Physics Book - User contributions [en]

File:Frictioncharge.gif

2020-04-20T02:04:37Z

Ftariq6:

Charge Transfer

2020-04-20T01:35:00Z

Ftariq6:

claimed by '''Fehmeen Tariq Spring 2020'''

If a charged conductor comes in contact, or is in close enough proximity, with another conductor, it is possible to transfer this charge to the second conductor. This process is called '''charge transfer'''. However, the ''Law of Conservation of Charge'' states that charges cannot be created or destroyed. Charge cannot be created, the presence of a negative charge is merely the effect of an object gaining electrons from another material. Since charge cannot be created or destroyed, and is just the transfer of electrons between materials, the magnitude of the charge transfer between two objects will be equivalent. For however much the charge of one object increases during charge transfer, the other must decrease the same amount. There are multiple ways that charge can be transferred such as through direct contact (ie friction), induction, and conduction.

==The Main Idea==
'''Insulators vs Conductors'''

[[File:inschargedist.gif|thumb|500px|Charges transferred to an insulator remains at the location of transfer.]]

In an '''insulator''', electrons are bounded tightly to atoms, which prevents charged particles from moving through the material. If charge is transferred to an insulator at a given location, the charge will remain at the location that the transfer occurred.

Within '''conductors''', on the other hand, electrons are able to flow freely from particle to particle. When charge is transferred to a conductor, the charge is distributed evenly across the surface of the object via ''electron movement''. The electrons will be distributed until the repelling force between the excess electrons is minimized. This is the main difference between insulators and conductors: insulators do not have mobile charged particles whereas conductors have mobile charged particles that allow for charge transfer through the free movement of electrons. Examples of insulators include rubber and air and examples of conductors include metals and salt water.

'''Charge by Friction'''

Certain objects and materials have a greater attraction to electrons than others. For example, rubber is highly attracted to electrons, whereas fur or hair has a lower attraction. Therefore, if you take a balloon and rub it on your head, electrons previously in the atoms of the hair will be pulled to the atoms of the rubber balloon. This creates an electron imbalance which makes the balloon negatively charged and the hair positively charged. This creates the effect where the hair will stand up and be pulled towards the balloon since the opposing charges make the two objects attracted. Likewise, two charged balloons will repel each other. Another example of this is rubbing a glass rod with silk. The glass rod will become positively charged and the silk will become negatively charged; this means that electrons were transferred from the glass rod to the silk, since protons are not removed from the nuclei. Rubbing two objects together is not necessary for charge transfer, but because rubbing creates more points of contact between two objects, it facilitates charge transfer.

'''Transfer Charges by Conduction'''

As shown in the previous section, electrons move from one object to another through points of contact; this is especially true among metals. Charging by conduction requires the contact of a charged metal, positive or negative, to a neutral metal. If a negatively charged object touches the neutral metal, the excess electrons will flow through the neutral object. Since electrons repel each other and the negatively charged metal has a buildup of electrons, a certain number of excess electrons will flow out and spread across the neutral object when given an outlet in the form of the neutral metal. This process leaves both metals negatively charged. The same process occurs with positively charged objects touching neutral metals. Additionally, note that this process only works between two conductors an insulator cannot undergo conduction.

[[File:Conductiontransfer.gif|450 px]]

'''Transfer Charges by Induction'''

Unlike the transfer of charges by conduction, objects can transfer charges by induction without making contact. When an object is charged, it has an electric field. This electric field will repel or attract electrons in another object. This electron movement is called transfer of charges by induction. A neutral object can be charged by another charged object through a process called '''polarization'''. This is when electrons in the object are repelled or attracted to one side of the object by the charged second object. For example, if a negatively charged sphere is placed near a neutral sphere, the electrons in the neutral sphere will be repelled by the charged sphere. The neutral sphere is now polarized, with one side of it being negatively charged and the other side being positively charged. The negatively charged side of the sphere can be removed through grounding or with a conductor. Once removed, the originally neutral sphere will now be positively charged. Another example of induction is the balloon and black pepper experiment. A balloon can be given a negative charge by rubbing it on hair. When the balloon is placed near grounded black pepper, the black pepper particles will be polarized so that they become positively charged on top and will be attracted to the negatively charged balloon.

[[File:Indtransfer.gif|500 px]]

===A Mathematical Model===

===A Computational Model===

This [https://phet.colorado.edu/en/simulation/balloons website] is a great model for charge transfer. It shows charge transfer between a sweater and a balloon to demonstrate static electricity. Try using it without the charges showing and guessing where they will go.

[[File:Charge_transfer_model_pic.PNG|400 px]]

==Examples==
'''Some question and answers from Matter and Interactions book referenced in references.'''

'''Q21. You take two invisible tapes of some unknown brand, stick them together, and discharge the pair before pulling them apart and hanging them from the edge of your desk. When you bring an uncharged plastic pen within 10 cm of either the U tape or the L tape you see a slight attraction. Next you rub the pen through your hair, which is known to charge the pen negatively. Now you find that if you bring the charged pen within 8 cm of the L tape you see a slight repulsion, and if you bring the pen within 12 cm of the U tape you see a slight attraction. Briefly explain all of your observations.
'''

If a plastic pen is rubbed through your hair, it gets negatively charged. There is a slight repulsion with L-tape and negatively charged plastic pen. So the charge on the L-tape should be positive. There is a slight attraction with U-tape and negatively charged plastic pen. So the charge on the U-tape should be positive.

'''P61 You run your finger along the slick side of a positively charged tape, and then observe that the tape is no longer attracted to your hand. Which of the following are not plausible explanations for this observation? Check all that apply. (1) Sodium ions (Na+) from the salt water on your skin move onto the tape, leaving the tape with a zero (or very small) net charge. (2) Electrons from the mobile electron sea in your hand move onto the tape, leaving the tape with a zero (or very small) net charge. (3) Chloride ions (Cl-) from the salt water on your skin move onto the tape, leaving the tape with a zero (or very small) net charge. (4) Protons are pulled out of the nuclei of atoms in the tape and move onto your finger.'''

When Sodium ions (Na+) from the salt water on your skin move onto the tape, they provide additional positive charge to the already positive charged tape. So 1) is not a plausible explanation.
Human hand is considered neutral so no electrons travel from the hand to the positively charged tape so the net charge remains the same as before. So 2) is not a plausible explanation.
Chlorine particles are negative and they neutralize the positive tape. So tape then has not net charge and can't attract the hand. So 3) is a plausible explanation.
There is no chance of protons getting pulled out of the atoms nucleus as it's not that easy. So 4) is not a plausible explanation.

===Simple===

===Middling===

===Difficult===

==Connectedness==

==History==

==See also==

===Further Reading===
[[Charge Motion in Metals]]

[[Polarization]]

===External Links===
More on [https://byjus.com/physics/charge-transfer/ charge transfer].

More on [https://www.physicsclassroom.com/class/estatics/Lesson-2/Charging-by-Conduction charging by conduction].

This [https://www.brightstorm.com/science/physics/electricity/charge-transfer-electroscope/ video] talks about methods of charge transfer and using an electroscope to measure charge.

==References==

Internet resources on this topic:

http://www.physicsclassroom.com/class/estatics/Lesson-1/Conductors-and-Insulators

http://www.physicsclassroom.com/class/estatics/Lesson-1/Charge-Interactions

Chabay, Ruth W., Bruce Sherwood. Matter and Interactions, Volume II: Electric and Magnetic Interactions, 4th Edition. Wiley, 19/2015. VitalBook file.

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

Charge Transfer

2020-04-20T01:03:55Z

Ftariq6:

claimed by '''Fehmeen Tariq Spring 2020'''

If a charged conductor comes in contact, or is in close enough proximity, with another conductor, it is possible to transfer this charge to the second conductor. This process is called '''charge transfer'''. However, the ''Law of Conservation of Charge'' states that charges cannot be created or destroyed. Charge cannot be created, the presence of a negative charge is merely the effect of an object gaining electrons from another material. Since charge cannot be created or destroyed, and is just the transfer of electrons between materials, the magnitude of the charge transfer between two objects will be equivalent. For however much the charge of one object increases during charge transfer, the other must decrease the same amount. There are multiple ways that charge can be transferred such as through direct contact (ie friction), induction, and conduction.

==The Main Idea==
'''Insulators vs Conductors'''

[[File:inschargedist.gif|thumb|500px|Charges transferred to an insulator remains at the location of transfer.]]

In an '''insulator''', electrons are bounded tightly to atoms, which prevents charged particles from moving through the material. If charge is transferred to an insulator at a given location, the charge will remain at the location that the transfer occurred.

Within '''conductors''', on the other hand, electrons are able to flow freely from particle to particle. When charge is transferred to a conductor, the charge is distributed evenly across the surface of the object via ''electron movement''. The electrons will be distributed until the repelling force between the excess electrons is minimized. This is the main difference between insulators and conductors: insulators do not have mobile charged particles whereas conductors have mobile charged particles that allow for charge transfer through the free movement of electrons. Examples of insulators include rubber and air and examples of conductors include metals and salt water.

'''Charge by Friction'''

Certain objects and materials have a greater attraction to electrons than others. For example, rubber is highly attracted to electrons, whereas fur or hair has a lower attraction. Therefore, if you take a balloon and rub it on your head, electrons previously in the atoms of the hair will be pulled to the atoms of the rubber balloon. This creates an electron imbalance which makes the balloon negatively charged and the hair positively charged. This creates the effect where the hair will stand up and be pulled towards the balloon since the opposing charges make the two objects attracted. Likewise, two charged balloons will repel each other. Another example of this is rubbing a glass rod with silk. The glass rod will become positively charged and the silk will become negatively charged; this means that electrons were transferred from the glass rod to the silk, since protons are not removed from the nuclei. Rubbing two objects together is not necessary for charge transfer, but because rubbing creates more points of contact between two objects, it facilitates charge transfer.

'''Transfer Charges by Conduction'''

As shown in the previous section, electrons move from one object to another through points of contact; this is especially true among metals. Charging by conduction requires the contact of a charged metal, positive or negative, to a neutral metal. If a negatively charged object touches the neutral metal, the excess electrons will flow through the neutral object. Since electrons repel each other and the negatively charged metal has a buildup of electrons, a certain number of excess electrons will flow out and spread across the neutral object when given an outlet in the form of the neutral metal. This process leaves both metals negatively charged. The same process occurs with positively charged objects touching neutral metals. Additionally, note that this process only works between two conductors an insulator cannot undergo conduction.

[[File:Conductiontransfer.gif|450 px]]

'''Transfer Charges by Induction'''

Unlike the transfer of charges by conduction, objects can transfer charges by induction without making contact. When an object is charged, it has an electric field. This electric field will repel or attract electrons in another object. This electron movement is called transfer of charges by induction. A neutral object can be charged by another charged object through a process called '''polarization'''. This is when electrons in the object are repelled or attracted to one side of the object by the charged second object. For example, if a negatively charged sphere is placed near a neutral sphere, the electrons in the neutral sphere will be repelled by the charged sphere. The neutral sphere is now polarized, with one side of it being negatively charged and the other side being positively charged. The negatively charged side of the sphere can be removed through grounding or with a conductor. Once removed, the originally neutral sphere will now be positively charged. Another example of induction is the balloon and black pepper experiment. A balloon can be given a negative charge by rubbing it on hair. When the balloon is placed near grounded black pepper, the black pepper particles will be polarized so that they become positively charged on top and will be attracted to the negatively charged balloon.

[[File:Indtransfer.gif|500 px]]

===A Mathematical Model===

===A Computational Model===

This [https://phet.colorado.edu/en/simulation/balloons website] is a great model for charge transfer. It shows charge transfer between a sweater and a balloon to demonstrate static electricity. Try using it without the charges showing and guessing where they will go.

[[File:Charge_transfer_model_pic.PNG|400 px]]

==Examples==

===Simple===

===Middling===

===Difficult===

==Connectedness==

==History==

==See also==

===Further Reading===
[[Charge Motion in Metals]]

[[Polarization]]

===External Links===
More on [https://byjus.com/physics/charge-transfer/ charge transfer].

More on [https://www.physicsclassroom.com/class/estatics/Lesson-2/Charging-by-Conduction charging by conduction].

This [https://www.brightstorm.com/science/physics/electricity/charge-transfer-electroscope/ video] talks about methods of charge transfer and using an electroscope to measure charge.

==References==

Internet resources on this topic:

http://www.physicsclassroom.com/class/estatics/Lesson-1/Conductors-and-Insulators

http://www.physicsclassroom.com/class/estatics/Lesson-1/Charge-Interactions

Chabay, Ruth W., Bruce Sherwood. Matter and Interactions, Volume II: Electric and Magnetic Interactions, 4th Edition. Wiley, 19/2015. VitalBook file.

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

Length and Stiffness of an Interatomic Bond

2019-11-24T05:25:32Z

Ftariq6:

'''Claimed by Fehmeen Tariq Fall 2019'''

This topic covers find the length and stiffness of an Interatomic Bond.

==The Main Idea==

A solid object is made up of a large amount of atoms that are held together by chemical bonds in a lattice-like structure as seen in ''Figure 1''. Each atom is connected to its neighboring atoms by these chemical bonds. One way to approximate this solid object model is to imagine the atoms as balls and the chemical bonds as springs-- a ball and spring model! The relaxed length of the microscopic spring between two atoms is just the distance from the center of one atom to the center of the other atom, <math>d</math>. For our object, the distance is just twice the radius of one of the atoms since the electron cloud of the atom fills in the extra space. If the atoms are in a cubic arrangement that means that the volume of each atom would be <math>d × d × d</math> and would have a cross-sectional area of <math>d × d</math>.

''Figure 1''
[[File:ballcube.jpg|center]]

Since the interatomic bonds are modeled as springs, they have a stiffness, <math>k_{si}</math>, that relates the interatomic force to the stretch of the interatomic bonds. We use <math>s</math> for the microscopic stretch.

With this information, we can determine the stiffness of an interatomic bond. In order to determine stiffness, we must determine the length of an interatomic bond in a particular material. For different materials, bond lengths will vary slightly depending on the size of the atoms. The length of one interatomic bond is defined as the center-to-center distance between two adjacent atoms, which is just the distance of the two atoms' radii added together since we use the space-filling model. To find the radius, we divide the diameter of a single atom in half. So, the center-to-center distance is essentially equal to the diameter of a single atom since in the space-filling model there is no space between each atom. If we can calculate the length of the interatomic bond (the diameter of a single atom), we can use this data to find the stiffness of the interatomic bond, modeled as a spring.

==Length of an Interatomic Bond==

The length of an interatomic bond is defined as the center-to-center distance between adjacent atoms. This is the same as the diameter of an atom (including the full electron cloud).

[[File:electroncloudstuff.jpg|center]]

We can calculate atomic diameters for crystals of particular elements by using the measured density of the material in kilograms per cubic meter and Avogadro's number (the number of atoms in one mole of the material), 6.02 X 10^23 atoms per 1 mol.

The mass of one atom can be determined using the mass of one mole and dividing it by <math>6.022 x 10^{23} </math> atoms (Avogadro's number)

==The Stiffness of an Interatomic Bond==

It is difficult to measure the stiffness of an interatomic bond directly, so instead we can analyze data from macroscopic experiments to determine this quantity. We will consider the stiffness of an interatomic bond as a spring.

The equation for stiffness is:
<math>|F| = k_{si}|s|</math>

===Springs in Series===

Springs in series refers to when springs are linked end-to-end.

Two identical springs linked end to end stretch twice as much as one spring when the same force is applied. The combined spring therefore is only half as stiff as the individual springs.

[[File:springsinseries.jpg]]

===Springs in Parallel===

Springs in parallel is when springs are linked side-by-side.

We can think of the two springs as a single, wider spring. Two springs side by side are effectively twice as stuff as a single spring.

[[File:springsinparallel.jpg]]

===Cross-Sectional Area===

The cross-sectional area of an object is the area of a flat surface made by slicing through the object.

For example, the cross-sectional area of a cylinder is the area of a cicle and the cross-sectional area of a rectangular solid is the area of a rectangle.

[[File:crosssectionalarea.jpg]]

== Mathematical Model ==
'''Formulas to Know'''

In many problems, you will be given the density of the solid that you are being questioned on. It is important to remember that <math>{density} = \frac{mass}{volume}</math>
Number of bonds in one "chain" of the solid:
<math>n_{bonds}={\frac{L}{d_{atomic}}}</math>

Spring Constant for one chain in the overall material:
<math>k_{chain}({\frac{k_{atomic}}{n_{bonds}}})</math>

Number of "chains" in the solid:
<math>n_{chains}={\frac{A_{wire}}{A_{atom}}}={\frac{A_{wire}}{{(d_{atomic})}^2}}</math>

Spring constant for the overall material:
<math>k_{wire}=k_{chain}×n_{chains} = k_{atomic} ({\frac{n_{chains}}{n_{bonds}}})</math>
<math>k_{wire}=k_{atomic}( {\frac{A_{wire}}{L_{wire}}})({\frac{1}{d_{atomic}}})</math>

==Examples==

===Example 1===

The US Penny is actually made of zinc. A typical penny has a diameter of 1.905 cm and an average thickness fo 1.228 mm. The density of zinc is 7140 kg/m^3 and its atomic weight is 65.4 amu = 65.4 g/mol. Young's modulus of zinc is 1.022 e11 N/m^2.

''a. Determine the mass of a typical penny.''

<div align ="center">

<math align = 'center'>ρ = {\frac{m}{v}}</math>

<math>m = ρ v = ρ(hπr^2) = ρ(hπ({\frac{d}{2}})^2)</math>

<math>m = (7140)(1.228e-3)(π)({\frac{1.905e-2}{2}}^2)</math>

<math>m = 0.002499\ kg = 2.499\ g</math>

</div>

''b. What is the diameter of a single zinc atom?''

<div align ="center">

<math> n_{atoms} = (2.499\ g)({\frac{1\ mol}{65.4\ g}})({\frac{6.022e23\ atoms}{1\ mol}})=2.3e22\ atoms</math>
<math>m_{atom} = {\frac{0.002499\ kg}{2.3e22\ atoms}} = 1.09e-25\ kg*m^3 </math>

<math>ρ = {\frac{m_{atom}}{v_{atom}}}</math>

<math> v_{atom} = {\frac{m_{atom}}{ρ_{atom}}} = {\frac{1.09e-25\ kg*m^3}{7140\ kg}}=1.53e-29\ m^3</math>

<math> v_{atom} = (d_{atom})^3</math>

<math> d_{atom} = (v_{atom})^{1/3}=(1.53e-29\ m^3)^{1/3} = 2.48e-10\ m</math>

</div>

''c. How many zinc atoms make up one side of the penny?''

<div align ="center">

<math>{\frac{area\ of\ penny\ face}{cross-sectional\ area\ of\ one\ atom}}={\frac{π({\frac{d}{2}})^2}{π({\frac{d_{atom}}{2}})^2}}={\frac{(1.095e-2)^2}{(2.48e-10)^2}} = 1.95e15\ atoms</math>

</div>

''d. Calculate the interatomic spring stiffness for zinc.''

<div align ="center">

<math>y = {\frac{k_{si}}{d_{atom}}}</math>

<math>k_{si} = y*d_{atom} = (1.077e11)(2.48e-10) = 26.7096\ {\frac{N}{m}}</math>

</div>

===Example 2===

A copper wire is 2m long. The wire has a square cross section. Each side of the wire is 1mm in width. Making sure the wire is straight, you hang a 10 kg mass on the end of the wire. The wrie is now 1.67 mm longer. Determine the stiffness of one interatomic bond in copper.

'''Step 1'''
Spring Stiffness of Entire Wire
<div align ="center">

<math>Δp_y = 0 = (k_{s,\ wire}*s-mg)Δt</math>

<math>k_{s, wire} = {\frac{mg}{s}} = {\frac{(10\ kg)(9.8\ N/kg)}{(1.67e-3\ m)}}= 5.87e4\ N/m</math>

</div>

'''Step 2'''
Cross-Sectional Area of Wire

<div align ="center">

<math>A_{wire} = (1e-3\ m)^2 = 1.6e-6\ m^2</math>

</div>

'''Step 3'''
Cross-Sectional Area of One Atom

<div align ="center">

<math>A_{atom} = (2.28e-10\ m)^2 = 5.20e-20\ m^2 </math>

</div>

'''Step 4'''
Number of Chains

<div align ="center">

<math>n_{chains} = {\frac{A_{wire}}{A_{atom}}} = {\frac{1e-6\ m^2}{5.2e-20\ m^2}} = 1.92e13</math>

</div>

'''Step 5'''
Number of Bonds in One Chain

<div align ="center">

<math>n_{bonds} = {\frac{L_{wire}}{d}} = {\frac{2\ m}{2.28e-10\ m}} = 8.77e9</math>

</div>

'''Step 6'''
Stiffness of One Interatomic Spring

<div align ="center">

<math>k_{s, wire} = {\frac{(k_{si})(n_{chains})}{n_{bonds}}}</math>

<math>k_{si} = {\frac{(k_{s,wire})(n_{bonds})}{n_{chains}}}={\frac{(5.87e4)(8.77e9)}{1.92e13}} = 26.8\ N/m</math>

</div>

===Example 3===

If a chain of 50 identical short springs linked end to end has a stiffness of 270 N/m, what is the stiffness of one short spring?

<div align ="center">

<math> Total\ number\ of\ springs\ = n = 50</math>

<math> Effective\ stiffness\ constant\ = k_{eff} = 270\ N/m</math>

<math> Stiffness\ constant\ of\ each\ spring\ = k_{s}</math>

<math> {\frac{1}{k_{eff}}}={\frac{n}{k_{s}}} </math>

<math> k_{s}=n*k_{eff}</math>

<math>k_{s} = 50 * 270\ N/m</math>

<math>k_{s} = 13500\ N/m </math>

</div>

===Example 4===

Forty-five identical springs are placed side by side and connected in parallel to a large massive block. The stiffness of the 45-spring combination is 20,250 N/m. What is the stiffness of one of the individual springs?

<div align ="center">

<math> Total\ number\ of\ springs\ = n = 45</math>

<math> Effective\ stiffness\ constant\ = k_{eff} = 20250\ N/m</math>

<math> Stiffness\ constant\ of\ each\ spring\ = k_{s}</math>

<math> k_{eff}=n*k_{s}</math>

<math>k_{s} = {\frac{k_{eff}}{n}}</math>

<math>k_{s} = {\frac{20250\ N/m}{45}}</math>

<math>k_{s} = 450\ N/m </math>

</div>

===Example 5===

Five identical springs, each with stiffness 390 N/m, are attached in parallel (that is side by side) to hold up a heavy weight. If these springs were replaced by an equivalent single spring, what should be the stiffness of this single spring?

<div align ="center">

<math> k_{eff}=n*k_{s}</math>

<math> k_{s}=390\ N/m</math>

<math> n = 5</math>

<math>k_{eff} = {5}*{390\ N/m}</math>

<math>k_{eff} = {1950\ N/m}</math>

</div>
==Connectedness==

#How is this topic connected to something that you are interested in?
Materials are all made up of atoms and their connections. Its interesting to see how we can apply macroscopic observations to a microscopic topic.

#How is it connected to your major?
Biomedical Engineers must always chose the right materials when creating devices for humans. Depending on their use, a material might not need to be very durable, or it may need to be the opposite, and be extremely durable.

#Is there an interesting industrial application?
Bonds are found in every material and object, so how these different bonds are connected create different objects and its really cool.

== See also ==

To further your reading and increase your knowledge, check out [[Young's Modulus]]. Like density and interatomic spring stiffness, Young's Modulus is a property of a particular material and is independent of the share or size of a particular object made of that material.

==References==

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 4

http://www.webassign.net/question_assets/ncsucalcphysmechl3/lab_10_1/manual.html

Created by Emily Milburn

Length and Stiffness of an Interatomic Bond

2019-11-24T05:05:42Z

Ftariq6:

'''Claimed by Fehmeen Tariq Fall 2019'''

This topic covers find the length and stiffness of an Interatomic Bond.

==The Main Idea==

A solid object is made up of a large amount of atoms that are held together by chemical bonds in a lattice-like structure as seen in ''Figure 1''. Each atom is connected to its neighboring atoms by these chemical bonds. One way to approximate this solid object model is to imagine the atoms as balls and the chemical bonds as springs-- a ball and spring model! The relaxed length of the microscopic spring between two atoms is just the distance from the center of one atom to the center of the other atom, <math>d</math>. For our object, the distance is just twice the radius of one of the atoms since the electron cloud of the atom fills in the extra space. If the atoms are in a cubic arrangement that means that the volume of each atom would be <math>d × d × d</math> and would have a cross-sectional area of <math>d × d</math>.

''Figure 1''
[[File:ballcube.jpg|center]]

Since the interatomic bonds are modeled as springs, they have a stiffness, <math>k_{si}</math>, that relates the interatomic force to the stretch of the interatomic bonds. We use <math>s</math> for the microscopic stretch.

With this information, we can determine the stiffness of an interatomic bond. In order to determine stiffness, we must determine the length of an interatomic bond in a particular material. For different materials, bond lengths will vary slightly depending on the size of the atoms. The length of one interatomic bond is defined as the center-to-center distance between two adjacent atoms, which is just the distance of the two atoms' radii added together since we use the space-filling model. To find the radius, we divide the diameter of a single atom in half. So, the center-to-center distance is essentially equal to the diameter of a single atom since in the space-filling model there is no space between each atom. If we can calculate the length of the interatomic bond (the diameter of a single atom), we can use this data to find the stiffness of the interatomic bond, modeled as a spring.

==Length of an Interatomic Bond==

The length of an interatomic bond is defined as the center-to-center distance between adjacent atoms. This is the same as the diameter of an atom (including the full electron cloud).

[[File:electroncloudstuff.jpg|center]]

We can calculate atomic diameters for crystals of particular elements by using the measured density of the material in kilograms per cubic meter and Avogadro's number (the number of atoms in one mole of the material), 6.02 X 10^23 atoms per 1 mol.

The mass of one atom can be determined using the mass of one mole and dividing it by <math>6.022 x 10^{23} </math> atoms (Avogadro's number)

==The Stiffness of an Interatomic Bond==

It is difficult to measure the stiffness of an interatomic bond directly, so instead we can analyze data from macroscopic experiments to determine this quantity. We will consider the stiffness of an interatomic bond as a spring.

The equation for stiffness is:
<math>|F| = k_{si}|s|</math>

===Springs in Series===

Springs in series refers to when springs are linked end-to-end.

Two identical springs linked end to end stretch twice as much as one spring when the same force is applied. The combined spring therefore is only half as stiff as the individual springs.

[[File:springsinseries.jpg]]

===Springs in Parallel===

Springs in parallel is when springs are linked side-by-side.

We can think of the two springs as a single, wider spring. Two springs side by side are effectively twice as stuff as a single spring.

[[File:springsinparallel.jpg]]

===Cross-Sectional Area===

The cross-sectional area of an object is the area of a flat surface made by slicing through the object.

For example, the cross-sectional area of a cylinder is the area of a cicle and the cross-sectional area of a rectangular solid is the area of a rectangle.

[[File:crosssectionalarea.jpg]]

== Mathematical Model ==
'''Formulas to Know'''

In many problems, you will be given the density of the solid that you are being questioned on. It is important to remember that <math>{density} = \frac{mass}{volume}</math>
Number of bonds in one "chain" of the solid:
<math>n_{bonds}={\frac{L}{d_{atomic}}}</math>

Spring Constant for one chain in the overall material:
<math>k_{chain}({\frac{k_{atomic}}{n_{bonds}}})</math>

Number of "chains" in the solid:
<math>n_{chains}={\frac{A_{wire}}{A_{atom}}}={\frac{A_{wire}}{{(d_{atomic})}^2}}</math>

Spring constant for the overall material:
<math>k_{wire}=k_{chain}×n_{chains} = k_{atomic} ({\frac{n_{chains}}{n_{bonds}}})</math>
<math>k_{wire}=k_{atomic}( {\frac{A_{wire}}{L_{wire}}})({\frac{1}{d_{atomic}}})</math>

==Examples==

===Example 1===

The US Penny is actually made of zinc. A typical penny has a diameter of 1.905 cm and an average thickness fo 1.228 mm. The density of zinc is 7140 kg/m^3 and its atomic weight is 65.4 amu = 65.4 g/mol. Young's modulus of zinc is 1.022 e11 N/m^2.

''a. Determine the mass of a typical penny.''

<div align ="center">

<math align = 'center'>ρ = {\frac{m}{v}}</math>

<math>m = ρ v = ρ(hπr^2) = ρ(hπ({\frac{d}{2}})^2)</math>

<math>m = (7140)(1.228e-3)(π)({\frac{1.905e-2}{2}}^2)</math>

<math>m = 0.002499\ kg = 2.499\ g</math>

</div>

''b. What is the diameter of a single zinc atom?''

<div align ="center">

<math> n_{atoms} = (2.499\ g)({\frac{1\ mol}{65.4\ g}})({\frac{6.022e23\ atoms}{1\ mol}})=2.3e22\ atoms</math>
<math>m_{atom} = {\frac{0.002499\ kg}{2.3e22\ atoms}} = 1.09e-25\ kg*m^3 </math>

<math>ρ = {\frac{m_{atom}}{v_{atom}}}</math>

<math> v_{atom} = {\frac{m_{atom}}{ρ_{atom}}} = {\frac{1.09e-25\ kg*m^3}{7140\ kg}}=1.53e-29\ m^3</math>

<math> v_{atom} = (d_{atom})^3</math>

<math> d_{atom} = (v_{atom})^{1/3}=(1.53e-29\ m^3)^{1/3} = 2.48e-10\ m</math>

</div>

''c. How many zinc atoms make up one side of the penny?''

<div align ="center">

<math>{\frac{area\ of\ penny\ face}{cross-sectional\ area\ of\ one\ atom}}={\frac{π({\frac{d}{2}})^2}{π({\frac{d_{atom}}{2}})^2}}={\frac{(1.095e-2)^2}{(2.48e-10)^2}} = 1.95e15\ atoms</math>

</div>

''d. Calculate the interatomic spring stiffness for zinc.''

<div align ="center">

<math>y = {\frac{k_{si}}{d_{atom}}}</math>

<math>k_{si} = y*d_{atom} = (1.077e11)(2.48e-10) = 26.7096\ {\frac{N}{m}}</math>

</div>

===Example 2===

A copper wire is 2m long. The wire has a square cross section. Each side of the wire is 1mm in width. Making sure the wire is straight, you hang a 10 kg mass on the end of the wire. The wrie is now 1.67 mm longer. Determine the stiffness of one interatomic bond in copper.

'''Step 1'''
Spring Stiffness of Entire Wire
<div align ="center">

<math>Δp_y = 0 = (k_{s,\ wire}*s-mg)Δt</math>

<math>k_{s, wire} = {\frac{mg}{s}} = {\frac{(10\ kg)(9.8\ N/kg)}{(1.67e-3\ m)}}= 5.87e4\ N/m</math>

</div>

'''Step 2'''
Cross-Sectional Area of Wire

<div align ="center">

<math>A_{wire} = (1e-3\ m)^2 = 1.6e-6\ m^2</math>

</div>

'''Step 3'''
Cross-Sectional Area of One Atom

<div align ="center">

<math>A_{atom} = (2.28e-10\ m)^2 = 5.20e-20\ m^2 </math>

</div>

'''Step 4'''
Number of Chains

<div align ="center">

<math>n_{chains} = {\frac{A_{wire}}{A_{atom}}} = {\frac{1e-6\ m^2}{5.2e-20\ m^2}} = 1.92e13</math>

</div>

'''Step 5'''
Number of Bonds in One Chain

<div align ="center">

<math>n_{bonds} = {\frac{L_{wire}}{d}} = {\frac{2\ m}{2.28e-10\ m}} = 8.77e9</math>

</div>

'''Step 6'''
Stiffness of One Interatomic Spring

<div align ="center">

<math>k_{s, wire} = {\frac{(k_{si})(n_{chains})}{n_{bonds}}}</math>

<math>k_{si} = {\frac{(k_{s,wire})(n_{bonds})}{n_{chains}}}={\frac{(5.87e4)(8.77e9)}{1.92e13}} = 26.8\ N/m</math>

</div>

===Example 3===

Forty-five identical springs are placed side by side and connected in parallel to a large massive block. The stiffness of the 45-spring combination is 20,250 N/m. What is the stiffness of one of the individual springs?

<div align ="center">

<math> Total\ number\ of\ springs\ = n = 45</math>

<math> Effective\ stiffness\ constant\ = k_{eff} = 20250\ N/m</math>

<math> Stiffness\ constant\ of\ each\ spring\ = k_{s}</math>

<math> k_{eff}=n*k_{s}</math>

<math>k_{s} = {\frac{k_{eff}}{n}}</math>

<math>k_{s} = {\frac{20250\ N/m}{45}}</math>

<math>k_{s} = 450\ N/m </math>

</div>

===Example 4===

Five identical springs, each with stiffness 390 N/m, are attached in parallel (that is side by side) to hold up a heavy weight. If these springs were replaced by an equivalent single spring, what should be the stiffness of this single spring?

<div align ="center">

<math> k_{eff}=n*k_{s}</math>

<math> k_{s}=390\ N/m</math>

<math> n = 5</math>

<math>k_{eff} = {5}*{390\ N/m}</math>

<math>k_{eff} = {1950\ N/m}</math>

</div>
==Connectedness==

#How is this topic connected to something that you are interested in?
Materials are all made up of atoms and their connections. Its interesting to see how we can apply macroscopic observations to a microscopic topic.

#How is it connected to your major?
Biomedical Engineers must always chose the right materials when creating devices for humans. Depending on their use, a material might not need to be very durable, or it may need to be the opposite, and be extremely durable.

#Is there an interesting industrial application?
Bonds are found in every material and object, so how these different bonds are connected create different objects and its really cool.

== See also ==

To further your reading and increase your knowledge, check out [[Young's Modulus]]. Like density and interatomic spring stiffness, Young's Modulus is a property of a particular material and is independent of the share or size of a particular object made of that material.

==References==

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 4

http://www.webassign.net/question_assets/ncsucalcphysmechl3/lab_10_1/manual.html

Created by Emily Milburn

Length and Stiffness of an Interatomic Bond

2019-11-24T04:55:01Z

Ftariq6:

'''Claimed by Fehmeen Tariq Fall 2019'''

This topic covers find the length and stiffness of an Interatomic Bond.

==The Main Idea==

A solid object is made up of a large amount of atoms that are held together by chemical bonds in a lattice-like structure as seen in ''Figure 1''. Each atom is connected to its neighboring atoms by these chemical bonds. One way to approximate this solid object model is to imagine the atoms as balls and the chemical bonds as springs-- a ball and spring model! The relaxed length of the microscopic spring between two atoms is just the distance from the center of one atom to the center of the other atom, <math>d</math>. For our object, the distance is just twice the radius of one of the atoms since the electron cloud of the atom fills in the extra space. If the atoms are in a cubic arrangement that means that the volume of each atom would be <math>d × d × d</math> and would have a cross-sectional area of <math>d × d</math>.

''Figure 1''
[[File:ballcube.jpg|center]]

Since the interatomic bonds are modeled as springs, they have a stiffness, <math>k_{si}</math>, that relates the interatomic force to the stretch of the interatomic bonds. We use <math>s</math> for the microscopic stretch.

With this information, we can determine the stiffness of an interatomic bond. In order to determine stiffness, we must determine the length of an interatomic bond in a particular material. For different materials, bond lengths will vary slightly depending on the size of the atoms. The length of one interatomic bond is defined as the center-to-center distance between two adjacent atoms, which is just the distance of the two atoms' radii added together since we use the space-filling model. To find the radius, we divide the diameter of a single atom in half. So, the center-to-center distance is essentially equal to the diameter of a single atom since in the space-filling model there is no space between each atom. If we can calculate the length of the interatomic bond (the diameter of a single atom), we can use this data to find the stiffness of the interatomic bond, modeled as a spring.

==Length of an Interatomic Bond==

The length of an interatomic bond is defined as the center-to-center distance between adjacent atoms. This is the same as the diameter of an atom (including the full electron cloud).

[[File:electroncloudstuff.jpg|center]]

We can calculate atomic diameters for crystals of particular elements by using the measured density of the material in kilograms per cubic meter and Avogadro's number (the number of atoms in one mole of the material), 6.02 X 10^23 atoms per 1 mol.

The mass of one atom can be determined using the mass of one mole and dividing it by <math>6.022 x 10^{23} </math> atoms (Avogadro's number)

==The Stiffness of an Interatomic Bond==

It is difficult to measure the stiffness of an interatomic bond directly, so instead we can analyze data from macroscopic experiments to determine this quantity. We will consider the stiffness of an interatomic bond as a spring.

The equation for stiffness is:
<math>|F| = k_{si}|s|</math>

===Springs in Series===

Springs in series refers to when springs are linked end-to-end.

Two identical springs linked end to end stretch twice as much as one spring when the same force is applied. The combined spring therefore is only half as stiff as the individual springs.

[[File:springsinseries.jpg]]

===Springs in Parallel===

Springs in parallel is when springs are linked side-by-side.

We can think of the two springs as a single, wider spring. Two springs side by side are effectively twice as stuff as a single spring.

[[File:springsinparallel.jpg]]

===Cross-Sectional Area===

The cross-sectional area of an object is the area of a flat surface made by slicing through the object.

For example, the cross-sectional area of a cylinder is the area of a cicle and the cross-sectional area of a rectangular solid is the area of a rectangle.

[[File:crosssectionalarea.jpg]]

== Mathematical Model ==
'''Formulas to Know'''

In many problems, you will be given the density of the solid that you are being questioned on. It is important to remember that <math>{density} = \frac{mass}{volume}</math>
Number of bonds in one "chain" of the solid:
<math>n_{bonds}={\frac{L}{d_{atomic}}}</math>

Spring Constant for one chain in the overall material:
<math>k_{chain}({\frac{k_{atomic}}{n_{bonds}}})</math>

Number of "chains" in the solid:
<math>n_{chains}={\frac{A_{wire}}{A_{atom}}}={\frac{A_{wire}}{{(d_{atomic})}^2}}</math>

Spring constant for the overall material:
<math>k_{wire}=k_{chain}×n_{chains} = k_{atomic} ({\frac{n_{chains}}{n_{bonds}}})</math>
<math>k_{wire}=k_{atomic}( {\frac{A_{wire}}{L_{wire}}})({\frac{1}{d_{atomic}}})</math>

==Examples==

===Example 1===

The US Penny is actually made of zinc. A typical penny has a diameter of 1.905 cm and an average thickness fo 1.228 mm. The density of zinc is 7140 kg/m^3 and its atomic weight is 65.4 amu = 65.4 g/mol. Young's modulus of zinc is 1.022 e11 N/m^2.

''a. Determine the mass of a typical penny.''

<div align ="center">

<math align = 'center'>ρ = {\frac{m}{v}}</math>

<math>m = ρ v = ρ(hπr^2) = ρ(hπ({\frac{d}{2}})^2)</math>

<math>m = (7140)(1.228e-3)(π)({\frac{1.905e-2}{2}}^2)</math>

<math>m = 0.002499\ kg = 2.499\ g</math>

</div>

''b. What is the diameter of a single zinc atom?''

<div align ="center">

<math> n_{atoms} = (2.499\ g)({\frac{1\ mol}{65.4\ g}})({\frac{6.022e23\ atoms}{1\ mol}})=2.3e22\ atoms</math>
<math>m_{atom} = {\frac{0.002499\ kg}{2.3e22\ atoms}} = 1.09e-25\ kg*m^3 </math>

<math>ρ = {\frac{m_{atom}}{v_{atom}}}</math>

<math> v_{atom} = {\frac{m_{atom}}{ρ_{atom}}} = {\frac{1.09e-25\ kg*m^3}{7140\ kg}}=1.53e-29\ m^3</math>

<math> v_{atom} = (d_{atom})^3</math>

<math> d_{atom} = (v_{atom})^{1/3}=(1.53e-29\ m^3)^{1/3} = 2.48e-10\ m</math>

</div>

''c. How many zinc atoms make up one side of the penny?''

<div align ="center">

<math>{\frac{area\ of\ penny\ face}{cross-sectional\ area\ of\ one\ atom}}={\frac{π({\frac{d}{2}})^2}{π({\frac{d_{atom}}{2}})^2}}={\frac{(1.095e-2)^2}{(2.48e-10)^2}} = 1.95e15\ atoms</math>

</div>

''d. Calculate the interatomic spring stiffness for zinc.''

<div align ="center">

<math>y = {\frac{k_{si}}{d_{atom}}}</math>

<math>k_{si} = y*d_{atom} = (1.077e11)(2.48e-10) = 26.7096\ {\frac{N}{m}}</math>

</div>

===Example 2===

A copper wire is 2m long. The wire has a square cross section. Each side of the wire is 1mm in width. Making sure the wire is straight, you hang a 10 kg mass on the end of the wire. The wrie is now 1.67 mm longer. Determine the stiffness of one interatomic bond in copper.

'''Step 1'''
Spring Stiffness of Entire Wire
<div align ="center">

<math>Δp_y = 0 = (k_{s,\ wire}*s-mg)Δt</math>

<math>k_{s, wire} = {\frac{mg}{s}} = {\frac{(10\ kg)(9.8\ N/kg)}{(1.67e-3\ m)}}= 5.87e4\ N/m</math>

</div>

'''Step 2'''
Cross-Sectional Area of Wire

<div align ="center">

<math>A_{wire} = (1e-3\ m)^2 = 1.6e-6\ m^2</math>

</div>

'''Step 3'''
Cross-Sectional Area of One Atom

<div align ="center">

<math>A_{atom} = (2.28e-10\ m)^2 = 5.20e-20\ m^2 </math>

</div>

'''Step 4'''
Number of Chains

<div align ="center">

<math>n_{chains} = {\frac{A_{wire}}{A_{atom}}} = {\frac{1e-6\ m^2}{5.2e-20\ m^2}} = 1.92e13</math>

</div>

'''Step 5'''
Number of Bonds in One Chain

<div align ="center">

<math>n_{bonds} = {\frac{L_{wire}}{d}} = {\frac{2\ m}{2.28e-10\ m}} = 8.77e9</math>

</div>

'''Step 6'''
Stiffness of One Interatomic Spring

<div align ="center">

<math>k_{s, wire} = {\frac{(k_{si})(n_{chains})}{n_{bonds}}}</math>

<math>k_{si} = {\frac{(k_{s,wire})(n_{bonds})}{n_{chains}}}={\frac{(5.87e4)(8.77e9)}{1.92e13}} = 26.8\ N/m</math>

</div>

===Example 3===

Forty-five identical springs are placed side by side and connected in parallel to a large massive block. The stiffness of the 45-spring combination is 20,250 N/m. What is the stiffness of one of the individual springs?

<div align ="center">

<math> Total\ number\ of\ springs\ = n = 45</math>

<math> Effective\ stiffness\ constant\ = k_{eff} = 20250</math>

<math> Stiffness\ constant\ of\ each\ spring\ = k_{s}</math>

<math> k_{eff}=45k_{s}</math>

<math>k_{s} = {\frac{k_{eff}}{45}}</math>

<math>k_{s} = {\frac{20250\ N/m}{45}}</math>

<math>k_{s} = 450\ N/m </math>

</div>
==Connectedness==

#How is this topic connected to something that you are interested in?
Materials are all made up of atoms and their connections. Its interesting to see how we can apply macroscopic observations to a microscopic topic.

#How is it connected to your major?
Biomedical Engineers must always chose the right materials when creating devices for humans. Depending on their use, a material might not need to be very durable, or it may need to be the opposite, and be extremely durable.

#Is there an interesting industrial application?
Bonds are found in every material and object, so how these different bonds are connected create different objects and its really cool.

== See also ==

To further your reading and increase your knowledge, check out [[Young's Modulus]]. Like density and interatomic spring stiffness, Young's Modulus is a property of a particular material and is independent of the share or size of a particular object made of that material.

==References==

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 4

http://www.webassign.net/question_assets/ncsucalcphysmechl3/lab_10_1/manual.html

Created by Emily Milburn

Length and Stiffness of an Interatomic Bond

2019-11-24T04:47:58Z

Ftariq6:

'''Claimed by Fehmeen Tariq Fall 2019'''
Claimed by Andy Stevens Spring 2017
Claimed by Nicole Tansey Fall 2017

This topic covers find the length and stiffness of an Interatomic Bond.

==The Main Idea==

A solid object is made up of a large amount of atoms that are held together by chemical bonds in a lattice-like structure as seen in ''Figure 1''. Each atom is connected to its neighboring atoms by these chemical bonds. One way to approximate this solid object model is to imagine the atoms as balls and the chemical bonds as springs-- a ball and spring model! The relaxed length of the microscopic spring between two atoms is just the distance from the center of one atom to the center of the other atom, <math>d</math>. For our object, the distance is just twice the radius of one of the atoms since the electron cloud of the atom fills in the extra space. If the atoms are in a cubic arrangement that means that the volume of each atom would be <math>d × d × d</math> and would have a cross-sectional area of <math>d × d</math>.

''Figure 1''
[[File:ballcube.jpg|center]]

Since the interatomic bonds are modeled as springs, they have a stiffness, <math>k_{si}</math>, that relates the interatomic force to the stretch of the interatomic bonds. We use <math>s</math> for the microscopic stretch.

With this information, we can determine the stiffness of an interatomic bond. In order to determine stiffness, we must determine the length of an interatomic bond in a particular material. For different materials, bond lengths will vary slightly depending on the size of the atoms. The length of one interatomic bond is defined as the center-to-center distance between two adjacent atoms, which is just the distance of the two atoms' radii added together since we use the space-filling model. To find the radius, we divide the diameter of a single atom in half. So, the center-to-center distance is essentially equal to the diameter of a single atom since in the space-filling model there is no space between each atom. If we can calculate the length of the interatomic bond (the diameter of a single atom), we can use this data to find the stiffness of the interatomic bond, modeled as a spring.

==Length of an Interatomic Bond==

The length of an interatomic bond is defined as the center-to-center distance between adjacent atoms. This is the same as the diameter of an atom (including the full electron cloud).

[[File:electroncloudstuff.jpg|center]]

We can calculate atomic diameters for crystals of particular elements by using the measured density of the material in kilograms per cubic meter and Avogadro's number (the number of atoms in one mole of the material), 6.02 X 10^23 atoms per 1 mol.

The mass of one atom can be determined using the mass of one mole and dividing it by <math>6.022 x 10^{23} </math> atoms (Avogadro's number)

==The Stiffness of an Interatomic Bond==

It is difficult to measure the stiffness of an interatomic bond directly, so instead we can analyze data from macroscopic experiments to determine this quantity. We will consider the stiffness of an interatomic bond as a spring.

The equation for stiffness is:
<math>|F| = k_{si}|s|</math>

===Springs in Series===

Springs in series refers to when springs are linked end-to-end.

Two identical springs linked end to end stretch twice as much as one spring when the same force is applied. The combined spring therefore is only half as stiff as the individual springs.

[[File:springsinseries.jpg]]

===Springs in Parallel===

Springs in parallel is when springs are linked side-by-side.

We can think of the two springs as a single, wider spring. Two springs side by side are effectively twice as stuff as a single spring.

[[File:springsinparallel.jpg]]

===Cross-Sectional Area===

The cross-sectional area of an object is the area of a flat surface made by slicing through the object.

For example, the cross-sectional area of a cylinder is the area of a cicle and the cross-sectional area of a rectangular solid is the area of a rectangle.

[[File:crosssectionalarea.jpg]]

== Mathematical Model ==
'''Formulas to Know'''

In many problems, you will be given the density of the solid that you are being questioned on. It is important to remember that <math>{density} = \frac{mass}{volume}</math>
Number of bonds in one "chain" of the solid:
<math>n_{bonds}={\frac{L}{d_{atomic}}}</math>

Spring Constant for one chain in the overall material:
<math>k_{chain}({\frac{k_{atomic}}{n_{bonds}}})</math>

Number of "chains" in the solid:
<math>n_{chains}={\frac{A_{wire}}{A_{atom}}}={\frac{A_{wire}}{{(d_{atomic})}^2}}</math>

Spring constant for the overall material:
<math>k_{wire}=k_{chain}×n_{chains} = k_{atomic} ({\frac{n_{chains}}{n_{bonds}}})</math>
<math>k_{wire}=k_{atomic}( {\frac{A_{wire}}{L_{wire}}})({\frac{1}{d_{atomic}}})</math>

==Examples==

===Example 1===

The US Penny is actually made of zinc. A typical penny has a diameter of 1.905 cm and an average thickness fo 1.228 mm. The density of zinc is 7140 kg/m^3 and its atomic weight is 65.4 amu = 65.4 g/mol. Young's modulus of zinc is 1.022 e11 N/m^2.

''a. Determine the mass of a typical penny.''

<div align ="center">

<math align = 'center'>ρ = {\frac{m}{v}}</math>

<math>m = ρ v = ρ(hπr^2) = ρ(hπ({\frac{d}{2}})^2)</math>

<math>m = (7140)(1.228e-3)(π)({\frac{1.905e-2}{2}}^2)</math>

<math>m = 0.002499\ kg = 2.499\ g</math>

</div>

''b. What is the diameter of a single zinc atom?''

<div align ="center">

<math> n_{atoms} = (2.499\ g)({\frac{1\ mol}{65.4\ g}})({\frac{6.022e23\ atoms}{1\ mol}})=2.3e22\ atoms</math>
<math>m_{atom} = {\frac{0.002499\ kg}{2.3e22\ atoms}} = 1.09e-25\ kg*m^3 </math>

<math>ρ = {\frac{m_{atom}}{v_{atom}}}</math>

<math> v_{atom} = {\frac{m_{atom}}{ρ_{atom}}} = {\frac{1.09e-25\ kg*m^3}{7140\ kg}}=1.53e-29\ m^3</math>

<math> v_{atom} = (d_{atom})^3</math>

<math> d_{atom} = (v_{atom})^{1/3}=(1.53e-29\ m^3)^{1/3} = 2.48e-10\ m</math>

</div>

''c. How many zinc atoms make up one side of the penny?''

<div align ="center">

<math>{\frac{area\ of\ penny\ face}{cross-sectional\ area\ of\ one\ atom}}={\frac{π({\frac{d}{2}})^2}{π({\frac{d_{atom}}{2}})^2}}={\frac{(1.095e-2)^2}{(2.48e-10)^2}} = 1.95e15\ atoms</math>

</div>

''d. Calculate the interatomic spring stiffness for zinc.''

<div align ="center">

<math>y = {\frac{k_{si}}{d_{atom}}}</math>

<math>k_{si} = y*d_{atom} = (1.077e11)(2.48e-10) = 26.7096\ {\frac{N}{m}}</math>

</div>

===Example 2===

A copper wire is 2m long. The wire has a square cross section. Each side of the wire is 1mm in width. Making sure the wire is straight, you hang a 10 kg mass on the end of the wire. The wrie is now 1.67 mm longer. Determine the stiffness of one interatomic bond in copper.

'''Step 1'''
Spring Stiffness of Entire Wire
<div align ="center">

<math>Δp_y = 0 = (k_{s,\ wire}*s-mg)Δt</math>

<math>k_{s, wire} = {\frac{mg}{s}} = {\frac{(10\ kg)(9.8\ N/kg)}{(1.67e-3\ m)}}= 5.87e4\ N/m</math>

</div>

'''Step 2'''
Cross-Sectional Area of Wire

<div align ="center">

<math>A_{wire} = (1e-3\ m)^2 = 1.6e-6\ m^2</math>

</div>

'''Step 3'''
Cross-Sectional Area of One Atom

<div align ="center">

<math>A_{atom} = (2.28e-10\ m)^2 = 5.20e-20\ m^2 </math>

</div>

'''Step 4'''
Number of Chains

<div align ="center">

<math>n_{chains} = {\frac{A_{wire}}{A_{atom}}} = {\frac{1e-6\ m^2}{5.2e-20\ m^2}} = 1.92e13</math>

</div>

'''Step 5'''
Number of Bonds in One Chain

<div align ="center">

<math>n_{bonds} = {\frac{L_{wire}}{d}} = {\frac{2\ m}{2.28e-10\ m}} = 8.77e9</math>

</div>

'''Step 6'''
Stiffness of One Interatomic Spring

<div align ="center">

<math>k_{s, wire} = {\frac{(k_{si})(n_{chains})}{n_{bonds}}}</math>

<math>k_{si} = {\frac{(k_{s,wire})(n_{bonds})}{n_{chains}}}={\frac{(5.87e4)(8.77e9)}{1.92e13}} = 26.8\ N/m</math>

</div>

===Example 3===

Forty-five identical springs are placed side by side and connected in parallel to a large massive block. The stiffness of the 45-spring combination is 20,250 N/m. What is the stiffness of one of the individual springs?

<div align ="center">

<math> Total\ number\ of\ springs\ = n = 45</math>

<math> Effective\ stiffness\ constant\ = k_{eff} = 20250</math>

<math> Stiffness\ constant\ of\ each\ spring\ = k_{s}</math>

<math> k_{eff}=45k_{s}</math>

<math>k_{s} = {\frac{k_{eff}}{45}}</math>

<math>k_{s} = {\frac{20250\ N/m}{45}}</math>

<math>k_{s} = 450\ N/m </math>

</div>
==Connectedness==

#How is this topic connected to something that you are interested in?
Materials are all made up of atoms and their connections. Its interesting to see how we can apply macroscopic observations to a microscopic topic.

#How is it connected to your major?
Biomedical Engineers must always chose the right materials when creating devices for humans. Depending on their use, a material might not need to be very durable, or it may need to be the opposite, and be extremely durable.

#Is there an interesting industrial application?
Bonds are found in every material and object, so how these different bonds are connected create different objects and its really cool.

== See also ==

To further your reading and increase your knowledge, check out [[Young's Modulus]]. Like density and interatomic spring stiffness, Young's Modulus is a property of a particular material and is independent of the share or size of a particular object made of that material.

==References==

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 4

http://www.webassign.net/question_assets/ncsucalcphysmechl3/lab_10_1/manual.html

Created by Emily Milburn