cannot be used. In fact, at high temperatures, Einstein's expression of specific heat reduces to the Dulong-Petit mathematical expression.

The Einstein Debye Equation is below.

[[File:Einstein Debye Equation.gif]]

For high temperatures it may be reduced like this:

[[File:Einstein Debye for High Temperatures.gif]]

This actually reduces to the Dulong-Petit Formula for Specific Heat:

[[File:Dulong Petit.gif]]

===Specific Heats of Gases===

The specific heats of gases are usually expressed as molar specific heats. For an ideal gas, the internal energy is all in the form of kinetic energy. The expression for the internal energy is

[[File:U.png]]

There are two specific heats for gases, one for gases at a constant volume and one gases at a constant pressure. In the formula below, the gas has a constant volume:

[[File:constant volume.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For an ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant volume.gif]]

For a constant pressure, specific heat can be derived as:

[[File:Constant Pressure.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For and ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant pressure.gif]]

The molar specific heats of gases all gravitate towards these ranges depending on the conditions the gas is kept in.

== Specific Heat in Thermodynamics ==

While working on thermodynamic processes specific heat is differentiated between specific heat at a constant temperate <Cp> and specific heat at a constant volume <Cv>. For the scope of Physics 2211 it is not necessary to note this distinction. However, <Cp> and <Cv> can be indicative to the value of the other because

<Cp> = <Cv> + R

where R is the gas constant.

While dealing with in Thermodynamic processes it is also important to recognize that <Cp> values change through out different temperature ranges. To account for this the follow equation is utilized:

[[File:cpr.png]]

Where R is the gas constant, and A, B, C, and D are the heat capacity constants for the specific substance of interest which can be found in the back of thermodynamic textbooks or online.

'''Example of Heat Capacities Table'''

[[File:cpcon.png]]

You may notice is in the particular table there are no D values listed. That is because D values are equal to 0 for gases, and C values are equal to 0 in liquids and solids.

It is easiest to find the Cp value by plugging in the constants into a program.

If working with vPython or glowscript you would simply have to write down the constants. Below is an example of what the program may look like.

[[File:glow.png]]

The units which you should use for your temperature will be specified in your table where you get you A, B, C, and D values. From the table you will also note the units to determine which gas constant R is appropriate for the equation.

==Connectedness==

Specific heat and thermodynamics are used often in chemistry, nuclear engineering, aerodynamics, and mechanical engineering. It is also used in everyday life in the radiator and cooling system of a car.

Specific heat can have a lot to do with prosthetic manufacturing, which is a focus in Biomedical Engineering. Prosthetics materials must be durable and easy to manipulate in a normal range of temperatures. In order to created medical devices, specific heats must be known, especially for welding or molding things, which require a specific temperature to be effective. At higher temperatures, the Dulong-Petit law must be used to calculate the specific heat of an object. Especially for solid metal objects, which would be used in prosthetics, Dulong-Petit is very useful.

Cooking materials such as pots and pans are made to have a low specific heat so that they need less heat to raise their temperature. This allows for faster cooking processes. The handles of these cooking utensils are made of substances with high specific heats so that their temperature won’t rise too much if a large amount of heat is absorbed.

Have you ever noticed that sand on the beach can burn your feet but the ocean water is cool and refreshing? Sand has a lower specific heat than ocean water. So when the sun is beating down, the temperature of the land increases faster than that of the sea.

Insulation is made of materials with high specific heat so that they won't change temperature easily. For example, wood has a high specific heat. A wooden house helps keep the inside cooler during summer because it requires lots of heat to change its temperature. Builders can choose certain materials which allows us to build houses for specific locations and altitudes.

== See also ==

'''Further reading'''

[http://www.physicsbook.gatech.edu/Kinds_of_Matter Kinds of Matter]

[http://www.physicsbook.gatech.edu/Boiling_Point Boiling Point]

[http://www.physicsbook.gatech.edu/Melting_Point Melting Point]

[http://www.physicsbook.gatech.edu/Thermal_Energy Thermal Energy]

[http://www.physicsbook.gatech.edu/Heat_Capacity Heat Capacity]

[http://www.physicsbook.gatech.edu/Specific_Heat_Capacity Specific Heat Capacity]

[http://www.physicsbook.gatech.edu/First_Law_of_Thermodynamics Thermodynamics]

'''External Links'''

[http://www.wisegeek.org/what-is-specific-heat.htm WiseGeek.org]

[https://www.khanacademy.org/science/biology/water-acids-and-bases/water-as-a-solid-liquid-and-gas/v/specific-heat-of-water KhanAcademy.org]

==References==

This section contains the references I used while writing this page:

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html

http://scienceworld.wolfram.com/physics/SpecificHeat.html

http://www.wikihow.com/Calculate-Specific-Heat

https://www.khanacademy.org/

http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/heat/heatcapacityapplication.html

http://brainly.in/question/40990

http://www.tutorvista.com/content/physics/physics-iii/heat-and-thermodynamics/dulong-and-petit-law.php

Matter & Interactions Vol I. Chabay Sherwood

http://www.personal.utulsa.edu/~geoffrey-price/Courses/ChE7023/HeatCapacity-HeatOfFormation.pdf

https://engineering.ucsb.edu/~shell/che110a/heatcapacitycalculations.pdf

http://faculty.uca.edu/saddison/ThermalPhysics/Heat%20Capacity.pdf

https://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm

Introduction to Chemical Engineering Thermodynamics Seventh Edition. J. M. Smith, H. C. Van Ness, Michael M. Abbott

This page was last modified on 27 November 2016, by Panna Rasania.

Specific Heat

2016-11-28T03:34:11Z

Bradak3: /* Common Specific Heats and Noticeable Trends */

'''Improved by Ben Radak Fall 2016'''

Specific heat, also known as the specific heat capacity, is defined as the amount of energy required to raise the temperature of a unit mass by one degree Celsius. This unit mass is one gram. The idea of heat capacity was first speculated by Albert Einstein is 1907 with his specific heat of solids lattice vibrations model, and later expanded by Peter Debye. The units for specific heat are Joules per gram-degree Celsius (J / g °C). Specific heat is important as it can determine the thermal interaction a material has with other materials. We can test the validity of models with specific heat since it is experimentally measurable. The study of thermodynamics was sparked by the research done on specific heat. Thermodynamics is the study of the conversion of energy involving heat and temperature change of a system.

==The Main Idea==

The most common definition is that specific heat is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius. The specific heat of a substance depends on its phase (solid, liquid, or gas) and its molecular structure. The relationship between heat and temperature change is best defined by the constant "c" in the equation below. Specific Heat is an intensive property, meaning that the amount of substance does not affect this property, only the composition of the subtance.

[[File:Specific Heat Equation.gif]]

where:

m = the mass of the system (grams, g)

Q = the amount of heat added to the system (Joules, J)

T = the temperature change (in Celsius ºC or Kelvin ºK)

c = the specific heat of the system ([Joules/(gram x degree celsius)], J/gºC)

The formula below is a reconfigured version of the relationship above. It shows how to calculate "c" specifically from the basic principle.

[[File:specific heat formula.png|300px|thumb|left|]]

This equation does not apply if a phase change occurs (say from a liquid state to a gaseous state). This is because the amount of heat added or removed during a phase change does not change the overall temperature of the substance. So we disregard this relationship when phase changes take place.

The specific heat for solid can be calculated by the change in energy of the atoms over the change in temperature. The change in energy of the atoms is calculated by dividing the change in the energy of the system by the number of atoms in the substance.

The specific heat most commonly known is the specific heat for water, which is 4.186 J/g°C or 1 calorie/g°C. The specific heat of water is higher than any other common substance. Water has a very large specific heat on a per-gram basis, meaning that it takes a lot more added heat to cause a change in its temperature. Since the specific heat of water is so high, water can be used for temperature regulation. Due to the difference in atomic structures, the specific heat per gram of water is much higher than that of a metal substance. It is possible to predict the specific heat of any material, as long as you know about its atomic structure, as a rise in temperature is the increase in energy at the atomic level of substances. Generally, it is more more useful to compare molar specific heats of substances.

There are two models to determine the specific heats of substances at an atomic level. These are the Dulong-Petit Law and the Einstein-Debye model. The Dulong-Petit Law states that the molar specific heats of most solids (at room temperature or above) are almost constant. The Einstein-Debye model of specific heat states that specific heats drop at lower temperatures, as atomic processes become more relevant.

Here is an example of how to calculate specific heat.

'''Example: Find the specific heat of 350 g of an unknown substance when 34,700 Joules of heat are applied, and the temperature rises from 22ºC to 173ºC with no phase change.'''

'''We know that'''

m = 350 g

Q = 34,700 J

T initial = 22ºC

T final = 173ºC

c = ?

'''Using the formula above, c = Q / (mΔT)'''

c = 34,700 / (350*(173-22))

c = 34,700 / (350*151)

c = 34,700 / 52,850

'''c = 0.657 J/(gºC)'''

== Common Specific Heats and Noticeable Trends ==

[[File:Airspecific.PNG]]

Although with small temperature changes, the change in specific heat with temperature is ignored, it is worth noting that it does change with temperature as evident with air.

The temperatures represented in the graph above are accessible. At extremely low temperatures, the specific heats of materials approaches 0.

[[File:Specific-heat-capacity.PNG]]

These are

http://physics.tutorcircle.com/heat/specific-heat.html

== Using Specific Heat (Problem Solving) ==

More often, the specific heat of a substance can be looked up in a table or online, and it is unnecessary to find the specific heat. Instead, you will be asked to find the final temperature of a system. This can be done using the principle of the conservation of Energy

'''Example: Coffee at 93ºC with a specific heat of 4.2 J/gºC is mixed with 55.3 grams of creams at 5ºC (specific heat of 3.8). What is the final temperature?'''

[[File:Specifc_heat_in_equation.gif]]

[[File:Eq3.gif]]

[[File:Eq4.gif]]

[[File:Eq5.gif]]

[[File:Eq6.gif]]

The above solution was completed using the Principle of the Conservation of Energy. This principle states that the change of energy in a system should be equal to zero because energy cannot be created nor destroyed, which gives us the 0 on the right hand side of the equation. Assuming their is no heat transfer from the surroundings, this principle applies. For both substances, the masses are multiplied by the specific heats and the difference in temperature, and this is done to find their change in internal energy. Through the concept of convection we recognize that the substance with the higher temperature will lose energy, and this energy will be absorbed by the other substance, giving us a net energy change of 0. To convince yourself of this you can look at the units, the mass multiplied by the specific heat multiplied by the change in temperature.

g * J/gºC * ºC = J

leaves us with the energy unit Joules. Since the net change must equal zero we recognize that the energy lost by one substance is gained by the other, and the final temperature of the combined substances can be found.

In all cases, the two objects at different temperatures will reach the same final temperature. This is Thermal Equilibrium.

== Law of Dulong and Petit ==

The Law of Dulong and Petit is a thermodynamic law discovered in 1819 by the French physicists Pierre Louis Dulong and Alexis Thérèse Petit. It yields the expression for the molar specific heat capacity of certain chemical elements. They found, through experiments, that the mass specific heat capacity for many elements was close to a constant value, after it had been adjusted to reflect the relative atomic weight of the element.

Basically, Dulong and Petit found that the heat capacity of a mole of numerous solid elements is about 3R, where R is the universal gas constant. Dulong and Petit were unaware of the relationship to R, since it had not yet been defined. The value of 3R is about 25 joules per kelvin, and Dulong and Petit found that this was the approximate heat capacity of some solid elements per mole of atoms they contained.

'''Example''' The specific heat of copper is 0.389 J/gm K. The specific heat of lead is 0.128 J/gm K. Why are the values so different in these two metals? Did you notice that they are expressed as energy per unit mass? If you express each as energy per mole, they are actually very similar. The Law of Dulong and Petit addresses this similarity in molar specific heats. It can be accounted for by applying equipartition of energy to the atoms of solids.

[[File:Dulong.gif]]

When observed on a molar basis, the specific heats of copper and lead are as follows:

Copper: 0.386 J/gm K * 63.6 gm/mol = 24.6 J/mol K

Lead: 0.128 J/gm K * 207 gm/mol = 26.5 J/mol K

Other molar specific heats of metals are shown below:

Aluminum: 24.3 J/mol K

Gold: 25.6 J/mol K

Silver: 24.9 J/mol K

Zinc: 25.2 J/mol K

As you can see, molar specific heats of many metals are around 25 J/mol K and are really very similar.

== Einstein Debye Model ==

Einstein and Debye had developed models for specific heat separately. Einstein's model stated that low energy excitation of a solid material was caused by oscillation of a single atom, whereas Debye's model stated that phonons or collective modes iterating through a material caused excitations. However, these two models are able to be put together to find the specific heat given by the following formula.

[[File:einstein debye.png]]

For low temperatures, Einstein and Debye found that the Law of Dulong and Petit was not applicable. At lower temperatures, it was found that atomic interactions were deemed significant in calculating the molar specific heat of an object.

[[File: Einstein Debye Graphs.gif]]

According to the Einstein Debye Model for Copper and Aluminum, specific heat varies a lot at lower temperatures and goes much below the Dulong-Petit Model. This is due to increased effects on specific heat by interatomic forces. However, for very high temperatures, the Einstein-Debye Model cannot be used. In fact, at high temperatures, Einstein's expression of specific heat reduces to the Dulong-Petit mathematical expression.

The Einstein Debye Equation is below.

[[File:Einstein Debye Equation.gif]]

For high temperatures it may be reduced like this:

[[File:Einstein Debye for High Temperatures.gif]]

This actually reduces to the Dulong-Petit Formula for Specific Heat:

[[File:Dulong Petit.gif]]

===Specific Heats of Gases===

The specific heats of gases are usually expressed as molar specific heats. For an ideal gas, the internal energy is all in the form of kinetic energy. The expression for the internal energy is

[[File:U.png]]

There are two specific heats for gases, one for gases at a constant volume and one gases at a constant pressure. In the formula below, the gas has a constant volume:

[[File:constant volume.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For an ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant volume.gif]]

For a constant pressure, specific heat can be derived as:

[[File:Constant Pressure.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For and ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant pressure.gif]]

The molar specific heats of gases all gravitate towards these ranges depending on the conditions the gas is kept in.

== Specific Heat in Thermodynamics ==

While working on thermodynamic processes specific heat is differentiated between specific heat at a constant temperate <Cp> and specific heat at a constant volume <Cv>. For the scope of Physics 2211 it is not necessary to note this distinction. However, <Cp> and <Cv> can be indicative to the value of the other because

<Cp> = <Cv> + R

where R is the gas constant.

While dealing with in Thermodynamic processes it is also important to recognize that <Cp> values change through out different temperature ranges. To account for this the follow equation is utilized:

[[File:cpr.png]]

Where R is the gas constant, and A, B, C, and D are the heat capacity constants for the specific substance of interest which can be found in the back of thermodynamic textbooks or online.

'''Example of Heat Capacities Table'''

[[File:cpcon.png]]

You may notice is in the particular table there are no D values listed. That is because D values are equal to 0 for gases, and C values are equal to 0 in liquids and solids.

It is easiest to find the Cp value by plugging in the constants into a program.

If working with vPython or glowscript you would simply have to write down the constants. Below is an example of what the program may look like.

[[File:glow.png]]

The units which you should use for your temperature will be specified in your table where you get you A, B, C, and D values. From the table you will also note the units to determine which gas constant R is appropriate for the equation.

==Connectedness==

Specific heat and thermodynamics are used often in chemistry, nuclear engineering, aerodynamics, and mechanical engineering. It is also used in everyday life in the radiator and cooling system of a car.

Specific heat can have a lot to do with prosthetic manufacturing, which is a focus in Biomedical Engineering. Prosthetics materials must be durable and easy to manipulate in a normal range of temperatures. In order to created medical devices, specific heats must be known, especially for welding or molding things, which require a specific temperature to be effective. At higher temperatures, the Dulong-Petit law must be used to calculate the specific heat of an object. Especially for solid metal objects, which would be used in prosthetics, Dulong-Petit is very useful.

Cooking materials such as pots and pans are made to have a low specific heat so that they need less heat to raise their temperature. This allows for faster cooking processes. The handles of these cooking utensils are made of substances with high specific heats so that their temperature won’t rise too much if a large amount of heat is absorbed.

Have you ever noticed that sand on the beach can burn your feet but the ocean water is cool and refreshing? Sand has a lower specific heat than ocean water. So when the sun is beating down, the temperature of the land increases faster than that of the sea.

Insulation is made of materials with high specific heat so that they won't change temperature easily. For example, wood has a high specific heat. A wooden house helps keep the inside cooler during summer because it requires lots of heat to change its temperature. Builders can choose certain materials which allows us to build houses for specific locations and altitudes.

== See also ==

'''Further reading'''

[http://www.physicsbook.gatech.edu/Kinds_of_Matter Kinds of Matter]

[http://www.physicsbook.gatech.edu/Boiling_Point Boiling Point]

[http://www.physicsbook.gatech.edu/Melting_Point Melting Point]

[http://www.physicsbook.gatech.edu/Thermal_Energy Thermal Energy]

[http://www.physicsbook.gatech.edu/Heat_Capacity Heat Capacity]

[http://www.physicsbook.gatech.edu/Specific_Heat_Capacity Specific Heat Capacity]

[http://www.physicsbook.gatech.edu/First_Law_of_Thermodynamics Thermodynamics]

'''External Links'''

[http://www.wisegeek.org/what-is-specific-heat.htm WiseGeek.org]

[https://www.khanacademy.org/science/biology/water-acids-and-bases/water-as-a-solid-liquid-and-gas/v/specific-heat-of-water KhanAcademy.org]

==References==

This section contains the references I used while writing this page:

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html

http://scienceworld.wolfram.com/physics/SpecificHeat.html

http://www.wikihow.com/Calculate-Specific-Heat

https://www.khanacademy.org/

http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/heat/heatcapacityapplication.html

http://brainly.in/question/40990

http://www.tutorvista.com/content/physics/physics-iii/heat-and-thermodynamics/dulong-and-petit-law.php

Matter & Interactions Vol I. Chabay Sherwood

http://www.personal.utulsa.edu/~geoffrey-price/Courses/ChE7023/HeatCapacity-HeatOfFormation.pdf

https://engineering.ucsb.edu/~shell/che110a/heatcapacitycalculations.pdf

http://faculty.uca.edu/saddison/ThermalPhysics/Heat%20Capacity.pdf

https://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm

Introduction to Chemical Engineering Thermodynamics Seventh Edition. J. M. Smith, H. C. Van Ness, Michael M. Abbott

This page was last modified on 27 November 2016, by Panna Rasania.

File:Airspecific.PNG

2016-11-28T02:59:19Z

Bradak3:

File:Specific-heat-capacity.PNG

2016-11-28T02:57:34Z

Bradak3:

Specific Heat

2016-11-28T02:47:34Z

Bradak3:

'''Improved by Ben Radak Fall 2016'''

Specific heat, also known as the specific heat capacity, is defined as the amount of energy required to raise the temperature of a unit mass by one degree Celsius. This unit mass is one gram. The idea of heat capacity was first speculated by Albert Einstein is 1907 with his specific heat of solids lattice vibrations model, and later expanded by Peter Debye. The units for specific heat are Joules per gram-degree Celsius (J / g °C). Specific heat is important as it can determine the thermal interaction a material has with other materials. We can test the validity of models with specific heat since it is experimentally measurable. The study of thermodynamics was sparked by the research done on specific heat. Thermodynamics is the study of the conversion of energy involving heat and temperature change of a system.

==The Main Idea==

The most common definition is that specific heat is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius. The specific heat of a substance depends on its phase (solid, liquid, or gas) and its molecular structure. The relationship between heat and temperature change is best defined by the constant "c" in the equation below. Specific Heat is an intensive property, meaning that the amount of substance does not affect this property, only the composition of the subtance.

[[File:Specific Heat Equation.gif]]

where:

m = the mass of the system (grams, g)

Q = the amount of heat added to the system (Joules, J)

T = the temperature change (in Celsius ºC or Kelvin ºK)

c = the specific heat of the system ([Joules/(gram x degree celsius)], J/gºC)

The formula below is a reconfigured version of the relationship above. It shows how to calculate "c" specifically from the basic principle.

[[File:specific heat formula.png|300px|thumb|left|]]

This equation does not apply if a phase change occurs (say from a liquid state to a gaseous state). This is because the amount of heat added or removed during a phase change does not change the overall temperature of the substance. So we disregard this relationship when phase changes take place.

The specific heat for solid can be calculated by the change in energy of the atoms over the change in temperature. The change in energy of the atoms is calculated by dividing the change in the energy of the system by the number of atoms in the substance.

The specific heat most commonly known is the specific heat for water, which is 4.186 J/g°C or 1 calorie/g°C. The specific heat of water is higher than any other common substance. Water has a very large specific heat on a per-gram basis, meaning that it takes a lot more added heat to cause a change in its temperature. Since the specific heat of water is so high, water can be used for temperature regulation. Due to the difference in atomic structures, the specific heat per gram of water is much higher than that of a metal substance. It is possible to predict the specific heat of any material, as long as you know about its atomic structure, as a rise in temperature is the increase in energy at the atomic level of substances. Generally, it is more more useful to compare molar specific heats of substances.

There are two models to determine the specific heats of substances at an atomic level. These are the Dulong-Petit Law and the Einstein-Debye model. The Dulong-Petit Law states that the molar specific heats of most solids (at room temperature or above) are almost constant. The Einstein-Debye model of specific heat states that specific heats drop at lower temperatures, as atomic processes become more relevant.

Here is an example of how to calculate specific heat.

'''Example: Find the specific heat of 350 g of an unknown substance when 34,700 Joules of heat are applied, and the temperature rises from 22ºC to 173ºC with no phase change.'''

'''We know that'''

m = 350 g

Q = 34,700 J

T initial = 22ºC

T final = 173ºC

c = ?

'''Using the formula above, c = Q / (mΔT)'''

c = 34,700 / (350*(173-22))

c = 34,700 / (350*151)

c = 34,700 / 52,850

'''c = 0.657 J/(gºC)'''

== Common Specific Heats and Noticeable Trends ==

== Using Specific Heat (Problem Solving) ==

More often, the specific heat of a substance can be looked up in a table or online, and it is unnecessary to find the specific heat. Instead, you will be asked to find the final temperature of a system. This can be done using the principle of the conservation of Energy

'''Example: Coffee at 93ºC with a specific heat of 4.2 J/gºC is mixed with 55.3 grams of creams at 5ºC (specific heat of 3.8). What is the final temperature?'''

[[File:Specifc_heat_in_equation.gif]]

[[File:Eq3.gif]]

[[File:Eq4.gif]]

[[File:Eq5.gif]]

[[File:Eq6.gif]]

The above solution was completed using the Principle of the Conservation of Energy. This principle states that the change of energy in a system should be equal to zero because energy cannot be created nor destroyed, which gives us the 0 on the right hand side of the equation. Assuming their is no heat transfer from the surroundings, this principle applies. For both substances, the masses are multiplied by the specific heats and the difference in temperature, and this is done to find their change in internal energy. Through the concept of convection we recognize that the substance with the higher temperature will lose energy, and this energy will be absorbed by the other substance, giving us a net energy change of 0. To convince yourself of this you can look at the units, the mass multiplied by the specific heat multiplied by the change in temperature.

g * J/gºC * ºC = J

leaves us with the energy unit Joules. Since the net change must equal zero we recognize that the energy lost by one substance is gained by the other, and the final temperature of the combined substances can be found.

In all cases, the two objects at different temperatures will reach the same final temperature. This is Thermal Equilibrium.

== Law of Dulong and Petit ==

The Law of Dulong and Petit is a thermodynamic law discovered in 1819 by the French physicists Pierre Louis Dulong and Alexis Thérèse Petit. It yields the expression for the molar specific heat capacity of certain chemical elements. They found, through experiments, that the mass specific heat capacity for many elements was close to a constant value, after it had been adjusted to reflect the relative atomic weight of the element.

Basically, Dulong and Petit found that the heat capacity of a mole of numerous solid elements is about 3R, where R is the universal gas constant. Dulong and Petit were unaware of the relationship to R, since it had not yet been defined. The value of 3R is about 25 joules per kelvin, and Dulong and Petit found that this was the approximate heat capacity of some solid elements per mole of atoms they contained.

'''Example''' The specific heat of copper is 0.389 J/gm K. The specific heat of lead is 0.128 J/gm K. Why are the values so different in these two metals? Did you notice that they are expressed as energy per unit mass? If you express each as energy per mole, they are actually very similar. The Law of Dulong and Petit addresses this similarity in molar specific heats. It can be accounted for by applying equipartition of energy to the atoms of solids.

[[File:Dulong.gif]]

When observed on a molar basis, the specific heats of copper and lead are as follows:

Copper: 0.386 J/gm K * 63.6 gm/mol = 24.6 J/mol K

Lead: 0.128 J/gm K * 207 gm/mol = 26.5 J/mol K

Other molar specific heats of metals are shown below:

Aluminum: 24.3 J/mol K

Gold: 25.6 J/mol K

Silver: 24.9 J/mol K

Zinc: 25.2 J/mol K

As you can see, molar specific heats of many metals are around 25 J/mol K and are really very similar.

== Einstein Debye Model ==

Einstein and Debye had developed models for specific heat separately. Einstein's model stated that low energy excitation of a solid material was caused by oscillation of a single atom, whereas Debye's model stated that phonons or collective modes iterating through a material caused excitations. However, these two models are able to be put together to find the specific heat given by the following formula.

[[File:einstein debye.png]]

For low temperatures, Einstein and Debye found that the Law of Dulong and Petit was not applicable. At lower temperatures, it was found that atomic interactions were deemed significant in calculating the molar specific heat of an object.

[[File: Einstein Debye Graphs.gif]]

According to the Einstein Debye Model for Copper and Aluminum, specific heat varies a lot at lower temperatures and goes much below the Dulong-Petit Model. This is due to increased effects on specific heat by interatomic forces. However, for very high temperatures, the Einstein-Debye Model cannot be used. In fact, at high temperatures, Einstein's expression of specific heat reduces to the Dulong-Petit mathematical expression.

The Einstein Debye Equation is below.

[[File:Einstein Debye Equation.gif]]

For high temperatures it may be reduced like this:

[[File:Einstein Debye for High Temperatures.gif]]

This actually reduces to the Dulong-Petit Formula for Specific Heat:

[[File:Dulong Petit.gif]]

===Specific Heats of Gases===

The specific heats of gases are usually expressed as molar specific heats. For an ideal gas, the internal energy is all in the form of kinetic energy. The expression for the internal energy is

[[File:U.png]]

There are two specific heats for gases, one for gases at a constant volume and one gases at a constant pressure. In the formula below, the gas has a constant volume:

[[File:constant volume.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For an ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant volume.gif]]

For a constant pressure, specific heat can be derived as:

[[File:Constant Pressure.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For and ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant pressure.gif]]

The molar specific heats of gases all gravitate towards these ranges depending on the conditions the gas is kept in.

== Specific Heat in Thermodynamics ==

While working on thermodynamic processes specific heat is differentiated between specific heat at a constant temperate <Cp> and specific heat at a constant volume <Cv>. For the scope of Physics 2211 it is not necessary to note this distinction. However, <Cp> and <Cv> can be indicative to the value of the other because

<Cp> = <Cv> + R

where R is the gas constant.

While dealing with in Thermodynamic processes it is also important to recognize that <Cp> values change through out different temperature ranges. To account for this the follow equation is utilized:

[[File:cpr.png]]

Where R is the gas constant, and A, B, C, and D are the heat capacity constants for the specific substance of interest which can be found in the back of thermodynamic textbooks or online.

'''Example of Heat Capacities Table'''

[[File:cpcon.png]]

You may notice is in the particular table there are no D values listed. That is because D values are equal to 0 for gases, and C values are equal to 0 in liquids and solids.

It is easiest to find the Cp value by plugging in the constants into a program.

If working with vPython or glowscript you would simply have to write down the constants. Below is an example of what the program may look like.

[[File:glow.png]]

The units which you should use for your temperature will be specified in your table where you get you A, B, C, and D values. From the table you will also note the units to determine which gas constant R is appropriate for the equation.

==Connectedness==

Specific heat and thermodynamics are used often in chemistry, nuclear engineering, aerodynamics, and mechanical engineering. It is also used in everyday life in the radiator and cooling system of a car.

Specific heat can have a lot to do with prosthetic manufacturing, which is a focus in Biomedical Engineering. Prosthetics materials must be durable and easy to manipulate in a normal range of temperatures. In order to created medical devices, specific heats must be known, especially for welding or molding things, which require a specific temperature to be effective. At higher temperatures, the Dulong-Petit law must be used to calculate the specific heat of an object. Especially for solid metal objects, which would be used in prosthetics, Dulong-Petit is very useful.

Cooking materials such as pots and pans are made to have a low specific heat so that they need less heat to raise their temperature. This allows for faster cooking processes. The handles of these cooking utensils are made of substances with high specific heats so that their temperature won’t rise too much if a large amount of heat is absorbed.

Have you ever noticed that sand on the beach can burn your feet but the ocean water is cool and refreshing? Sand has a lower specific heat than ocean water. So when the sun is beating down, the temperature of the land increases faster than that of the sea.

Insulation is made of materials with high specific heat so that they won't change temperature easily. For example, wood has a high specific heat. A wooden house helps keep the inside cooler during summer because it requires lots of heat to change its temperature. Builders can choose certain materials which allows us to build houses for specific locations and altitudes.

== See also ==

'''Further reading'''

[http://www.physicsbook.gatech.edu/Kinds_of_Matter Kinds of Matter]

[http://www.physicsbook.gatech.edu/Boiling_Point Boiling Point]

[http://www.physicsbook.gatech.edu/Melting_Point Melting Point]

[http://www.physicsbook.gatech.edu/Thermal_Energy Thermal Energy]

[http://www.physicsbook.gatech.edu/Heat_Capacity Heat Capacity]

[http://www.physicsbook.gatech.edu/Specific_Heat_Capacity Specific Heat Capacity]

[http://www.physicsbook.gatech.edu/First_Law_of_Thermodynamics Thermodynamics]

'''External Links'''

[http://www.wisegeek.org/what-is-specific-heat.htm WiseGeek.org]

[https://www.khanacademy.org/science/biology/water-acids-and-bases/water-as-a-solid-liquid-and-gas/v/specific-heat-of-water KhanAcademy.org]

==References==

This section contains the references I used while writing this page:

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html

http://scienceworld.wolfram.com/physics/SpecificHeat.html

http://www.wikihow.com/Calculate-Specific-Heat

https://www.khanacademy.org/

http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/heat/heatcapacityapplication.html

http://brainly.in/question/40990

http://www.tutorvista.com/content/physics/physics-iii/heat-and-thermodynamics/dulong-and-petit-law.php

Matter & Interactions Vol I. Chabay Sherwood

http://www.personal.utulsa.edu/~geoffrey-price/Courses/ChE7023/HeatCapacity-HeatOfFormation.pdf

https://engineering.ucsb.edu/~shell/che110a/heatcapacitycalculations.pdf

http://faculty.uca.edu/saddison/ThermalPhysics/Heat%20Capacity.pdf

https://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm

Introduction to Chemical Engineering Thermodynamics Seventh Edition. J. M. Smith, H. C. Van Ness, Michael M. Abbott

This page was last modified on 27 November 2016, by Panna Rasania.

Specific Heat

2016-11-28T02:21:40Z

Bradak3:

'''Improved by Ben Radak Fall 2016'''

Specific heat, also known as the specific heat capacity, is defined as the amount of energy required to raise the temperature of a unit mass by one degree Celsius. This unit mass is one gram. The idea of heat capacity was first speculated by Albert Einstein is 1907 with his specific heat of solids lattice vibrations model, and later expanded by Peter Debye. The units for specific heat are Joules per gram-degree Celsius (J / g °C). Specific heat is important as it can determine the thermal interaction a material has with other materials. We can test the validity of models with specific heat since it is experimentally measurable. The study of thermodynamics was sparked by the research done on specific heat. Thermodynamics is the study of the conversion of energy involving heat and temperature change of a system.

==The Main Idea==

The most common definition is that specific heat is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius. The specific heat of a substance depends on its phase (solid, liquid, or gas) and its molecular structure. The relationship between heat and temperature change is best defined by the constant "c" in the equation below. Specific Heat is an intensive property, meaning that the amount of substance does not affect this property, only the composition of the subtance.

[[File:Specific Heat Equation.gif]]

where:

m = the mass of the system (grams, g)

Q = the amount of heat added to the system (Joules, J)

T = the temperature change (in Celsius ºC or Kelvin ºK)

c = the specific heat of the system ([Joules/(gram x degree celsius)], J/gºC)

The formula below is a reconfigured version of the relationship above. It shows how to calculate "c" specifically from the basic principle.

[[File:specific heat formula.png|300px|thumb|left|]]

This equation does not apply if a phase change occurs (say from a liquid state to a gaseous state). This is because the amount of heat added or removed during a phase change does not change the overall temperature of the substance. So we disregard this relationship when phase changes take place.

The specific heat for solid can be calculated by the change in energy of the atoms over the change in temperature. The change in energy of the atoms is calculated by dividing the change in the energy of the system by the number of atoms in the substance.

The specific heat most commonly known is the specific heat for water, which is 4.186 J/g°C or 1 calorie/g°C. The specific heat of water is higher than any other common substance. Water has a very large specific heat on a per-gram basis, meaning that it takes a lot more added heat to cause a change in its temperature. Since the specific heat of water is so high, water can be used for temperature regulation. Due to the difference in atomic structures, the specific heat per gram of water is much higher than that of a metal substance. It is possible to predict the specific heat of any material, as long as you know about its atomic structure, as a rise in temperature is the increase in energy at the atomic level of substances. Generally, it is more more useful to compare molar specific heats of substances.

There are two models to determine the specific heats of substances at an atomic level. These are the Dulong-Petit Law and the Einstein-Debye model. The Dulong-Petit Law states that the molar specific heats of most solids (at room temperature or above) are almost constant. The Einstein-Debye model of specific heat states that specific heats drop at lower temperatures, as atomic processes become more relevant.

Here is an example of how to calculate specific heat.

'''Example: Find the specific heat of 350 g of an unknown substance when 34,700 Joules of heat are applied, and the temperature rises from 22ºC to 173ºC with no phase change.'''

'''We know that'''

m = 350 g

Q = 34,700 J

T initial = 22ºC

T final = 173ºC

c = ?

'''Using the formula above, c = Q / (mΔT)'''

c = 34,700 / (350*(173-22))

c = 34,700 / (350*151)

c = 34,700 / 52,850

'''c = 0.657 J/(gºC)'''

== Using Specific Heat (Problem Solving) ==

More often, the specific heat of a substance can be looked up in a table or online, and it is unnecessary to find the specific heat. Instead, you will be asked to find the final temperature of a system. This can be done using the principle of the conservation of Energy

'''Example: Coffee at 93ºC with a specific heat of 4.2 J/gºC is mixed with 55.3 grams of creams at 5ºC (specific heat of 3.8). What is the final temperature?'''

[[File:Specifc_heat_in_equation.gif]]

[[File:Eq3.gif]]

[[File:Eq4.gif]]

[[File:Eq5.gif]]

[[File:Eq6.gif]]

(350)(4.2)(Tf-93)+(55.3)(3.8)(Tf-5) = 0

1470Tf - 136710 + 210.14Tf - 1050.7 = 0

1680Tf = 137760.7

Tf = 82ºC

The above solution was completed using the Principle of the Conservation of Energy. This principle states that the change of energy in a system should be equal to zero because energy cannot be created nor destroyed, which gives us the 0 on the right hand side of the equation. For both substances, their masses are multiplied by the specific heats and the difference in temperature, and this is done to find their change in internal energy. Through the concept of convection we recognize that the substance with the higher temperature will lose energy, and this energy will be absorbed by the other substance, giving us a net energy change of 0. To convince yourself of this you can look at the units, the mass multiplied by the specific heat multiplied by the change in temperature

g * J/gºC * ºC = J

leaves us with the energy unit Joules. Since the net change must equal zero we recognize that the energy lost by one substance is gained by the other, and the final temperature of the combined substances can be found.

== Law of Dulong and Petit ==

The Law of Dulong and Petit is a thermodynamic law discovered in 1819 by the French physicists Pierre Louis Dulong and Alexis Thérèse Petit. It yields the expression for the molar specific heat capacity of certain chemical elements. They found, through experiments, that the mass specific heat capacity for many elements was close to a constant value, after it had been adjusted to reflect the relative atomic weight of the element.

Basically, Dulong and Petit found that the heat capacity of a mole of numerous solid elements is about 3R, where R is the universal gas constant. Dulong and Petit were unaware of the relationship to R, since it had not yet been defined. The value of 3R is about 25 joules per kelvin, and Dulong and Petit found that this was the approximate heat capacity of some solid elements per mole of atoms they contained.

'''Example''' The specific heat of copper is 0.389 J/gm K. The specific heat of lead is 0.128 J/gm K. Why are the values so different in these two metals? Did you notice that they are expressed as energy per unit mass? If you express each as energy per mole, they are actually very similar. The Law of Dulong and Petit addresses this similarity in molar specific heats. It can be accounted for by applying equipartition of energy to the atoms of solids.

[[File:Dulong.gif]]

When observed on a molar basis, the specific heats of copper and lead are as follows:

Copper: 0.386 J/gm K * 63.6 gm/mol = 24.6 J/mol K

Lead: 0.128 J/gm K * 207 gm/mol = 26.5 J/mol K

Other molar specific heats of metals are shown below:

Aluminum: 24.3 J/mol K

Gold: 25.6 J/mol K

Silver: 24.9 J/mol K

Zinc: 25.2 J/mol K

As you can see, molar specific heats of many metals are around 25 J/mol K and are really very similar.

== Einstein Debye Model ==

Einstein and Debye had developed models for specific heat separately. Einstein's model stated that low energy excitation of a solid material was caused by oscillation of a single atom, whereas Debye's model stated that phonons or collective modes iterating through a material caused excitations. However, these two models are able to be put together to find the specific heat given by the following formula.

[[File:einstein debye.png]]

For low temperatures, Einstein and Debye found that the Law of Dulong and Petit was not applicable. At lower temperatures, it was found that atomic interactions were deemed significant in calculating the molar specific heat of an object.

[[File: Einstein Debye Graphs.gif]]

According to the Einstein Debye Model for Copper and Aluminum, specific heat varies a lot at lower temperatures and goes much below the Dulong-Petit Model. This is due to increased effects on specific heat by interatomic forces. However, for very high temperatures, the Einstein-Debye Model cannot be used. In fact, at high temperatures, Einstein's expression of specific heat reduces to the Dulong-Petit mathematical expression.

The Einstein Debye Equation is below.

[[File:Einstein Debye Equation.gif]]

For high temperatures it may be reduced like this:

[[File:Einstein Debye for High Temperatures.gif]]

This actually reduces to the Dulong-Petit Formula for Specific Heat:

[[File:Dulong Petit.gif]]

===Specific Heats of Gases===

The specific heats of gases are usually expressed as molar specific heats. For an ideal gas, the internal energy is all in the form of kinetic energy. The expression for the internal energy is

[[File:U.png]]

There are two specific heats for gases, one for gases at a constant volume and one gases at a constant pressure. In the formula below, the gas has a constant volume:

[[File:constant volume.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For an ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant volume.gif]]

For a constant pressure, specific heat can be derived as:

[[File:Constant Pressure.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For and ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant pressure.gif]]

The molar specific heats of gases all gravitate towards these ranges depending on the conditions the gas is kept in.

== Specific Heat in Thermodynamics ==

While working on thermodynamic processes specific heat is differentiated between specific heat at a constant temperate <Cp> and specific heat at a constant volume <Cv>. For the scope of Physics 2211 it is not necessary to note this distinction. However, <Cp> and <Cv> can be indicative to the value of the other because

<Cp> = <Cv> + R

where R is the gas constant.

While dealing with in Thermodynamic processes it is also important to recognize that <Cp> values change through out different temperature ranges. To account for this the follow equation is utilized:

[[File:cpr.png]]

Where R is the gas constant, and A, B, C, and D are the heat capacity constants for the specific substance of interest which can be found in the back of thermodynamic textbooks or online.

'''Example of Heat Capacities Table'''

[[File:cpcon.png]]

You may notice is in the particular table there are no D values listed. That is because D values are equal to 0 for gases, and C values are equal to 0 in liquids and solids.

It is easiest to find the Cp value by plugging in the constants into a program.

If working with vPython or glowscript you would simply have to write down the constants. Below is an example of what the program may look like.

[[File:glow.png]]

The units which you should use for your temperature will be specified in your table where you get you A, B, C, and D values. From the table you will also note the units to determine which gas constant R is appropriate for the equation.

==Connectedness==

Specific heat and thermodynamics are used often in chemistry, nuclear engineering, aerodynamics, and mechanical engineering. It is also used in everyday life in the radiator and cooling system of a car.

Specific heat can have a lot to do with prosthetic manufacturing, which is a focus in Biomedical Engineering. Prosthetics materials must be durable and easy to manipulate in a normal range of temperatures. In order to created medical devices, specific heats must be known, especially for welding or molding things, which require a specific temperature to be effective. At higher temperatures, the Dulong-Petit law must be used to calculate the specific heat of an object. Especially for solid metal objects, which would be used in prosthetics, Dulong-Petit is very useful.

Cooking materials such as pots and pans are made to have a low specific heat so that they need less heat to raise their temperature. This allows for faster cooking processes. The handles of these cooking utensils are made of substances with high specific heats so that their temperature won’t rise too much if a large amount of heat is absorbed.

Have you ever noticed that sand on the beach can burn your feet but the ocean water is cool and refreshing? Sand has a lower specific heat than ocean water. So when the sun is beating down, the temperature of the land increases faster than that of the sea.

Insulation is made of materials with high specific heat so that they won't change temperature easily. For example, wood has a high specific heat. A wooden house helps keep the inside cooler during summer because it requires lots of heat to change its temperature. Builders can choose certain materials which allows us to build houses for specific locations and altitudes.

== See also ==

'''Further reading'''

[http://www.physicsbook.gatech.edu/Kinds_of_Matter Kinds of Matter]

[http://www.physicsbook.gatech.edu/Boiling_Point Boiling Point]

[http://www.physicsbook.gatech.edu/Melting_Point Melting Point]

[http://www.physicsbook.gatech.edu/Thermal_Energy Thermal Energy]

[http://www.physicsbook.gatech.edu/Heat_Capacity Heat Capacity]

[http://www.physicsbook.gatech.edu/Specific_Heat_Capacity Specific Heat Capacity]

[http://www.physicsbook.gatech.edu/First_Law_of_Thermodynamics Thermodynamics]

'''External Links'''

[http://www.wisegeek.org/what-is-specific-heat.htm WiseGeek.org]

[https://www.khanacademy.org/science/biology/water-acids-and-bases/water-as-a-solid-liquid-and-gas/v/specific-heat-of-water KhanAcademy.org]

==References==

This section contains the references I used while writing this page:

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html

http://scienceworld.wolfram.com/physics/SpecificHeat.html

http://www.wikihow.com/Calculate-Specific-Heat

https://www.khanacademy.org/

http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/heat/heatcapacityapplication.html

http://brainly.in/question/40990

http://www.tutorvista.com/content/physics/physics-iii/heat-and-thermodynamics/dulong-and-petit-law.php

Matter & Interactions Vol I. Chabay Sherwood

http://www.personal.utulsa.edu/~geoffrey-price/Courses/ChE7023/HeatCapacity-HeatOfFormation.pdf

https://engineering.ucsb.edu/~shell/che110a/heatcapacitycalculations.pdf

http://faculty.uca.edu/saddison/ThermalPhysics/Heat%20Capacity.pdf

https://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm

Introduction to Chemical Engineering Thermodynamics Seventh Edition. J. M. Smith, H. C. Van Ness, Michael M. Abbott

This page was last modified on 27 November 2016, by Panna Rasania.

File:Eq6.gif

2016-11-28T02:19:40Z

Bradak3:

File:Eq5.gif

2016-11-28T02:19:19Z

Bradak3:

File:Eq4.gif

2016-11-28T02:18:25Z

Bradak3:

File:Eq3.gif

2016-11-28T02:18:00Z

Bradak3:

Specific Heat

2016-11-28T02:10:57Z

Bradak3:

'''Improved by Ben Radak Fall 2016'''

Specific heat, also known as the specific heat capacity, is defined as the amount of energy required to raise the temperature of a unit mass by one degree Celsius. This unit mass is one gram. The idea of heat capacity was first speculated by Albert Einstein is 1907 with his specific heat of solids lattice vibrations model, and later expanded by Peter Debye. The units for specific heat are Joules per gram-degree Celsius (J / g °C). Specific heat is important as it can determine the thermal interaction a material has with other materials. We can test the validity of models with specific heat since it is experimentally measurable. The study of thermodynamics was sparked by the research done on specific heat. Thermodynamics is the study of the conversion of energy involving heat and temperature change of a system.

==The Main Idea==

The most common definition is that specific heat is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius. The specific heat of a substance depends on its phase (solid, liquid, or gas) and its molecular structure. The relationship between heat and temperature change is best defined by the constant "c" in the equation below. Specific Heat is an intensive property, meaning that the amount of substance does not affect this property, only the composition of the subtance.

[[File:Specific Heat Equation.gif]]

where:

m = the mass of the system (grams, g)

Q = the amount of heat added to the system (Joules, J)

T = the temperature change (in Celsius ºC or Kelvin ºK)

c = the specific heat of the system ([Joules/(gram x degree celsius)], J/gºC)

The formula below is a reconfigured version of the relationship above. It shows how to calculate "c" specifically from the basic principle.

[[File:specific heat formula.png|300px|thumb|left|]]

This equation does not apply if a phase change occurs (say from a liquid state to a gaseous state). This is because the amount of heat added or removed during a phase change does not change the overall temperature of the substance. So we disregard this relationship when phase changes take place.

The specific heat for solid can be calculated by the change in energy of the atoms over the change in temperature. The change in energy of the atoms is calculated by dividing the change in the energy of the system by the number of atoms in the substance.

The specific heat most commonly known is the specific heat for water, which is 4.186 J/g°C or 1 calorie/g°C. The specific heat of water is higher than any other common substance. Water has a very large specific heat on a per-gram basis, meaning that it takes a lot more added heat to cause a change in its temperature. Since the specific heat of water is so high, water can be used for temperature regulation. Due to the difference in atomic structures, the specific heat per gram of water is much higher than that of a metal substance. It is possible to predict the specific heat of any material, as long as you know about its atomic structure, as a rise in temperature is the increase in energy at the atomic level of substances. Generally, it is more more useful to compare molar specific heats of substances.

There are two models to determine the specific heats of substances at an atomic level. These are the Dulong-Petit Law and the Einstein-Debye model. The Dulong-Petit Law states that the molar specific heats of most solids (at room temperature or above) are almost constant. The Einstein-Debye model of specific heat states that specific heats drop at lower temperatures, as atomic processes become more relevant.

Here is an example of how to calculate specific heat.

'''Example: Find the specific heat of 350 g of an unknown substance when 34,700 Joules of heat are applied, and the temperature rises from 22ºC to 173ºC with no phase change.'''

'''We know that'''

m = 350 g

Q = 34,700 J

T initial = 22ºC

T final = 173ºC

c = ?

'''Using the formula above, c = Q / (mΔT)'''

c = 34,700 / (350*(173-22))

c = 34,700 / (350*151)

c = 34,700 / 52,850

'''c = 0.657 J/(gºC)'''

== Using Specific Heat (Problem Solving) ==

More often, the specific heat of a substance can be looked up in a table or online, and it is unnecessary to find the specific heat. Instead, you will be asked to find the final temperature of a system. This can be done using the principle of the conservation of Energy

'''Example: Coffee at 93ºC with a specific heat of 4.2 J/gºC is mixed with 55.3 grams of creams at 5ºC (specific heat of 3.8). What is the final temperature?'''

[[File:Specifc_heat_in_equation.gif]]

(350)(4.2)(Tf-93)+(55.3)(3.8)(Tf-5) = 0

1470Tf - 136710 + 210.14Tf - 1050.7 = 0

1680Tf = 137760.7

Tf = 82ºC

The above solution was completed using the Principle of the Conservation of Energy. This principle states that the change of energy in a system should be equal to zero because energy cannot be created nor destroyed, which gives us the 0 on the right hand side of the equation. For both substances, their masses are multiplied by the specific heats and the difference in temperature, and this is done to find their change in internal energy. Through the concept of convection we recognize that the substance with the higher temperature will lose energy, and this energy will be absorbed by the other substance, giving us a net energy change of 0. To convince yourself of this you can look at the units, the mass multiplied by the specific heat multiplied by the change in temperature

g * J/gºC * ºC = J

leaves us with the energy unit Joules. Since the net change must equal zero we recognize that the energy lost by one substance is gained by the other, and the final temperature of the combined substances can be found.

== Law of Dulong and Petit ==

The Law of Dulong and Petit is a thermodynamic law discovered in 1819 by the French physicists Pierre Louis Dulong and Alexis Thérèse Petit. It yields the expression for the molar specific heat capacity of certain chemical elements. They found, through experiments, that the mass specific heat capacity for many elements was close to a constant value, after it had been adjusted to reflect the relative atomic weight of the element.

Basically, Dulong and Petit found that the heat capacity of a mole of numerous solid elements is about 3R, where R is the universal gas constant. Dulong and Petit were unaware of the relationship to R, since it had not yet been defined. The value of 3R is about 25 joules per kelvin, and Dulong and Petit found that this was the approximate heat capacity of some solid elements per mole of atoms they contained.

'''Example''' The specific heat of copper is 0.389 J/gm K. The specific heat of lead is 0.128 J/gm K. Why are the values so different in these two metals? Did you notice that they are expressed as energy per unit mass? If you express each as energy per mole, they are actually very similar. The Law of Dulong and Petit addresses this similarity in molar specific heats. It can be accounted for by applying equipartition of energy to the atoms of solids.

[[File:Dulong.gif]]

When observed on a molar basis, the specific heats of copper and lead are as follows:

Copper: 0.386 J/gm K * 63.6 gm/mol = 24.6 J/mol K

Lead: 0.128 J/gm K * 207 gm/mol = 26.5 J/mol K

Other molar specific heats of metals are shown below:

Aluminum: 24.3 J/mol K

Gold: 25.6 J/mol K

Silver: 24.9 J/mol K

Zinc: 25.2 J/mol K

As you can see, molar specific heats of many metals are around 25 J/mol K and are really very similar.

== Einstein Debye Model ==

Einstein and Debye had developed models for specific heat separately. Einstein's model stated that low energy excitation of a solid material was caused by oscillation of a single atom, whereas Debye's model stated that phonons or collective modes iterating through a material caused excitations. However, these two models are able to be put together to find the specific heat given by the following formula.

[[File:einstein debye.png]]

For low temperatures, Einstein and Debye found that the Law of Dulong and Petit was not applicable. At lower temperatures, it was found that atomic interactions were deemed significant in calculating the molar specific heat of an object.

[[File: Einstein Debye Graphs.gif]]

According to the Einstein Debye Model for Copper and Aluminum, specific heat varies a lot at lower temperatures and goes much below the Dulong-Petit Model. This is due to increased effects on specific heat by interatomic forces. However, for very high temperatures, the Einstein-Debye Model cannot be used. In fact, at high temperatures, Einstein's expression of specific heat reduces to the Dulong-Petit mathematical expression.

The Einstein Debye Equation is below.

[[File:Einstein Debye Equation.gif]]

For high temperatures it may be reduced like this:

[[File:Einstein Debye for High Temperatures.gif]]

This actually reduces to the Dulong-Petit Formula for Specific Heat:

[[File:Dulong Petit.gif]]

===Specific Heats of Gases===

The specific heats of gases are usually expressed as molar specific heats. For an ideal gas, the internal energy is all in the form of kinetic energy. The expression for the internal energy is

[[File:U.png]]

There are two specific heats for gases, one for gases at a constant volume and one gases at a constant pressure. In the formula below, the gas has a constant volume:

[[File:constant volume.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For an ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant volume.gif]]

For a constant pressure, specific heat can be derived as:

[[File:Constant Pressure.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For and ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant pressure.gif]]

The molar specific heats of gases all gravitate towards these ranges depending on the conditions the gas is kept in.

== Specific Heat in Thermodynamics ==

While working on thermodynamic processes specific heat is differentiated between specific heat at a constant temperate <Cp> and specific heat at a constant volume <Cv>. For the scope of Physics 2211 it is not necessary to note this distinction. However, <Cp> and <Cv> can be indicative to the value of the other because

<Cp> = <Cv> + R

where R is the gas constant.

While dealing with in Thermodynamic processes it is also important to recognize that <Cp> values change through out different temperature ranges. To account for this the follow equation is utilized:

[[File:cpr.png]]

Where R is the gas constant, and A, B, C, and D are the heat capacity constants for the specific substance of interest which can be found in the back of thermodynamic textbooks or online.

'''Example of Heat Capacities Table'''

[[File:cpcon.png]]

You may notice is in the particular table there are no D values listed. That is because D values are equal to 0 for gases, and C values are equal to 0 in liquids and solids.

It is easiest to find the Cp value by plugging in the constants into a program.

If working with vPython or glowscript you would simply have to write down the constants. Below is an example of what the program may look like.

[[File:glow.png]]

The units which you should use for your temperature will be specified in your table where you get you A, B, C, and D values. From the table you will also note the units to determine which gas constant R is appropriate for the equation.

==Connectedness==

Specific heat and thermodynamics are used often in chemistry, nuclear engineering, aerodynamics, and mechanical engineering. It is also used in everyday life in the radiator and cooling system of a car.

Specific heat can have a lot to do with prosthetic manufacturing, which is a focus in Biomedical Engineering. Prosthetics materials must be durable and easy to manipulate in a normal range of temperatures. In order to created medical devices, specific heats must be known, especially for welding or molding things, which require a specific temperature to be effective. At higher temperatures, the Dulong-Petit law must be used to calculate the specific heat of an object. Especially for solid metal objects, which would be used in prosthetics, Dulong-Petit is very useful.

Cooking materials such as pots and pans are made to have a low specific heat so that they need less heat to raise their temperature. This allows for faster cooking processes. The handles of these cooking utensils are made of substances with high specific heats so that their temperature won’t rise too much if a large amount of heat is absorbed.

Have you ever noticed that sand on the beach can burn your feet but the ocean water is cool and refreshing? Sand has a lower specific heat than ocean water. So when the sun is beating down, the temperature of the land increases faster than that of the sea.

Insulation is made of materials with high specific heat so that they won't change temperature easily. For example, wood has a high specific heat. A wooden house helps keep the inside cooler during summer because it requires lots of heat to change its temperature. Builders can choose certain materials which allows us to build houses for specific locations and altitudes.

== See also ==

'''Further reading'''

[http://www.physicsbook.gatech.edu/Kinds_of_Matter Kinds of Matter]

[http://www.physicsbook.gatech.edu/Boiling_Point Boiling Point]

[http://www.physicsbook.gatech.edu/Melting_Point Melting Point]

[http://www.physicsbook.gatech.edu/Thermal_Energy Thermal Energy]

[http://www.physicsbook.gatech.edu/Heat_Capacity Heat Capacity]

[http://www.physicsbook.gatech.edu/Specific_Heat_Capacity Specific Heat Capacity]

[http://www.physicsbook.gatech.edu/First_Law_of_Thermodynamics Thermodynamics]

'''External Links'''

[http://www.wisegeek.org/what-is-specific-heat.htm WiseGeek.org]

[https://www.khanacademy.org/science/biology/water-acids-and-bases/water-as-a-solid-liquid-and-gas/v/specific-heat-of-water KhanAcademy.org]

==References==

This section contains the references I used while writing this page:

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html

http://scienceworld.wolfram.com/physics/SpecificHeat.html

http://www.wikihow.com/Calculate-Specific-Heat

https://www.khanacademy.org/

http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/heat/heatcapacityapplication.html

http://brainly.in/question/40990

http://www.tutorvista.com/content/physics/physics-iii/heat-and-thermodynamics/dulong-and-petit-law.php

Matter & Interactions Vol I. Chabay Sherwood

http://www.personal.utulsa.edu/~geoffrey-price/Courses/ChE7023/HeatCapacity-HeatOfFormation.pdf

https://engineering.ucsb.edu/~shell/che110a/heatcapacitycalculations.pdf

http://faculty.uca.edu/saddison/ThermalPhysics/Heat%20Capacity.pdf

https://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm

Introduction to Chemical Engineering Thermodynamics Seventh Edition. J. M. Smith, H. C. Van Ness, Michael M. Abbott

This page was last modified on 27 November 2016, by Panna Rasania.

File:Specifc heat in equation.gif

2016-11-28T02:10:20Z

Bradak3:

Specific Heat

2016-11-28T01:35:26Z

Bradak3:

'''Ben Radak Fall 2016'''

Specific heat, also known as the specific heat capacity, is defined as the amount of energy required to raise the temperature of a unit mass by one degree Celsius. The idea of heat capacity was first speculated by Albert Einstein is 1907 with his specific heat of solids lattice vibrations model, and later expanded by Peter Debye. The units for specific heat are Joules per gram-degree Celsius (J / g °C). Specific heat is important as it can determine the thermal interaction a material has with other materials. We can test the validity of models with specific heat since it is experimentally measurable. The study of thermodynamics was sparked by the research done on specific heat. Thermodynamics is the study of the conversion of energy involving heat and temperature change of a system.

==The Main Idea==

The most common definition is that specific heat is the amount of heat needed to raise the temperature of a mass by 1 degree Celsius. The specific heat of a substance depends on its phase (solid, liquid, or gas) and its molecular structure. The relationship between heat and temperature change is best defined by the constant "c" in the equation below.

[[File:Specific Heat Equation.gif]]

where:

m = the mass of the system (grams, g)

Q = the amount of heat added to the system (Joules, J)

T = the temperature change (usually degrees Celsius, ºC)

c = the specific heat of the system ([Joules/(gram x degree celsius)], J/gºC)

The formula below is a reconfigured version of the relationship above. It shows how to calculate "c" specifically from the basic principle.

[[File:specific heat formula.png|300px|thumb|left|]]

This equation does not apply if a phase change occurs (say from a liquid state to a gaseous state). This is because the amount of heat added or removed during a phase change does not change the overall temperature of the substance. So we disregard this relationship when phase changes take place.

The specific heat for solid can be calculated by the change in energy of the atoms over the change in temperature. The change in energy of the atoms is calculated by dividing the change in the energy of the system by the number of atoms in the substance.

The specific heat most commonly known is the specific heat for water, which is 4.186 J/g°C or 1 calorie/g°C. The specific heat of water is higher than any other common substance. Water has a very large specific heat on a per-gram basis, meaning that it takes a lot more added heat to cause a change in its temperature. Since the specific heat of water is so high, water can be used for temperature regulation. Due to the difference in atomic structures, the specific heat per gram of water is much higher than that of a metal substance. It is possible to predict the specific heat of any material, as long as you know about its atomic structure, as a rise in temperature is the increase in energy at the atomic level of substances. Generally, it is more more useful to compare molar specific heats of substances.

There are two models to determine the specific heats of substances at an atomic level. These are the Dulong-Petit Law and the Einstein-Debye model. The Dulong-Petit Law states that the molar specific heats of most solids (at room temperature or above) are almost constant. The Einstein-Debye model of specific heat states that specific heats drop at lower temperatures, as atomic processes become more relevant.

Here is an example of how to calculate specific heat.

'''Example: Find the specific heat of 350 g of an unknown substance when 34,700 Joules of heat are applied, and the temperature rises from 22ºC to 173ºC with no phase change.'''

'''We know that'''

m = 350 g

Q = 34,700 J

T initial = 22ºC

T final = 173ºC

c = ?

'''Using the formula above, c = Q / (mΔT)'''

c = 34,700 / (350*(173-22))

c = 34,700 / (350*151)

c = 34,700 / 52,850

'''c = 0.657 J/(gºC)'''

== Using Specific Heat (Problem Solving) ==

More often, the specific heat of a substance can be looked up in a table or online, and it is unnecessary to find the specific heat. Instead, you will be asked to find the final temperature of a system. This can be done using the principle of the conservation of Energy

'''Example: Coffee at 93ºC with a specific heat of 4.2 J/gºC is mixed with 55.3 grams of creams at 5ºC (specific heat of 3.8). What is the final temperature?'''

m1c1(Tf-Ti) + m2c2(Tf-Ti) = 0

(350)(4.2)(Tf-93)+(55.3)(3.8)(Tf-5) = 0

1470Tf - 136710 + 210.14Tf - 1050.7 = 0

1680Tf = 137760.7

Tf = 82ºC

The above solution was completed using the Principle of the Conservation of Energy. This principle states that the change of energy in a system should be equal to zero because energy cannot be created nor destroyed, which gives us the 0 on the right hand side of the equation. For both substances, their masses are multiplied by the specific heats and the difference in temperature, and this is done to find their change in internal energy. Through the concept of convection we recognize that the substance with the higher temperature will lose energy, and this energy will be absorbed by the other substance, giving us a net energy change of 0. To convince yourself of this you can look at the units, the mass multiplied by the specific heat multiplied by the change in temperature

g * J/gºC * ºC = J

leaves us with the energy unit Joules. Since the net change must equal zero we recognize that the energy lost by one substance is gained by the other, and the final temperature of the combined substances can be found.

== Law of Dulong and Petit ==

The Law of Dulong and Petit is a thermodynamic law discovered in 1819 by the French physicists Pierre Louis Dulong and Alexis Thérèse Petit. It yields the expression for the molar specific heat capacity of certain chemical elements. They found, through experiments, that the mass specific heat capacity for many elements was close to a constant value, after it had been adjusted to reflect the relative atomic weight of the element.

Basically, Dulong and Petit found that the heat capacity of a mole of numerous solid elements is about 3R, where R is the universal gas constant. Dulong and Petit were unaware of the relationship to R, since it had not yet been defined. The value of 3R is about 25 joules per kelvin, and Dulong and Petit found that this was the approximate heat capacity of some solid elements per mole of atoms they contained.

'''Example''' The specific heat of copper is 0.389 J/gm K. The specific heat of lead is 0.128 J/gm K. Why are the values so different in these two metals? Did you notice that they are expressed as energy per unit mass? If you express each as energy per mole, they are actually very similar. The Law of Dulong and Petit addresses this similarity in molar specific heats. It can be accounted for by applying equipartition of energy to the atoms of solids.

[[File:Dulong.gif]]

When observed on a molar basis, the specific heats of copper and lead are as follows:

Copper: 0.386 J/gm K * 63.6 gm/mol = 24.6 J/mol K

Lead: 0.128 J/gm K * 207 gm/mol = 26.5 J/mol K

Other molar specific heats of metals are shown below:

Aluminum: 24.3 J/mol K

Gold: 25.6 J/mol K

Silver: 24.9 J/mol K

Zinc: 25.2 J/mol K

As you can see, molar specific heats of many metals are around 25 J/mol K and are really very similar.

== Einstein Debye Model ==

Einstein and Debye had developed models for specific heat separately. Einstein's model stated that low energy excitation of a solid material was caused by oscillation of a single atom, whereas Debye's model stated that phonons or collective modes iterating through a material caused excitations. However, these two models are able to be put together to find the specific heat given by the following formula.

[[File:einstein debye.png]]

For low temperatures, Einstein and Debye found that the Law of Dulong and Petit was not applicable. At lower temperatures, it was found that atomic interactions were deemed significant in calculating the molar specific heat of an object.

[[File: Einstein Debye Graphs.gif]]

According to the Einstein Debye Model for Copper and Aluminum, specific heat varies a lot at lower temperatures and goes much below the Dulong-Petit Model. This is due to increased effects on specific heat by interatomic forces. However, for very high temperatures, the Einstein-Debye Model cannot be used. In fact, at high temperatures, Einstein's expression of specific heat reduces to the Dulong-Petit mathematical expression.

The Einstein Debye Equation is below.

[[File:Einstein Debye Equation.gif]]

For high temperatures it may be reduced like this:

[[File:Einstein Debye for High Temperatures.gif]]

This actually reduces to the Dulong-Petit Formula for Specific Heat:

[[File:Dulong Petit.gif]]

===Specific Heats of Gases===

The specific heats of gases are usually expressed as molar specific heats. For an ideal gas, the internal energy is all in the form of kinetic energy. The expression for the internal energy is

[[File:U.png]]

There are two specific heats for gases, one for gases at a constant volume and one gases at a constant pressure. In the formula below, the gas has a constant volume:

[[File:constant volume.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For an ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant volume.gif]]

For a constant pressure, specific heat can be derived as:

[[File:Constant Pressure.gif]]

where Q is heat, n is number of moles, and delta T is change in Temperature.

For and ideal monatomic gas, the molar specific heat should be around:

[[File:ideal for constant pressure.gif]]

The molar specific heats of gases all gravitate towards these ranges depending on the conditions the gas is kept in.

== Specific Heat in Thermodynamics ==

While working on thermodynamic processes specific heat is differentiated between specific heat at a constant temperate <Cp> and specific heat at a constant volume <Cv>. For the scope of Physics 2211 it is not necessary to note this distinction. However, <Cp> and <Cv> can be indicative to the value of the other because

<Cp> = <Cv> + R

where R is the gas constant.

While dealing with in Thermodynamic processes it is also important to recognize that <Cp> values change through out different temperature ranges. To account for this the follow equation is utilized:

[[File:cpr.png]]

Where R is the gas constant, and A, B, C, and D are the heat capacity constants for the specific substance of interest which can be found in the back of thermodynamic textbooks or online.

'''Example of Heat Capacities Table'''

[[File:cpcon.png]]

You may notice is in the particular table there are no D values listed. That is because D values are equal to 0 for gases, and C values are equal to 0 in liquids and solids.

It is easiest to find the Cp value by plugging in the constants into a program.

If working with vPython or glowscript you would simply have to write down the constants. Below is an example of what the program may look like.

[[File:glow.png]]

The units which you should use for your temperature will be specified in your table where you get you A, B, C, and D values. From the table you will also note the units to determine which gas constant R is appropriate for the equation.

==Connectedness==

Specific heat and thermodynamics are used often in chemistry, nuclear engineering, aerodynamics, and mechanical engineering. It is also used in everyday life in the radiator and cooling system of a car.

Specific heat can have a lot to do with prosthetic manufacturing, which is a focus in Biomedical Engineering. Prosthetics materials must be durable and easy to manipulate in a normal range of temperatures. In order to created medical devices, specific heats must be known, especially for welding or molding things, which require a specific temperature to be effective. At higher temperatures, the Dulong-Petit law must be used to calculate the specific heat of an object. Especially for solid metal objects, which would be used in prosthetics, Dulong-Petit is very useful.

Cooking materials such as pots and pans are made to have a low specific heat so that they need less heat to raise their temperature. This allows for faster cooking processes. The handles of these cooking utensils are made of substances with high specific heats so that their temperature won’t rise too much if a large amount of heat is absorbed.

Have you ever noticed that sand on the beach can burn your feet but the ocean water is cool and refreshing? Sand has a lower specific heat than ocean water. So when the sun is beating down, the temperature of the land increases faster than that of the sea.

Insulation is made of materials with high specific heat so that they won't change temperature easily. For example, wood has a high specific heat. A wooden house helps keep the inside cooler during summer because it requires lots of heat to change its temperature. Builders can choose certain materials which allows us to build houses for specific locations and altitudes.

== See also ==

'''Further reading'''

[http://www.physicsbook.gatech.edu/Kinds_of_Matter Kinds of Matter]

[http://www.physicsbook.gatech.edu/Boiling_Point Boiling Point]

[http://www.physicsbook.gatech.edu/Melting_Point Melting Point]

[http://www.physicsbook.gatech.edu/Thermal_Energy Thermal Energy]

[http://www.physicsbook.gatech.edu/Heat_Capacity Heat Capacity]

[http://www.physicsbook.gatech.edu/Specific_Heat_Capacity Specific Heat Capacity]

[http://www.physicsbook.gatech.edu/First_Law_of_Thermodynamics Thermodynamics]

'''External Links'''

[http://www.wisegeek.org/what-is-specific-heat.htm WiseGeek.org]

[https://www.khanacademy.org/science/biology/water-acids-and-bases/water-as-a-solid-liquid-and-gas/v/specific-heat-of-water KhanAcademy.org]

==References==

This section contains the references I used while writing this page:

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html

http://scienceworld.wolfram.com/physics/SpecificHeat.html

http://www.wikihow.com/Calculate-Specific-Heat

https://www.khanacademy.org/

http://www.one-school.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/heat/heatcapacityapplication.html

http://brainly.in/question/40990

http://www.tutorvista.com/content/physics/physics-iii/heat-and-thermodynamics/dulong-and-petit-law.php

Matter & Interactions Vol I. Chabay Sherwood

http://www.personal.utulsa.edu/~geoffrey-price/Courses/ChE7023/HeatCapacity-HeatOfFormation.pdf

https://engineering.ucsb.edu/~shell/che110a/heatcapacitycalculations.pdf

http://faculty.uca.edu/saddison/ThermalPhysics/Heat%20Capacity.pdf

https://www.lhup.edu/~dsimanek/scenario/labman1/spheat.htm

Introduction to Chemical Engineering Thermodynamics Seventh Edition. J. M. Smith, H. C. Van Ness, Michael M. Abbott

This page was last modified on 27 November 2016, by Panna Rasania.