The Maxwell-Boltzmann Distribution - Revision history

Laurence12799: /* See also */

2019-08-07T04:16:06Z

Laurence12799: /* History */

2019-08-07T04:12:32Z

History

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	==History==		==History==


	In 1857, a scientist by the name of Rudolf Clausius figured out how to calculate the speed of atoms. A few years later, in 1860, Irish physicist James Maxwell published a paper, in which he ascertained a way to connect the physical properties of gases to the distribution of particle velocities. His research compounded off of Clausius's work, but he went further by describing the distribution of speeds in reference to the average speed. In 1872, Austrian physicist, Ludwig Boltzmann, substantiated Maxwell's research by adding the H-theorem, which quantified the approach to equilibrium at which the Maxwell distribution velocities exist. Because of Maxwell and Boltzmann's collaboration, they had come up with a function called the Maxwell-Boltzmann Distribution.		In 1857, a scientist by the name of Rudolf Clausius figured out how to calculate the speed of atoms. A few years later, in 1860, Irish physicist James Maxwell published a paper, in which he ascertained a way to connect the physical properties of gases to the distribution of particle velocities. His research compounded off of Clausius's work, but he went further by describing the distribution of speeds in reference to the average speed. In 1872, Austrian physicist, Ludwig Boltzmann, substantiated Maxwell's research by adding the H-theorem, which quantified the approach to equilibrium at which the Maxwell distribution velocities exist. Because of Maxwell and Boltzmann's collaboration, they had come up with a function called the Maxwell-Boltzmann Distribution.

Laurence12799: /* Connectedness */

2019-08-07T04:11:46Z

Connectedness

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	==Connectedness==		==Connectedness==

	This topic is particularly interesting to me because of its potential use in my father's work. My father is a ~~chemist~~, and often he has to induce reactions involving gases. When I read about this topic, it was really interesting to see how one can calculate the speeds of particles based on these distributions. This is important because from this one can calculate the energy of a single gas particle, and this could be important if a scientist like my father needed a specific energy level. It was ~~just~~ really interesting to see how mass, speed, and ~~temperature~~ of a system all have an important relationships with each other. This is not a huge connection to my major (Mechanical Engineering) as it seems to be more related to ~~chemical engineering~~, but I would like to learn more about this concept in other classes. There are some applications for scientists to use these Maxwell-Boltzmann Distributions as guides to induce reactions between particles.		This topic is particularly interesting to me because of its potential use in my father's work. My father is a Chemist, and often he has to induce reactions involving gases. When I read about this topic, it was really interesting to see how one can calculate the speeds of particles based on these distributions. This is important because from this, one can calculate the energy of a single gas particle, and this could be important if a scientist like my father needed a specific energy level. It was really interesting to see how mass, speed, and the Temperature of a system all have important relationships with each other. This is not a huge connection to my major (Mechanical Engineering) as it seems to be more related to Chemical Engineering, but I would like to learn more about this concept in other classes. There are some applications for scientists to use these Maxwell-Boltzmann Distributions as guides to induce reactions between particles.

	==History==		==History==

Laurence12799: /* Difficult */

2019-08-07T04:06:39Z

Difficult

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	:::<math>v_{rms} = \sqrt{\frac{3 \times 1.381 \times 10^{-23} \times 500}{6.65 \times 10^{-27}}} = 1,765.0 \ \frac{m}{s}</math>		:::<math>v_{rms} = \sqrt{\frac{3 \times 1.381 \times 10^{-23} \times 500}{6.65 \times 10^{-27}}} = 1,765.0 \ \frac{m}{s}</math>

			::As we can see, all these speeds are much greater than <math>325 \ \frac{m}{s}</math>
	~~We have been given the values of T, v, <math> m_{helium} </math>, and <math> k </math>. From here,~~ we can ~~plug~~ these ~~values into the formula.~~

	:<math> f(325~~) =~~ \~~sqrt{\left(~~\frac{~~(6.65 × 10^{-27})~~}{~~(2 \pi)(1.381 × 10^{-23~~}~~)(500)}\right)^3}\, 4\pi (325)^2 e^{- \frac{(6.65 × 10^{-27})(325)^2}{2(1.381 × 10^{-23})(500)}}, </math>~~

	~~:<math> f(325) = .0000755 </math>~~

	~~:Number of Helium Particles <math> = f(325)*(100,000) = 7.55~~ </math>


	~~Based on this fraction, if there were 100,000 helium particles, there would only be between 7 or 8 particles with this speed at this temperature. This is a very small fraction of helium particles.~~

	==Connectedness==		==Connectedness==

Laurence12799: /* Difficult */

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Difficult

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 ===Difficult===
-In a sample of <math>100,000</math> Helium particles, with a speed of <math>325 \ \frac{m}{s}</math> at <math>500&deg;K</math>.  The mass of a Helium particle is <math>m_{helium} = 6.65 \times 10^{-27} \ kg </math> and Boltzmann's constant <math>k = 1.381 × 10^{-23} m^2*kg*s^{-2}*K^{-1} </math>.
+A sample of <math>100,000</math> Helium particles at <math>500&deg;K</math> is in your possession. The mass of a Helium particle is <math>m_{helium} = 6.65 \times 10^{-27} \ kg </math> and Boltzmann's constant is <math>k = 1.381 \times 10^{-23} \left(\frac{m^2 \cdot kg}{s^2 \cdot K}\right)</math>.
 :'''a) What is the value of the Maxwell-Boltzmann function for the Helium particles at <math>325 \ \frac{m}{s}</math>
-Given the Maxwell-Boltzmann function:
+::The Maxwell-Boltzmann function for Helium is:
-:<math> f(v) = \sqrt{\left(\frac{m}{2 \pi kT}\right)^3}\, 4\pi v^2 e^{- \frac{mv^2}{2kT}}, </math>
+:::<math> f(v) = \sqrt{\left(\frac{m_{He}}{2 \pi kT}\right)^3}\, 4\pi v^2 e^{p(v)} </math>, where

Laurence12799: /* Difficult */

2019-08-06T22:23:02Z

Difficult

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	===Difficult===		===Difficult===
			In a sample of <math>100,000</math> Helium particles, with a speed of <math>325 \ \frac{m}{s}</math> at <math>500°K</math>. The mass of a Helium particle is <math>m_{helium} = 6.65 \times 10^{-27} \ kg </math> and Boltzmann's constant <math>k = 1.381 × 10^{-23} m^2kgs^{-2}*K^{-1} </math>.

			:'''a) What is the value of the Maxwell-Boltzmann function for the Helium particles at <math>325 \ \frac{m}{s}</math>
	~~Using~~ the Maxwell-Boltzmann function~~, calculate~~ the ~~number of helium gas particles in a sample of 100,000 helium~~ particles ~~with a speed of 325 ''m/s''~~ at ~~500K. Given that~~ <math> m_{~~helium} = 6.65 × 10^{-27~~} ~~kg </math> and <math>k = 1.381 × 10^~~{-23} m^2kgs~~^{-2}*K^{-1~~} </math>.

	Given the Maxwell-Boltzmann function:		Given the Maxwell-Boltzmann function:

Laurence12799: /* Middling */

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Middling

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	===Middling===		===Middling===
			:'''a) Looking at the graph above from the prior example and using some logic based on the Maxwell-Boltzmann Distributions, which will have a wider range of appreciable speeds:'''

			::''Helium at 500 K or Argon at 250 K?''

	~~Looking at~~ the graph ~~above from~~ the ~~prior example, what will have~~ a ~~larger speed~~ distribution, ~~helium at 500 K or argon at 250 K?~~ Helium ~~at 300 K or argon at 400 K?~~ Argon ~~at 400 K or argon at 900 K?~~		:::As we can see from the graph, Helium's distribution is much more spread out than Argon's when at the same Temperature. It is generally true that a particle's distribution will "flatten out" as the system's Temperature increases, as can see be seen in the demo in the Computational Model section. For us, this means Helium's distribution would become even more spread out compared to Argon's; it would have a much greater range of appreciable speeds than Argon.

			::''Helium at 300 K or Argon at 400 K?''

	~~This problem requires some intuition~~.		:::Even though Helium's Temperature is much less than Argon's, its much lighter particle mass still allows its distribution of speeds to have a larger appreciable range than Argon.

			::''Argon at 400 K or Argon at 900 K?''

	~~Looking at the first~~ case, ~~helium at 500 K will probably~~ have the ~~larger speed distribution because the temperature~~ is ~~much higher than argon's temperature, and looking at~~ the ~~graph, it can be inferred that helium particles are lighter than argon particles~~. ~~This allows helium particles to achieve~~ a larger range of speeds.		:::In this case, both systems have Argon. Thus, the main characteristic to look at is the Temperature of each system. The higher Temperature system will have a larger range of speeds than the lower Temperature system..


	~~For the second case, it seems to be that helium particles will still have a larger speed distribution even though the helium particles are exposed to a lower temperature (300 K)~~ than ~~argon particles (400 K). This is again due to the light helium particles that make up for~~ the lower ~~temperature~~.


	~~For the third case, the larger speed distribution will occur for argon at 900 K because higher temperatures will cause particles to move at faster speeds, thus forming a larger speed distribution~~.

	===Difficult===		===Difficult===

Laurence12799: /* A Computational Model */

2019-08-06T21:57:12Z

A Computational Model

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	Maxwell-Boltzmann Distributions are not only represented by functions, but they are also represented by graphs. These graphs can tell scientists a lot about the properties of a gas. They can detail gas particles' masses and speeds, and the graphs can also tell a scientist how the gas responds to variations in temperature.		Maxwell-Boltzmann Distributions are not only represented by functions, but they are also represented by graphs. These graphs can tell scientists a lot about the properties of a gas. They can detail gas particles' masses and speeds, and the graphs can also tell a scientist how the gas responds to variations in temperature.

	To visualize how a Maxwell-Boltzmann Distribution plot looks, the link below provides a demonstration to see how the Temperature of the system affects the particle speeds.		To visualize how a Maxwell-Boltzmann Distribution plot looks, the link below provides a demonstration to see how the Temperature of the system affects the particle speeds:

	[http://www.absorblearning.com/media/attachment.action?quick=10w&att=2645 Maxwell-Boltzmann Temperature Effect Demo]		*[http://www.absorblearning.com/media/attachment.action?quick=10w&att=2645 Maxwell-Boltzmann Temperature Effect Demo]

	The graph below displays the Maxwell-Boltzmann Distribution of <math> O_2 </math> at 300 Kelvin. As one can see, the most probable speed, <math> v_p </math>, is marked at the peak of the distribution curve. The root-mean-square speed, <math> v_{rms} </math>, is also displayed on the curve.		The graph below displays the Maxwell-Boltzmann Distribution of <math> O_2 </math> at 300 Kelvin. As one can see, the most probable speed, <math> v_p </math>, is marked at the peak of the distribution curve. The root-mean-square speed, <math> v_{rms} </math>, is also displayed on the curve.

Laurence12799: /* A Computational Model */

2019-08-06T21:55:17Z

A Computational Model

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	The graph below displays the Maxwell-Boltzmann Distribution of <math> O_2 </math> at 300 Kelvin. As one can see, the most probable speed, <math> v_p </math>, is marked at the peak of the distribution curve. The root-mean-square speed, <math> v_{rms} </math>, is also displayed on the curve.		The graph below displays the Maxwell-Boltzmann Distribution of <math> O_2 </math> at 300 Kelvin. As one can see, the most probable speed, <math> v_p </math>, is marked at the peak of the distribution curve. The root-mean-square speed, <math> v_{rms} </math>, is also displayed on the curve.

	[[File: maxbolts.jpg\|Maxwell-Boltzmann Distribution of Oxygen at 300K\|Source: chemwiki.ucdavis.edu\|center\|600px]]		[[File: maxbolts.jpg\|Maxwell-Boltzmann Distribution of Oxygen at 300K\|Source: chemwiki.ucdavis.edu\|center\|600px\|center\|thumb\|Credit for this image is given to [http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/Gas_Phase_Kinetics/Maxwell-Boltzmann_Distributions chemwiki.ucdavis.edu]]]

	Credit for this image is given to [http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/Gas_Phase_Kinetics/Maxwell-Boltzmann_Distributions chemwiki.ucdavis.edu]

	==Examples==		==Examples==

Laurence12799: /* Simple */

2019-08-06T21:54:38Z

Simple

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	===Simple===		===Simple===

			Examine the given Maxwell-Boltzmann Distributions of Helium, Neon, Argon, and Xenon.

	~~Looking at the~~ Maxwell-~~Boltzmann Distributions of several gases below, what can be said about the relationship between particle speed distributions and the mass of the particles?~~		[[File: Problem1.png\|center\|600px\|thumb\|Credit for this image is given to [http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/Gas_Phase_Kinetics/Maxwell-Boltzmann_Distributions chemwiki.ucdavis.edu]]]

	~~[[File~~: ~~Problem1.png ]]~~		:'''a) What can be said about the relationship between particle speed distributions and the mass of the particles?'''

			::The Maxwell-Boltzmann distributions show a general trend with the mass of the particles:

	~~Credit for this image is given to [http~~:/~~/chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/Gas_Phase_Kinetics/Maxwell-Boltzmann_Distributions chemwiki.ucdavis.edu]~~		:::''As particle mass increases, the distribution of speeds becomes more localized around the most probable speed (<math>v_{p}</math>)''

			::It is also worth noting heavier particles will have a lower average speed, as can be seen from the graph.
	It ~~can be concluded that the~~ heavier particles ~~move slower on average and~~ have a ~~smaller range of particle speeds~~, ~~whereas lighter particles are capable of faster speeds so they have much larger ranges of particle speeds~~.

	===Middling===		===Middling===

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	== See also ==		== See also ==

	~~[http://www.physicsbook.gatech.edu/The_Energy_Principle The Energy Principle]~~
	~~[http://www.physicsbook.gatech.edu/Kinetic_Energy Kinetic Energy]~~


	===Further reading===		===Further reading===

	Matter and Interactions, Volume I, Modern Mechanics		*Matter and Interactions, Volume I, Modern Mechanics

	===External links===		===External links===
	[http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/Gas_Phase_Kinetics/Maxwell-Boltzmann_Distributions]		*http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/Gas_Phase_Kinetics/Maxwell-Boltzmann_Distributions<br>
	[http://dwb.unl.edu/teacher/nsf/c09/c09links/www.kobold.demon.co.uk/kinetics/maxboltz.htm]		*http://dwb.unl.edu/teacher/nsf/c09/c09links/www.kobold.demon.co.uk/kinetics/maxboltz.htm<br>
			*[[Thermal Energy]]<br>
			*[[First Law of Thermodynamics]]<br>
			*[[Second Law of Thermodynamics and Entropy]]<br>
			*[[Multi-particle analysis of Momentum]]<br>
			*[[Potential Energy of a Multiparticle System]]<br>
			*[[Work and Energy for an Extended System]]<br>

	==References==		==References==