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phy141kk:lectures:31-18 [2018/11/25 10:46] (current)
kkumar created
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 +~~SLIDESHOW~~
 +
 +====== Fall 2018: Lecture 31 - Ideal Gases and Kinetic Theory ======
 +
 +/*
 +----
 +If you need a pdf version of these notes you can get it [[http://​www.ic.sunysb.edu/​class/​phy141md/​lecturepdfs/​141lecture32F11.pdf|here]]
 +
 +===== Video of lecture ====
 +
 +The video should play in any browser, but works best in anything that isn't Internet Explorer. If you are having trouble watching the video within the page you can [[http://​www.ic.sunysb.edu/​class/​phy141md/​lecturevids/​phy141lecture32f2011.mp4|download the video]] and play it in [[http://​www.apple.com/​quicktime/​download/​|Quicktime]].
 +
 +<​html>​
 +<video id="​video"​ width="​640"​ height="​360"​ controls="​true"/>​
 +<source src="​lecturevids/​phy141lecture32f2011.mp4"​ type="​video/​mp4"></​source>​
 +<source src="​lecturevids/​phy141lecture32f2011.ogv"​ type="​video/​ogg"></​source>​
 +
 +
 +<object width="​640"​ height="​384"​ type="​application/​x-shockwave-flash"​ data="​player.swf">​
 + <param name="​movie"​ value="​player.swf"​ />
 + <param name="​flashvars"​ value="​file=lecturevids/​phy141lecture32f2011.mp4"​ />
 +
 + </​object>​
 + 
 +
 +    </​video>​
 +</​html>​
 +
 +Note: In the messing around with the hotplate I forgot to start the screencapture. Thankfully I did remember to put the flip camera on..
 +
 +*/
 +===== Boyle'​s Law =====
 +
 +At constant temperature,​ it is found that the product of the pressure and volume of an ideal gas are constant
 +
 +$PV=\mathrm{constant}$
 +
 +This is named [[wp>​Boyle%27s_law|Boyle'​s Law]], after Robert Boyle who formulated it in 1662.
 +
 +{{Boyles_Law_animated.gif}}
 +
 +===== Charles'​ Laws =====
 +
 +Joesph Louis Gay-Lussac published [[wp>​Charles%27s_law|Charles'​ Law]] in 1802, attributing it to unpublished work of Jacques Charles in the 1780'​s ​ (Gay-Lussac has his own law..though it's not clear he should!).
 +
 +Charles'​ Law states that at constant pressure the volume of a gas is proportional to the temperature.
 +
 +$V\propto T$
 +
 +{{Charles_and_Gay-Lussac'​s_Law_animated.gif}}
 +
 +===== Gay-Lussac'​s law =====
 +
 +[[wp>​Gay-Lussac%27s_Law|Gay Lussac'​s Law]] states that for a fixed volume the pressure is proportional to the temperature
 +
 +$P\propto T$
 +
 +===== Ideal Gas Law =====
 +
 +The combination of the previous 3 laws implies that 
 +
 +$PV\propto T$
 +
 +Our previous laws were for systems of constant mass, but we can see that the amount of mass should effect the volume (at a given pressure) or the pressure (at a given volume).
 +
 +$PV\propto mT$
 +
 +Measuring the amount of mass in moles will allow us to write the ideal gas law in terms of a universal constant. A mole of gas is a given number of molecules, Avagadro'​s number, $N_{A}=6.02\times 10^{23}$. If we have a certain mass $m$ of a gas which has a certain [[wp>​Molecular_mass|molecular mass]] (measured in atomic mass units, $\mathrm{u}$,​ which are also the number of grams per mole.), the the number of moles $n$ is given by
 +
 +$n=\frac{m[\mathrm{g}]}{\textrm{molecular mass}[\mathrm{g/​mol}]}$
 +
 +and 
 +
 +$PV=nRT$ where $R=8.314\mathrm{J/​(mol.K)}$
 +
 +This equation is the [[wp>​Ideal_gas_law|ideal gas law]]
 +
 +===== Ideal Gas Law for a number of molecules =====
 +
 +
 +The ideal gas law can also be written in terms of the number of molecules $N$
 +
 +$PV=nRT=\frac{N}{N_{A}}RT=NkT$
 +
 +where $k=\frac{R}{N_{A}}=\frac{8.314\mathrm{J/​(mol.K)}}{6.02\times 10^{23}}=1.38\times 10^{-23}\mathrm{J/​K}$ is the [[wp>​Boltzmann_constant|Boltzmann Constant]].
 +
 +===== Using the Ideal Gas Law to determine Absolute Zero =====
 +
 +If $PV=nRT$ the absolute zero temperature occurs when $P=0$. In practice most gases will liquefy before this point, but we can measure the pressure of fixed volume of gas at a couple of reference points and extrapolate down to zero pressure to get an estimate for [[wp>​Absolute_zero|absolute zero]].
 +
 +/*
 +Through laser cooling and molecular trapping techniques it is now possible (but difficult!) for temperatures on the order of a $\mathrm{nK}$ to be achieved. Prof. [[http://​ultracold.physics.sunysb.edu/​index.html|Dominik Schneble]] produces ultra-cold ($\mu K$) Bose-Einstein condensates in the basement of this building! Prof. [[http://​www.stonybrook.edu/​metcalf/​hmetcalf.html|Hal Metcalf]] was one of the key players in the original development of laser cooling.
 +*/
 +
 +===== What makes a gas ideal?=====
 +
 +There are a number of conditions which must be satisfied for a gas to be considered ideal
 +
 +  - There must be a large number of molecules and they should move in random directions with a range of different speeds.
 +  - The spacing between molecules should be much greater than the size of the molecules.
 +  - Molecules are assumed to interact only through collisions.
 +  - The collisions are assumed to be elastic.
 +
 +===== Ideal gas law from a molecular perspective ​ =====
 +
 +We can work out the pressure exerted by an ideal gas on it's container by starting from the change in momentum of a molecule when it strikes the container wall
 +
 +{{kinetictheory.png}}
 +
 +The average force due to one molecule is then 
 +
 +$F=\frac{\Delta(mv)}{\Delta t}=\frac{2mv_{x}}{2l/​v_{x}}=\frac{mv_{x}^{2}}{l}$
 +
 +The net force on the wall will be the sum of the forces from all $N$ molecules
 +
 +$F=\frac{m}{l}\Sigma_{i=1..N} v_{xi}^{2}$
 +
 +$\frac{\Sigma_{i=1..N} v_{xi}^{2}}{N}=\bar{v_{x}^{2}}$ โ†’ $F=\frac{m}{l}N\bar{v_{x}^{2}}$
 +
 +$v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}$ โ†’ $\bar{v^{2}}=\bar{v_{x}^{2}}+\bar{v_{y}^{2}}+\bar{v_{z}^{2}}=\bar{3v_{x}^{2}}$
 +
 +$F=\frac{m}{l}N\frac{\bar{v^{2}}}{3}$
 +
 +$P=\frac{F}{A}=\frac{1}{3}\frac{Nm\bar{v^{2}}}{Al}=\frac{1}{3}\frac{Nm\bar{v^{2}}}{V}$
 +
 +$PV=\frac{2}{3}N(\frac{1}{2}m\bar{v^{2}})=NkT$
 +
 +$\bar{KE}=\frac{1}{2}m\bar{v^{2}}=\frac{3}{2}kT$
 +
 +
 +===== Maxwell Boltzmann Distribution =====
 +
 +If we look at a [[http://​www.chm.davidson.edu/​vce/​kineticmoleculartheory/​Maxwell.html|simulation]] of particles moving according to the kinetic theory we can get some idea of the distribution of the speeds of the particles.
 +
 +The [[wp>​Maxwell%E2%80%93Boltzmann_distribution|Maxwell-Boltzmann Distribution]] gives the probability $f(v)$ that a particular particle in an ideal gas has a given speed $v$. It was originally derived by [[wp>​James_Clerk_Maxwell|James Clerk Maxwell]] based on symmetry arguments, later [[wp>​Ludwig_Boltzmann|Ludwig Boltzmann]] derived it on a more general basis. We will follow Maxwell'​s reasoning.
 +
 +===== Velocity distribution =====
 +
 +
 +Suppose we have a gas of $N$ particles. The number of particles which have a velocity in the x direction between $v_{x}$ and $v_{x}+dv_{x}$ will be $Nf(v_{x})dv_{x}$
 +
 +$\int_{-\infty}^{\infty}f(v_{x})dv_{x}=1$
 +
 +The same function $f$ should describe the $y$ and $z$ direction as well, so
 +
 +$Nf(v_{x})f(v_{y})f(v_{z})dv_{x}dv_{y}dv_{z}$ ​
 +
 +gives the number of particles with velocity between $v_{x}$ and $v_{x}+dv_{x}$,​$v_{y}$ and $v_{y}+dv_{y}$,​ and, $v_{z}$ and $v_{z}+dv_{z}$
 +
 +Maxwell realized that all directions are equivalent the distribution function should depend only on the total speed of the particle
 +
 +$f(v_{x})f(v_{y})f(v_{z})=F(v_{x}^2+v_{y}^2+v_{z}^2)$ ​
 +
 +This implies that the form of $f$ must be
 +
 +$f(v_{x})=Ae^{-Bv_{x}^2}$
 +
 +Which is a [[wp>​Gaussian_function|Gaussian Function]].
 +
 +===== Mean velocity =====
 +
 +The fact that the velocity in a given direction is given by the function
 +
 +$f(v_{x})=Ae^{-Bv_{x}^2}$
 +
 +implies that the mean velocity in any given direction is zero. Does this also mean the mean particle speed is zero?
 +===== From velocity to speed =====
 +
 +We now know that
 +
 +$f(v_{x})=Ae^{-Bv_{x}^2}$
 +
 +But we need to find out what A and B are, and we also would like to express the function in terms of the speed of the molecules rather than the velocity in a given direction.
 +
 +Starting from the number of particles with velocity between $v_{x}$ and $v_{x}+dv_{x}$,​$v_{y}$ and $v_{y}+dv_{y}$,​ and, $v_{z}$ and $v_{z}+dv_{z}$
 +
 +$Nf(v_{x})f(v_{y})f(v_{z})dv_{x}dv_{y}dv_{z}=NA^{3}e^{-B(v_{x}^2+v_{y}^2+v_{z}^2)}dv_{x}dv_{y}dv_{z}$ ​
 +$=NA^{3}e^{-Bv^2}dv_{x}dv_{y}dv_{z}$
 +
 +To transform $dv_{x}dv_{y}dv_{z}$ to an integral in the speed we consider a sphere in velocity space and can reason that the equivalent volume to $dv_{x}dv_{y}dv_{z}$ is $4\pi v^{2}dv$
 +
 +So the probability distribution as a function of speed is 
 +
 +$f(v)dv=4\pi v^2A^{3}e^{-Bv^{2}}dv$
 +
 +===== Finding A and B =====
 +
 +We now have
 +
 +$f(v)dv=4\pi v^2A^{3}e^{-Bv^{2}}dv$
 +
 +We can use some of the information about the system to find out what $A$ and $B$ have to be. All particles have to have a velocity between 0 and $\infty$, so
 +
 +$\int_{0}^{\infty}f(v)dv=1$
 +
 +We saw before that 
 +
 +$\bar{K}=\frac{1}{2}m\bar{v^{2}}=\frac{3}{2}kT$
 +
 +so 
 +
 +$\int_{0}^{\infty}\frac{1}{2}mv^{2}f(v)dv=\frac{3}{2}kT$
 +
 +===== Gaussian Intergral =====
 +
 +We will need to use the [[wp>​Gaussian_integral|standard integral of a gaussian function]].
 +
 +$\int_{-\infty}^{\infty}e^{-ax^{2}}=\sqrt{\frac{\pi}{a}}$
 +
 +This is explained quite nicely on [[http://​galileo.phys.virginia.edu/​classes/​152.mf1i.spring02/​ExpIntegrals.htm|this page]]
 +
 +The integrals we need to do
 +
 +$\int_{0}^{\infty}f(v)dv$
 +
 +and
 +
 +$\int_{0}^{\infty}\frac{1}{2}mv^{2}f(v)dv$
 +
 +can be found using [[wp>​Integration_by_parts|integration by parts]].
 +
 +$\int u\,dv = uv - \int v\,du$
 +
 +===== Finding A =====
 +
 +
 +$\int_{0}^{\infty}f(v)dv=1$
 +
 +$4\pi A^{3}\int_{0}^{\infty}v^{2}e^{-Bv^{2}}dv=1$
 +
 +
 +$\int u\,dv = uv - \int v\,du$
 +
 +$u=v$<​html>&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;</​html>​ $dv=ve^{-Bv^{2}}dv$
 +
 +$du=dv$<​html>&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;</​html>​$v=-\frac{1}{2B}e^{-Bv^{2}}$
 +
 +$4\pi A^{3}\int_{0}^{\infty}v^{2}e^{-Bv^{2}}dv=4\pi A^{3}([(-\frac{v}{2B}e^{-Bv^{2}})]_{0}^{\infty}+\int_{0}^{\infty}\frac{1}{2B}e^{-Bv^{2}}dv)$
 +
 +$=\frac{4\pi A^{3}}{2B}\int_{0}^{\infty}e^{-Bv^{2}}dv=\frac{4\pi A^{3}}{2B}\frac{1}{2}\sqrt{\frac{\pi}{B}}=1$
 +
 +$4\pi A^{3}=\frac{4B^{3/​2}}{\sqrt{\pi}}$ ​
 +
 +===== Finding B =====
 +
 +$\bar{K}=\frac{1}{2}m\bar{v^{2}}=\frac{3}{2}kT$
 +
 +$\int_{0}^{\infty}\frac{1}{2}mv^{2}f(v)dv=\frac{3}{2}kT$
 +
 +$2m\pi A^{3}\int_{0}^{\infty}v^{4}e^{-Bv^{2}}dv=\frac{3}{2}kT$
 +
 +$u=v^{3}$<​html>&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;</​html>​ $dv=ve^{-Bv^{2}}dv$
 +
 +$du=3v^{2}dv$<​html>&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;&​nbsp;</​html>​ $v=-\frac{1}{2B}e^{-Bv^{2}}$
 +
 +
 +$2m\pi A^{3}\int_{0}^{\infty}v^{4}e^{-Bv^{2}}dv=2m\pi A^{3}([(-\frac{v^3}{2B}e^{-Bv^{2}})])_{0}^{\infty}+\int_{0}^{\infty}\frac{3v^2}{2B}e^{-Bv^{2}}dv)$
 +
 +$=\frac{3m}{4B}(4\pi A^3)\int_{0}^{\infty}v^{2} e^{-Bv^{2}}dv=\frac{3m}{4B}$
 +
 +$\frac{3m}{4B}=\frac{3}{2}kT$
 +
 +$B=\frac{m}{2kT}$
 +
 +===== And finally .. the Maxwell-Boltzmann distribution =====
 +
 +$4\pi A^{3}=\frac{4B^{3/​2}}{\sqrt{\pi}}$ ​
 +
 +$B=\frac{m}{2kT}$
 +
 +
 +$f(v)dv=4\pi v^2A^{3}e^{-Bv^{2}}dv$
 +
 +$f(v)=4\pi (\frac{m}{2\pi k T})^{\frac{3}{2}}v^{2}e^{-\frac{1}{2}\frac{mv^{2}}{kT}}$
 +
 +Now that we have this we can find various useful quantities, such as the average velocity
 +
 +$\bar{v}=\int_{0}^{\infty}vf(v)dv=2\sqrt{\frac{2}{\pi}}\sqrt{\frac{kT}{m}}\approx 1.6 \sqrt{\frac{kT}{m}}$
 +
 +or the most probable speed of a particle, from the condition $\frac{df}{dv}=0$ which gives
 +
 +$v=\sqrt{\frac{2kT}{m}}\approx 1.41 \sqrt{\frac{kT}{m}}$
 +
 +A third measure of velocity that we could already obtain before deriving the Maxwell Boltzmann distribution,​ is the root-mean-square,​ or rms velocity. Here we take the square root of the average of the squared velocity
 +
 +$\frac{1}{2}m\bar{v^{2}}=\frac{3}{2}kT$ โ†’ $\sqrt{\bar{v^{2}}}=\sqrt{\frac{3kT}{m}}\approx 1.73 \sqrt{\frac{kT}{m}}$
  
phy141kk/lectures/31-18.txt ยท Last modified: 2018/11/25 10:46 by kkumar
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