{ "cells": [ { "cell_type": "markdown", "id": "4aac6b7a-c642-42ac-9692-edcaf8b9cea6", "metadata": { "tags": [] }, "source": [ "# Introduction to Thermodynamics and Statistical Physics" ] }, { "cell_type": "markdown", "id": "b434c8a5-e4ac-433f-8957-50e1b8802a74", "metadata": {}, "source": [ "In this lecture, we are going to discuss:\n", "\n", "* Functions of state of an ideal gas.\n", "* The Maxwell-Boltzmann distribution." ] }, { "cell_type": "markdown", "id": "2ef4d89a-53ee-463d-9f94-e9afcab884b0", "metadata": {}, "source": [ "## Functions of state of an ideal gas" ] }, { "cell_type": "markdown", "id": "6b5acae1-86ca-4364-9f28-212dd2d6fc38", "metadata": {}, "source": [ "So now that we know how to calculate the partition function for N indistinguishable particles, let's apply the techniques we have developed to understand an ideal gas. The partition function for N particles in a gas is\n", "$$\n", " Z_N = \\frac{1}{N!} \\left( \\frac{V}{\\lambda_{\\rm th}^3}\\right)^N\n", "$$\n", "Using the expression for $\\lambda_{\\rm th}$, we then get that\n", "$$\n", " Z_N = \\frac{1}{N!} V^N \\left( \\frac{ 2\\pi m k_{\\rm B} T}{h^2}\\right)^{3N/2}\n", "$$\n", "This means that\n", "$$\n", " \\ln{Z_N} = N \\ln{V} + \\frac{3}{2} N \\ln{T} + C\n", "$$\n", "The internal energy is then given by\n", "$$\n", " U = -\\frac{\\partial \\ln Z_N}{\\partial \\beta} = k_{\\rm B} T^2 \\frac{\\partial \\ln Z_N}{\\partial T} = \\frac{3}{2}N k_{\\rm B}T\n", "$$\n", "The Helmholtz free energy is\n", "$$\n", " F = -k_{\\rm B} T \\ln Z_N = -k_{\\rm B} T N \\ln{V} -k_{\\rm B} T \\frac{3}{2} N \\ln{T} -k_{\\rm B} T C\n", "$$\n", "such that if we use (from Lecture 7)\n", "$$\n", " P = -\\left( \\frac{\\partial F}{\\partial V} \\right)_T = \\frac{N k_{\\rm B} T}{V}\n", "$$\n", "which is the equation of state for an ideal gas!\n", "In order to work out the other thermodynamic quantities, we need to be a bit more careful with those constants in earlier equations. Returning to the partition function, we can write it instead as.\n", "$$\n", " \\ln{Z_N} = N \\ln{V} - 3 N \\ln {\\lambda_{\\rm th}} - N \\ln N + N\n", "$$\n", "after using Stirlings approximation of $\\ln{N!} \\sim N\\ln {N} - N$. This further simplifies to\n", "$$\n", " \\ln{Z_N} = N \\ln \\left(\\frac{V {\\rm e} }{N \\lambda_{\\rm th}^3}\\right)\n", "$$\n", "and so the Helmholtz free energy is given by\n", "\\begin{align}\n", " F &= -N k_{\\rm B} T \\ln \\left(\\frac{V {\\rm e} }{N \\lambda_{\\rm th}^3}\\right)\\\\\n", " &= N k_{\\rm B} T[ \\ln (n \\lambda_{\\rm th}^3)-1]\n", "\\end{align}\n", "from which all other thermodynamic quantities of an ideal gas can be derived." ] }, { "cell_type": "markdown", "id": "cfbd29b7-e31c-4662-9c98-0570dbe9fa73", "metadata": {}, "source": [ "## The Maxwell Boltzmann Distribution" ] }, { "cell_type": "markdown", "id": "6299545e-eb0a-4bb4-84c8-b88b5a8af5da", "metadata": {}, "source": [ "The next question we want to ask is, given a gas at a certain temperature, what is the distribution of particle velocities within that gas?\n", "\n", "First, we will ignore all rotational or vibrational energies associated with any particles (that is, we will assume a monatomic gas where the energy only arises due to translational motion of the particles, see Lecture 9). As such, the energy of a single particle can be written as\n", "$$\n", " \\epsilon = \\frac{p^2}{2m}\n", "$$\n", "The probability that a particle is in a particular state $s$ with translational energy $\\epsilon$ is then given by\n", "$$\n", " P_s = \\frac{1}{Z_1}e^{-\\beta \\epsilon}\n", "$$\n", "If the gas consisits of $N$ particles, the mean number $\\bar{n}_s$ of particles which are in the translational state $s$ is given by\n", "$$\n", " \\bar{n}_s = NP_s\n", "$$\n", "Now, substituting for $Z_1$ and $\\epsilon$ then gives\n", "$$\n", " \\bar{n}_s(p) = \\frac{\\hbar^3 N}{V} \\left(\\frac{2\\pi}{mk_{\\rm B}T}\\right)^{3/2} \\exp \\left( \\frac{- p^2}{2mk_{\\rm B}T} \\right)\n", "$$\n", "We now need to figure out what the appropriate expression is for particles which have momenta between ${\\bf p}$ and ${\\bf p}+{\\rm d}{\\bf p}$. Fortunately, we can get this from the density of states equation\n", "$$\n", " g(k){\\rm d}k = \\frac{V k^2 {\\rm d} k}{2\\pi^2}\n", "$$\n", "and using the relation that $p=\\hbar k$. This gives \n", "$$\n", " f(p){\\rm d}p = \\frac{V p^2 }{2\\pi^2 \\hbar^3} {\\rm d} p\n", "$$\n", "This means that the probability that a particle will possess a momentum with magnitude between ${\\bf p}$ and ${\\bf p}+{\\rm d}{\\bf p}$ is given by\n", "$$\n", " P(p)dp = \\frac{1}{N} f(p) \\; dp \\;\\bar{n}(p)\n", "$$\n", "Plugging in our above expressions thus gives\n", "$$\n", " P(p)dp = \\frac{4\\pi p^2}{(2 \\pi m k_{\\rm B} T)^{3/2}} \\exp \\left(\\frac{- p^2}{2mk_{\\rm B}T}\\right) dp\n", "$$\n", "The normalisation for this equation is the condition that\n", "$$\n", " \\int_{0}^{\\infty} P(p)dp = 1\n", "$$\n", "So, what does this equation tell us? If we let $p=mv$, then the expression above becomes\n", "$$\n", " P(v)dv = 4\\pi v^2\\left( \\frac{m}{2 \\pi k_{\\rm B} T}\\right)^{3/2} \\exp \\left(\\frac{- mv ^2}{2k_{\\rm B}T}\\right) dv\n", "$$\n", "which, if we then let $u=v / \\left(\\frac{2 k_{\\rm B} T}{m}\\right)^{1/2}$, becomes \n", "$$\n", " P(u)du = \\frac{4}{\\sqrt{\\pi}} u^2 \\exp (-u^2) du = F_1(u) du\n", "$$\n", "This equation describes a Maxwell velocity distribution, and is plotted below." ] }, { "cell_type": "code", "execution_count": 1, "id": "1fb5a646-9547-48b9-9f15-6b5cf6c9578d", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "u = np.arange(0,3,0.01) #in units of eps\n", "F = 4/ np.pi**(1/2) * u**2 * np.exp(-u**2)\n", "\n", "fig, ax = plt.subplots(ncols=1,figsize=[3,2],dpi=150)\n", "ax.plot(u,F)\n", "ax.set_xlabel(r\"$u = v / \\left(\\frac{2 k_{\\rm B} T}{m}\\right)^{1/2}$\")\n", "ax.set_ylabel(r\"$F_1(u)$\")\n", "ax.set_xlim(0,3)\n", "ax.set_ylim(0,0.85)\n", "ax.axvline(1,linestyle='-',color='grey',label=r'$v_{\\rm max}$')\n", "ax.axvline(2/np.sqrt(np.pi),linestyle='--',color='grey',label=r'$\\langle v \\rangle$')\n", "ax.axvline(np.sqrt(3/2),linestyle='-.',color='grey',label=r'$v_{\\rm rms}$')\n", "plt.legend()\n", "plt.tight_layout()\n", "plt.savefig(\"Figures/Maxwell_Distribution.png\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "06df9641-8795-4dd5-a3f0-964586410013", "metadata": {}, "source": [ "First, the maximum of the distribution occurs at\n", "\n", "$$\n", " v_{\\rm max}=\\left(\\frac{2 k_{\\rm B} T}{m}\\right)^{1/2}\n", "$$\n", "\n", "Now, the mean value for the velocity, $\\langle v \\rangle$, can be found using\n", "$$\n", " \\langle u \\rangle = \\int_0^\\infty u P(u) du = \\frac{4}{\\sqrt{\\pi}} \\int_0^\\infty u^3 e^{-u^2} du\n", "$$\n", "That last term is a known intergral, and is $\\int_0^\\infty u^3 e^{-u^2} du = \\frac{1}{2}$. This gives\n", "$$\n", " \\langle u \\rangle = \\frac{2}{\\sqrt{\\pi}}\n", "$$\n", "which, substituting in for $\\langle v \\rangle$, gives\n", "$$\n", " \\langle v \\rangle = \\left(\\frac{8 k_{\\rm B} T}{\\pi m}\\right)^{1/2} = \\frac{2}{\\sqrt{\\pi}} v_{\\rm max}\n", "$$\n", "The mean value for the squared-velocity can be found using a similar procedure, and gives\n", "$$\n", " \\langle v^2 \\rangle = \\frac{3 k_{\\rm B} T}{m}\n", "$$\n", "which means the root-mean-square (RMS) of the velocity is\n", "$$\n", " v_{\\rm RMS} = \\sqrt{\\langle v^2 \\rangle} = \\sqrt{\\frac{3 k_{\\rm B} T}{ m}} = \\sqrt{\\frac{3}{2}} v_{\\rm max}\n", "$$\n", "This means that $v_{\\rm max} \\langle \\langle v \\rangle\\langle v_{\\rm RMS}$ - so the velocity which has the highest probability of occuring is lower than the mean velocity, which is lower than the RMS velocity. This is because the long tail to our distribution. Remembering this is important when discussing the kinematics of gases (to come next year)\n", "This means that the average kinetic energy of a particle in this gas is\n", "$$\n", " \\langle E \\rangle= \\frac{1}{2} m \\langle v^2 \\rangle= \\frac{3}{2} k_{\\rm B} T\n", "$$\n", "in agreement with our previous result for the an ideal gas (if we multiple this by N particles)." ] }, { "cell_type": "code", "execution_count": null, "id": "39b42001-e5e4-4688-870c-81de429f82a8", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 5 }