Maxwell_Boltzmann.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/davis689/binder/blob/master/Maxwell_Boltzmann.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "C7IcEU6Wwecu"
      },
      "outputs": [],
      "source": [
        "import sympy as sp\n",
        "from sympy import oo, exp, sin, pi\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "sp.init_printing()\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WN4JK4GXwWeg"
      },
      "source": [
        "# Maxwell-Boltzmann distribution"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "3H_3NQ6awUXN"
      },
      "outputs": [],
      "source": [
        "eps,m,T,K=sp.symbols('varepsilon,m,T,K',positive=True,real=True)\n",
        "vx,vy,vz,fvx,fex,fv=sp.symbols('v_x,v_y,v_z,f_{v_x},f_x,f_v',real=True)\n",
        "kb=sp.symbols('k_B',constant=True,positive=True,real=True)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wpUPxp-kgVhZ"
      },
      "source": [
        "#One dimension# \n",
        "We begin with the Boltzmann distribution in one dimension. $$f(\\varepsilon_i)=\\left(\\frac{1}{\\sum_{j=0}^{\\infty} e^{-ɛ_j/kT} }\\right)e^{-\\varepsilon_i/kT}$$ The term in parentheses is just a number and since we'll be normalizing our expression later, let's simplify this as just $K$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "BdJDfQymHlBW",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 40
        },
        "outputId": "1011213a-0ab6-4f7e-d0ff-7fadb7d18f7e"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "             -varepsilon \n",
              "             ────────────\n",
              "                T⋅k_B    \n",
              "f_{v_x} = K⋅ℯ            "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle f_{v_x} = K e^{- \\frac{\\varepsilon}{T k_{B}}}$"
          },
          "metadata": {},
          "execution_count": 57
        }
      ],
      "source": [
        "mbx=sp.Eq(fvx,K*sp.exp(-eps/kb/T))\n",
        "mbx"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NLODOvH7h_nf"
      },
      "source": [
        "The energy levels we're going to consider here are the energies of motion, $\\frac{1}{2} m v_x^2$. Let's substitute this into our expression."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "HvNX4k-3IYU1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 48
        },
        "outputId": "0cbfe440-00c6-414c-e060-05162abe8d93"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                  2 \n",
              "             -m⋅vₓ  \n",
              "             ───────\n",
              "             2⋅T⋅k_B\n",
              "f_{v_x} = K⋅ℯ       "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle f_{v_x} = K e^{- \\frac{m v_{x}^{2}}{2 T k_{B}}}$"
          },
          "metadata": {},
          "execution_count": 58
        }
      ],
      "source": [
        "mbx=mbx.subs(eps,m*vx**2/2)\n",
        "mbx"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "b4HpfdM-iVKm"
      },
      "source": [
        "Since we're deriving an expression for the *fraction* of molecules, we can normalize this expression to determine an expression for $K$. Since the sum of fractions over the whole range of velocities will be unity, our expression will be $$\\int^\\infty_{-\\infty}K e^{-\\frac{m v_x^2}{2k_B T}}=1$$ For the purposes *sympy*, to solve for K we need the expression to be solved to be equal to zero so we'll move the 1 over to the left hand side of the equation.\n",
        "\n",
        "Here we'll solve for the normalization coefficient and call it *norm*. We'll then substitute it for K in the Maxwell-Boltzmann expression."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "9R3DShuwIVaI",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 62
        },
        "outputId": "0414a145-0fde-4bd5-ef71-730d05e2017f"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "     √2⋅√m     \n",
              "───────────────\n",
              "          _____\n",
              "2⋅√π⋅√T⋅╲╱ k_B "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\frac{\\sqrt{2} \\sqrt{m}}{2 \\sqrt{\\pi} \\sqrt{T} \\sqrt{k_{B}}}$"
          },
          "metadata": {},
          "execution_count": 59
        }
      ],
      "source": [
        "norm=sp.solve(sp.integrate(mbx.rhs,(vx,-oo,oo))-1,K)[0] # solve returns a list. There is only one solution so [0] will get it.\n",
        "norm"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "DgJ2m-2FLXgp",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 71
        },
        "outputId": "8b08158f-3378-46c8-df74-0ed7dd2dcbd8"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                       2 \n",
              "                  -m⋅vₓ  \n",
              "                  ───────\n",
              "                  2⋅T⋅k_B\n",
              "           √2⋅√m⋅ℯ       \n",
              "f_{v_x} = ───────────────\n",
              "                    _____\n",
              "          2⋅√π⋅√T⋅╲╱ k_B "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle f_{v_x} = \\frac{\\sqrt{2} \\sqrt{m} e^{- \\frac{m v_{x}^{2}}{2 T k_{B}}}}{2 \\sqrt{\\pi} \\sqrt{T} \\sqrt{k_{B}}}$"
          },
          "metadata": {},
          "execution_count": 60
        }
      ],
      "source": [
        "mbx=mbx.subs(K,norm) # substitute the normalization coefficient into the distribution\n",
        "mbx"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3GM6O0uQ5bgI"
      },
      "source": [
        "To make sure this expression is indeed normalized, we can integrate it over the whole range and we should get 1. "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "2DG_oJgczhb_",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        },
        "outputId": "5ead49ba-e8d0-46aa-b1b8-59ddc80e0422"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1"
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFUKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSxoPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA6XoX52TB9a4AAAAASUVORK5CYII=\n",
            "text/latex": "$\\displaystyle 1$"
          },
          "metadata": {},
          "execution_count": 61
        }
      ],
      "source": [
        "sp.integrate(mbx.rhs,(vx,-oo,oo))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eB6uNZNirKdJ"
      },
      "source": [
        "One disadvange of using the computer to do our math is that the final form might not match what we would do if we did the work on paper. In the case of the Maxwell-Boltzmann distribution, we can combine terms under one square root and obtain $$f_{v_x} = \\sqrt{\\dfrac{m}{2 \\pi k_B T}} e^ {-\\frac{m v_x^2}{2 k_B T}}$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eLh-MVdM6QID"
      },
      "source": [
        "#Three dimensions#\n",
        "To change to three dimensions, we realize that there is no real difference between molecules moving in any of the three dimensions so the $v_y$ function should look the same except with $v_y$ substituted for $v_x$. The overall distribution function is just the product of the three one-dimenional functions."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "FhfonGnb5eRB",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        },
        "outputId": "dbc242d1-c99d-4abb-fc24-67319b0dc887"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                   2         2         2 \n",
              "              -m⋅vₓ    -m⋅v_y    -m⋅v_z  \n",
              "              ───────  ────────  ────────\n",
              "         3/2  2⋅T⋅k_B  2⋅T⋅k_B   2⋅T⋅k_B \n",
              "     √2⋅m   ⋅ℯ       ⋅ℯ        ⋅ℯ        \n",
              "fᵥ = ────────────────────────────────────\n",
              "                 3/2  3/2    3/2         \n",
              "              4⋅π   ⋅T   ⋅k_B            "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle f_{v} = \\frac{\\sqrt{2} m^{\\frac{3}{2}} e^{- \\frac{m v_{x}^{2}}{2 T k_{B}}} e^{- \\frac{m v_{y}^{2}}{2 T k_{B}}} e^{- \\frac{m v_{z}^{2}}{2 T k_{B}}}}{4 \\pi^{\\frac{3}{2}} T^{\\frac{3}{2}} k_{B}^{\\frac{3}{2}}}$"
          },
          "metadata": {},
          "execution_count": 62
        }
      ],
      "source": [
        "mb3d=sp.Eq(fv,mbx.rhs*mbx.subs(vx,vy).rhs*mbx.subs(vx,vz).rhs)\n",
        "mb3d"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iNdqBouZtH6j"
      },
      "source": [
        "Of course, if we're looking at motion in three dimensions, we probably are much more interested in velocity in any direction rather than the three components of velocity along the axes. We can substitute $v^2=v_x^2+v_y^2+v_z^2$. "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "AeugUDxCM9s9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        },
        "outputId": "2d462584-f742-4214-9a87-43166d8e26fb"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                   2         2         2 \n",
              "              -m⋅vₓ    -m⋅v_y    -m⋅v_z  \n",
              "              ───────  ────────  ────────\n",
              "         3/2  2⋅T⋅k_B  2⋅T⋅k_B   2⋅T⋅k_B \n",
              "     √2⋅m   ⋅ℯ       ⋅ℯ        ⋅ℯ        \n",
              "fᵥ = ────────────────────────────────────\n",
              "                 3/2  3/2    3/2         \n",
              "              4⋅π   ⋅T   ⋅k_B            "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle f_{v} = \\frac{\\sqrt{2} m^{\\frac{3}{2}} e^{- \\frac{m v_{x}^{2}}{2 T k_{B}}} e^{- \\frac{m v_{y}^{2}}{2 T k_{B}}} e^{- \\frac{m v_{z}^{2}}{2 T k_{B}}}}{4 \\pi^{\\frac{3}{2}} T^{\\frac{3}{2}} k_{B}^{\\frac{3}{2}}}$"
          },
          "metadata": {},
          "execution_count": 63
        }
      ],
      "source": [
        "v,dv=sp.symbols('v,dv',real=True,positive=True)\n",
        "mb3d.subs(vx**2+vy**2+vz**2,v**2)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "CUSPnt2XVjmf",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        },
        "outputId": "e342d681-9a67-4446-f85f-42f4d9d37fbe"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                   2         2         2 \n",
              "              -m⋅vₓ    -m⋅v_y    -m⋅v_z  \n",
              "              ───────  ────────  ────────\n",
              "         3/2  2⋅T⋅k_B  2⋅T⋅k_B   2⋅T⋅k_B \n",
              "     √2⋅m   ⋅ℯ       ⋅ℯ        ⋅ℯ        \n",
              "fᵥ = ────────────────────────────────────\n",
              "                 3/2  3/2    3/2         \n",
              "              4⋅π   ⋅T   ⋅k_B            "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle f_{v} = \\frac{\\sqrt{2} m^{\\frac{3}{2}} e^{- \\frac{m v_{x}^{2}}{2 T k_{B}}} e^{- \\frac{m v_{y}^{2}}{2 T k_{B}}} e^{- \\frac{m v_{z}^{2}}{2 T k_{B}}}}{4 \\pi^{\\frac{3}{2}} T^{\\frac{3}{2}} k_{B}^{\\frac{3}{2}}}$"
          },
          "metadata": {},
          "execution_count": 64
        }
      ],
      "source": [
        "sp.powsimp(mb3d.subs(vx**2+vy**2+vz**2,v**2),force=True)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KZmJNTC0whjQ"
      },
      "source": [
        "It's not clear that we can make the substitution that we want for $v$ in sympy. We'll have to do it another way. \n",
        "\n",
        "Since $v^2=v_x^2+v_y^2+v_z^2$ and since there's no difference between the values of $v_x$, $v_y$, and $v_z$, we can just write $v^2=3 v_x^2$ or $v_x=v/\\sqrt{3}$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "o92UyePlYAt3",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 74
        },
        "outputId": "d9834538-5b17-44b2-8aa2-3cbf0396a07a"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                    2  \n",
              "                -m⋅v   \n",
              "               ─────── \n",
              "          3/2  2⋅T⋅k_B \n",
              "      √2⋅m   ⋅ℯ        \n",
              "fᵥ = ──────────────────\n",
              "        3/2  3/2    3/2\n",
              "     4⋅π   ⋅T   ⋅k_B   "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle f_{v} = \\frac{\\sqrt{2} m^{\\frac{3}{2}} e^{- \\frac{m v^{2}}{2 T k_{B}}}}{4 \\pi^{\\frac{3}{2}} T^{\\frac{3}{2}} k_{B}^{\\frac{3}{2}}}$"
          },
          "metadata": {},
          "execution_count": 65
        }
      ],
      "source": [
        "mb3d=sp.Eq(fv,mb3d.rhs.subs(vx,v/sp.sqrt(3)).subs(vy,v/sp.sqrt(3)).subs(vz,v/sp.sqrt(3)))\n",
        "mb3d"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "In actuality our function equation is multiplied on both sides by infinitessimals. Our fraction in terms of velocity components should actually be written as $$f_vdv_xdv_ydv_z=\\dfrac{\\sqrt{2}m^{3/2}e^{mv_x^2/2k_BT}e^{mv_y^2/2k_BT}e^{mv_z^2/2k_BT}}{(4\\pi k_BT)^{3/2}}dv_xdv_ydv_z$$\n",
        "When we switch over to $v$, we're effectively switching to spherical polar coordinates. Two of our three variables become the angles $\\theta$ and $\\phi$.\n",
        "To allow us to focus on only the magnitude of $v$, we integrate over all possible angles and fold this into $f_v$."
      ],
      "metadata": {
        "id": "-L10OoqeoYTi"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "JzEjRvGWWJcW",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        },
        "outputId": "a8c11aa9-ba28-4706-f9d5-9586d438b020"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "     2\n",
              "4⋅π⋅v "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle 4 \\pi v^{2}$"
          },
          "metadata": {},
          "execution_count": 66
        }
      ],
      "source": [
        "theta,phi=sp.symbols('theta, phi')\n",
        "anglefactor=sp.integrate(sp.integrate(v**2*sp.sin(theta),(theta,0,sp.pi)),(phi,0,2*sp.pi))\n",
        "anglefactor"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "mb3d=sp.Eq(mb3d.lhs,mb3d.rhs*anglefactor)\n",
        "mb3d"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 74
        },
        "id": "8GlU5op-qpua",
        "outputId": "6a751fd1-d161-43d4-f69a-8ad32ae97981"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                      2 \n",
              "                  -m⋅v  \n",
              "                 ───────\n",
              "         3/2  2  2⋅T⋅k_B\n",
              "     √2⋅m   ⋅v ⋅ℯ       \n",
              "fᵥ = ───────────────────\n",
              "            3/2    3/2  \n",
              "        √π⋅T   ⋅k_B     "
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAAArCAYAAACXSwEOAAAABHNCSVQICAgIfAhkiAAAB4tJREFUeJzt3HuwVVUdwPEPIDEIpUKNIoklplAoBGKJoJlFgSOm0VT2kjtTOmZpjFOWTKFYoljaA6LHJJUR4aSVU2lpY28zBXxVVqNkLyGtrLyVPeiP397dfTf7nLvP815gf2f2nLPXWY/f3vu31/r91vqtQ0VFCwwbbAHaxI4my+0u199JTsNs3IxbBlmWil2UA/HGwRaiUyzGmC61tS/uwBbcq/imfh7zcCt+grvxigbaOKiFsmWZKHqTC/ApnIkbcADOxtwk36cxEivwpA7IMSQ4r4ttjcDeyfcxeBDjM78Px+2YgBlJ2gH4rfJK3krZsizEm5LvXxdD+dswHceJF+MknI5LcQmOKaporzYL1m32wZ+72N5/0Jt8HyVufNaOmoVN+H1ywMN4BOPweIk2WilblunYIO7f/cKGnII1QllPw7FYgvX1KhreRqEGg4X4Wpfb3Bd34TdYJR5wynx8I5d/lui5ft1EW62UrcczsRVH4p4kbRT+iT/iJULBmnVOdhnOLUh7J36Mv+APYmyf1oG298f3k8+Ub4m3OmUc7sOcJupvpWxqo+WPA5uoa7dlLHoK0m8SXe80HIHrxVAwrgMyrBFGPDxZGL8po/AdvK6JelspW1GDyXhO5nyxcm/UWGG7nJxLf7rool8peo5eMTRNwVHiAfYKo3hSUmZ/oShET3OvUFI4Be9Jvg8T3tjyAnm24Zxc2lQ8IYznemVTJuIzeFTYgF/Uvyesx442HXV5Cj4kxsknkgIXlBSwE4zEUqzNpL21ZNkJQv65ufSTkvRbcbxQhJ/hNqFQc4Wx+QCuSsocLYaHu4SLfWamvjX6PJW5+G+SNz1SRbs5yZvlS/hsibKE/bJNeEdThdJ9W/S0Q4YbxM39qnDhluPZgylQwg+FUTla+YmtjdiclMtyoXh7s2/uh4XdlHXNr8YXSrRzR0EbRVyp/1A3B//AwSXKEkP0+3JpLxI2XxkW4BPimk5M0nrEy/lg8nk6Playvp2YIpTnxmYr6CCX4QQxXEwaIC98AL/DIQW/bcQ1ubQv2/nG3YKVjYlZlx5sz5x/D+8vWfZg8Wx68bfM8XcxnDXCfvpf6yQx6hDD/kqhbBdrcLrn7ETIoTh1PRur8ZYSea8U8ylTavz+czvbIg/hjFzaI8JOahezxf0dL16EPylv4C/CYzi04CjzQmW5HM/NnL9M37UfJ4bxJcn5AlyET9ar8OVqG0xTGxSuk9ymbxa1Fh8UnlctuccIw/rYTNp4ca3TM2kHJWmHNSVpMXsnbb9AuOnvaKDsAvxb9BCtsAIvzqVdLOaFCHvzMqETxIu2FO+qV+kcYetsx7+S78uFZ1FmbO8WqxQPSSmrhT3wQrEUkB7Zm36MeIjZJYIThS2S7a5Pxl+1f8L1F/iBmCAc3UC5/YSNdr3oPSYLRVjdgIw9wjBfq/+LeK2+a18rvMHLhZGezrcty1eWD2cYIW7YL/VpY7OcJ2Zty7JFeCMDMVIoeC1quZgX6XONzxLyZYe38/Eq4cKnLMNL7ezBtcp1OFU8zKsbLHuUeLAzxfN6QMwaX9pOAXOsE57nNgMsbRwhHsC6NjS6VWPzCu1os6LL5K3rmcnnpjbU/Yw21FExxMmPm6kCbe62IBW7JkU90A5hj2TZJiz3j2TSpopx8eiC/HTOBqoYogwT3sv9Bb8NNP1exFbts4HatX5THe0//s/hScKG/NPT+vT7nshA4a+thr52q43SvFoo0NsLfmtl+n1PpV74aztCX7vVRl2yNlA9D+wePC0Rbq4Iq1jULiEGkR6xnNDstdTbFlQv/LUdoa/daqMttDL9PpT5rs7uNkjDX3vx5kz6hSLuOMssMQzlWYlvdriNrtDs9PtQZaZYAmiG+QY2Ll+fyZ8Pf20k9PVGsXwzEK200RWuEzdmyUAZdxGuElMQee5TWymWJ3lG67/O9hCuyKXle7Y0/LXR0NeHRXxOGZpto6JB9hJRfEUcJpRlgVCECcJe6FHc8+4jIghPzaXXCn8tG/oqaX+HvmC+McJL3iRm+tvRRsUAPE9E3707k7ZQ7N4oYp5QiHT1/lDxEGuFdJyQ/J6Px6kV/lo29JVYyO0V3tbhone8Rp8it6ONigF4Kt6AX+nzUD4nguqLOEdEI6QsFj1QrXCJ80WIRVnKhr4Sceg/EnE5jyq/C7eRNipKMEaEfh4vNgx8pU7ejwt7L+US8RBrsd7OmwnbxQax0e8xIXuepWKD4xYxB9SsU9A0u/rO1LI8LoKwXit6lGvr5D1SDAkpM3LneWbizlYFrMEMocwjFYe9ThPBXjOEnXSG1qMVK2owX8Qf36T2TOxw0VNlDeKfijmVIsYK+6ITywPp3NtsvCaRa2Yuz+0iKpGwee7ugBwVCSPEDo11dfI8SxjEkzNpG4X9saAg/7yC/O3i+SL+OTWYV4hliInJ+TAxtG0RGwS2a2/sdkUBq4TX1C7OFXvLOvFPZ2eJ3i9lmFDmO0XvdIjYYJCyDB/tgBwVuymLxB9ApZyiv/HfFfYUI3p3ZJq+2K0RIpqi+g/DitKsF0som8XM9HtVHUJFRUVFRUVFRVf4HwGGap9aJjI4AAAAAElFTkSuQmCC\n",
            "text/latex": "$\\displaystyle f_{v} = \\frac{\\sqrt{2} m^{\\frac{3}{2}} v^{2} e^{- \\frac{m v^{2}}{2 T k_{B}}}}{\\sqrt{\\pi} T^{\\frac{3}{2}} k_{B}^{\\frac{3}{2}}}$"
          },
          "metadata": {},
          "execution_count": 67
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "##Graphing Interlude\n",
        "\n",
        "You may already know how to plot in sympy. You may skip to the next section. If you know how to plot in matplotlib, that is different and you may find the following useful.\n",
        "\n",
        "In order to plot our functions we need to add a a plot function to our repertoire. `from sympy.plotting import plot` gets us this function.\n",
        "\n",
        "Enter that here."
      ],
      "metadata": {
        "id": "_nRSIV0argXF"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "5MkBTH5ttUBK"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now if we want to plot a equation, let's say, it's $y=a x^2 + b x + c$, we can do that. First define the variables. We can but don't have tell sympy to assume that they're real, positive, etc. That's useful if you're integrating complicated expressions or simplifying things. Here we'll skip that part and just define the symbols. Next define the equation. Enter the left-hand-side in front of the comma and the right-hand-side after it."
      ],
      "metadata": {
        "id": "rchw2AVBtZSP"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "y,x,a,b,c=sp.symbols('y,x,a,b,c')\n",
        "q_eq=sp.Eq( y, a*x**2+b*x+c) #named q_eq because it's a quadratic equation but we can call it whatever we want."
      ],
      "metadata": {
        "id": "6j5yqSAduCj7"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now we're ready to plot but we'll need to know what $a$, $b$, and $c$ are. If we just define these variables, we'll have ruined the generality of our equation until we undefine the variables. It's useful to have a function to substitute values into our equation without actually changing the equation. We can do that with `subs`. We can substitute a=2 into the right-hand-side of the equation with `q_eq.rhs.subs(a,2)`. If we want b=-1 and c=4, just append `.subs(b,-1).subs(c,4)` to the end of that. Do that below and call the new expression q_eq1 and display the results."
      ],
      "metadata": {
        "id": "Chf-43SqtUwa"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        },
        "id": "D4_n_X6Cv4wo",
        "outputId": "8eafd089-c818-4f8b-d7a2-8aae7faeee9a"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              " 2          \n",
              "x  + 4⋅x + 7"
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle x^{2} + 4 x + 7$"
          },
          "metadata": {},
          "execution_count": 103
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now that we have an expression, we can plot it with `plot(expression,(indep variable,lowerlimit,upperlimit))`\n",
        "\n",
        "Do this for our q_eq1 for $x$ from -3 to 3."
      ],
      "metadata": {
        "id": "v6YLJDE9wdKW"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "JIeZkxVawGOE"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "You notice some gobbledygook at the botton of the plot. We can get rid of this but just assigning our plot to a name (here let's name it `qplot`) and ending our plot code with `qplot.show()`. "
      ],
      "metadata": {
        "id": "5A0b2H8Mw3HQ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "qplot=plot() # complete the plot command\n",
        "qplot.show()"
      ],
      "metadata": {
        "id": "nU-oEprBxRdS"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "We don't actually have to define a new expression and substitute it before we can plot. We can do it all in the plot command. Here we'll add a second quadratic equation to our plot. Let's use a=1, b=4, and c=7. \n",
        "\n",
        "`plot(q_eq.rhs.subs()...`\n",
        "\n",
        "We can plot an additional curve on our original plot by putting our plot command inside the parentheses in `qplot.extend()`. Do this."
      ],
      "metadata": {
        "id": "VaPlL-lKxgod"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "aUVAeqXqyl2c"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "The plot command probably worked but made two plots instead of one. We can make it combine them by telling the plot function to hold off plotting until we're all done. Do this by adding `,show=False` after the plot command (but inside the parentheses). \n",
        "\n",
        "We can also make the plot colors distinct by appending `,line_color='r'` where the 'r' is for red. Other colors are \n",
        "\n",
        "*  b: blue\n",
        "*  g: green\n",
        "*  c: cyan\n",
        "*  m: magenta\n",
        "*  y: yellow\n",
        "*  k: black\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "POjHpUwszzvN"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Make these changes"
      ],
      "metadata": {
        "id": "OGgrtQn3085E"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "DhRUkMec08Ko"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "We can also do things like change the axis labels. `,xlabel('distance (m)')` would add information to the label. `,ylabel('velocity (m/s)'` does the same thing for the $y$-axis. We only need to do this in the first plot command.\n",
        "\n",
        "We can label each of our two plots with descriptive information using `,label='water at 298 K'` or whatever. The adding `,legend=True` will show the label for each graph."
      ],
      "metadata": {
        "id": "-eZqiYju1n03"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "##Graphing Maxwell-Boltzmann"
      ],
      "metadata": {
        "id": "x6Q2PUUarkWo"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from sympy.plotting import plot # We don't need this if we've already done it in the previous section but it doesn't hurt anything\n",
        "from scipy.constants import Avogadro, Boltzmann\n",
        "k_b=Boltzmann # we can use k_b (not kb) to get the value of the Boltzmann constant\n",
        "NA=Avogadro"
      ],
      "metadata": {
        "id": "LxAaf001ys-3"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now let's plot our distribution functions starting with the one-dimensional Maxwell-Boltzmann distribution. We called it `mbx`. Here it is."
      ],
      "metadata": {
        "id": "v5fpLfavyqM7"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "mbx"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 71
        },
        "id": "P0Sg2S96zJW7",
        "outputId": "7232a734-6482-432e-f546-28ea2a564b5d"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                       2 \n",
              "                  -m⋅vₓ  \n",
              "                  ───────\n",
              "                  2⋅T⋅k_B\n",
              "           √2⋅√m⋅ℯ       \n",
              "f_{v_x} = ───────────────\n",
              "                    _____\n",
              "          2⋅√π⋅√T⋅╲╱ k_B "
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAAxCAYAAAD5nxoYAAAABHNCSVQICAgIfAhkiAAAB6BJREFUeJzt3HmwVnUZwPEP0M0IkohGZZEiJpUUu13FzGgRigRSWmhvrGhzSpPImcBISTTJSq3GIiqjzSGXcBmnqGxRS7JYjJykTbRlsMYUCytb7I/nnLnnnnve9b7vfe+9c74zZ973POec5zzvOb/l+T2/5/dSUpIwptMGlAwqr8Cpyfe7O2lIydBgCt7eaSM6waNNbsOJqbgJK3E53okbcAjehbnJeV9CF9bisUWKRrXb0pK2swjTsAHfTPaX43uYgMPxp+T7bPwXN+K2ThjbKZZiXKeNGARWYYZ42Zcmss/iADwpkW1UR8Uf3R77hgTTsL+F+pbhuhbqaxUzsAdHY1ciOwD/wl/xEmwS3d8ivBpvGnQrO8gEvLnFOm8RfW2zfshQ8E2eiO934L4d5XU4qIX6enBZC/V1gtE4G6/FER22ZVA5s0C2Cj/FQ/iL8LiPqlPfpTiuNaY1zE78omCb0iF7hhXjRf+eZwveIgrAbGzGXuFkVeMx+GGFY9NEc/8a4b0/jDtEzTsWNyey2zG94Pqp+DLux4O4BgfXsKekCjNxZGZ/qfpqzXgxzDo5I3s2vo5zMrJFolUpYrEoDD/AC0QhuwtbReGYi2fid3o9/ZQZuA8XYha6RaHbXIftKYPiuxyITwov9ZHkgpUNGDlYdGEF1mdk76nz2snid83NyJ4sPO179A7FviZagCI+IGp0tjZ/SnRDkzKyL4pClmULPpyTvUh0Y/WyEJ9LdM8XLeJ6EWpej9eLYeaAuEE8qBtxPtbgGQNV2kZuE/MsY9Ufar0SO/SfnxmHv4uafiCur6HjqznZdfq/gJuwLrP/FPF8H07ulW7/EF1Go0zM3HO6qMhE67dOFJrzRJdXkaKDR+ClouQubsKwTnAzni9e3pY6zr9YtAhzRVeRZb9oqt8ouqCrqujp1vvgU56lbzdDdBUbcvsP4ZgCnY9UM7wCq/S2jj3Ynvl+HHYnNi3E8cJXeVteSVFhmJd8XtOEUZ3iahFXuEvtwNAlYnh1oujLi/iKaHqni5m+IsaJwrIjI5uEQ3OyQxN5Vvbv5Pq9okUYCGvxnYz+HvE8CCc2HUFJ7N0nAlJVeaXKDsasARo8GGzFO2qc8wnxAmr9njEinr+xyjnPEa1KNuQ9H//Ut5KdjL/pG+2dKPyKzaIlmYkXi1hGI1HhZWLouV7vb78qc//1wve5SLRi6ZB7dS3FJwjf4M+i5K5JtnMNj7yHj+JpVY5fJmrIPDGjl27jq+g7sYq+00RLlOUs/CwnW41bC64/Vow4HhSF5Q6VRy2tYiPeKxzLfuQnL8YIw34jYt3tYLkIi9bLTlxbx3ldohBXotIQ6kOi0JfkmC0e2sY23mOPxsbA7bSlJEPegexJPrfnT2whT22j7pIBkHdW0sKwI39iycinqGV4VPTT7aJdPkNJCxklvO3dBcfuw+k52SwRIOlu8D57tM5naGVuQbllODwRbCp46N/Fp3Oya0VwpmSEkO0mqjmPu0RwJOUEnCQKUMkIIetA1ioM2Ymqi0QQ55422TVSGHBz3UKdzd6vH3OSiyZhCR7QNynkRyIXAL4golwl7eHcThvweBGHfyHuxPtzx08ScfYVYn69pD2p+keK/M6O82v8GL8XeQN5tol5+67BNGoIs7wNOt+nsWF5QzQyQ7ZLzNSdI5IwsswR3cY+1ecHhirpmohW9bkTxARUq5nYJr0tY6qYcZshnM96M46HEumaiFbR6lR9ib4zCuTrRD7DgBnoiqqxYv78DJF3dyE+OFCjmqTZVPge/FxzGUaVOEikAsACtVuWUwt05Fks0hDzdGtvxHhY0mwqfKU1EXeq/PLWVNGXT9Ufq2/+xL34WE5WT6t0fgX5XhXyE0p6KUqFz1NtTcRh4sUvFC9sssiPXKavA91Iqv4E/A8vr2F7nseJTOw8hyQ2pjGgcSKCvF0Ts8MjeeHtE8TveyDZL1oTsQDfqnD9weJB3yJq3zgxxL5VrwPdJeIu2b58ikiZK6JHzAFtq2L3NHw8J5svsqLydCe27BbR4NvxHzxXzAGVJORT4RtdE3G6yPhKWSpahqIKVG+q/lnCn6nGItGVTM7Izqtw35X4ichfvV//4ewK/EH4FHcb/utFm+JiUTvzOZGNrInYgG9k9s8XD76Ij4h8ySWKl9GlXIFvVzmesk78AwtRcNdWOG+TWHa/T/ymPJeLgkIU1P0q53yOyG7iEjG0m6d/Knx2TcRS1ddEHC2GzCnduf0sVyf6potaXYke1buIlOvxsuT7MVWu6RYFtkuxo3yU3pFGN35r4Kn5w4Z6UuEXCD9ii8rh4tHioWUdvV8qduJSaqXqjxfO46uqnJO9/73C71klfJU86RTBHLwhsbcnc3yUaDF24ldiqHtYHfceEdSbCl/PmoinC+dxZkZ2peiXF1a4plaq/vMKdFbj86LgXFDh+PHCWUxHNmvxRxEElNiyNXP+anymznsPexqJCdRaE9EMteZkzhSh5Hr/VO0UsWywUmtzmmitUkaJArtNtBqniH94S1mirw9UMowYK5r+Zv+U4+xkI1rDTXh3C+wq6RBvHcC1Vwi/Y4cIQl1gZA4YSkpKSkpKSkpKOsX/AeQ4bn1ReUPvAAAAAElFTkSuQmCC\n",
            "text/latex": "$\\displaystyle f_{v_x} = \\frac{\\sqrt{2} \\sqrt{m} e^{- \\frac{m v_{x}^{2}}{2 T k_{B}}}}{2 \\sqrt{\\pi} \\sqrt{T} \\sqrt{k_{B}}}$"
          },
          "metadata": {},
          "execution_count": 105
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "There are a number of substitutions that need to be made. If we want to use molar masses instead of molecular masses, we can use $R$ (8.314 J/molK) instead of $k_B$. Remember to use kg for mass. Plot at 298 K for He between -3000 and 3000 m/s and add a plot for Ar in the same limits."
      ],
      "metadata": {
        "id": "IvulhYFyzLMo"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "kVsO8MImeKEn"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Do this again for He at 150 K, 300 K, and 450 K."
      ],
      "metadata": {
        "id": "w8VIMwaq5iUB"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "nmGKm_5z56SG"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now do a similar comparison series for the three-dimensional Maxwell-Boltzmann distribution, `mb3d`. Plot between 0 and 5000 m/s."
      ],
      "metadata": {
        "id": "5DdaqgEe5-qI"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "AgatZwO9HFPF"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "##making funtions on the fly\n"
      ],
      "metadata": {
        "id": "wfcou0uJ6Z61"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "He=sp.lambdify(v,mb3d.rhs.subs(kb,8.314).subs(T,298).subs(m,0.004).evalf())\n",
        "Ar=sp.lambdify(v,mb3d.rhs.subs(kb,8.314).subs(T,298).subs(m,0.040).evalf())\n",
        "speed=np.linspace(0,5000,200)"
      ],
      "metadata": {
        "id": "AJp3Co0lKhT-"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.figure(figsize=(18,3))\n",
        "plt.plot(speed,Ar(speed) )\n",
        "plt.plot(speed,He(speed))\n",
        "plt.xticks([])\n",
        "plt.yticks([])\n",
        "plt.savefig('mbbanner.jpg')\n",
        "plt.show()"
      ],
      "metadata": {
        "id": "9ouotFqhMUrc",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 197
        },
        "outputId": "35f94e6a-9387-4026-cff9-ab72ca70f00e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1296x216 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "fig,(ax1,ax2)=plt.subplots(1,2,sharey=True,figsize=(15,2))\n",
        "ax1.plot(speed,He(speed))\n",
        "ax2.plot(speed,Ar(speed))"
      ],
      "metadata": {
        "id": "ylKAdAuGJm59",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 174
        },
        "outputId": "2d226812-3966-4a44-e457-e9eb2a4ce752"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f77e7351bd0>]"
            ]
          },
          "metadata": {},
          "execution_count": 72
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1080x144 with 2 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bFhbRUO13VWp"
      },
      "source": [
        "Make a set of graphs for Ar at 298 K vs Ar at 500 K."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "cVQJONkIeJzo"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JYnl_xTbaXXJ"
      },
      "source": [
        "# Averages\n",
        "Now that we a distribution function that gives the fraction of molecules going each speed, we can use it to get average values.\n",
        "\n",
        "To obtain averages we need to weight each possible speed by the fraction of molecules going that speed and add up the results. In other words, $$\\left<v\\right> = \\int^{oo}_0v f(v)dv$$"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "To integrate use `sp.integrate(expression,(indepvariable,lowerlimit,upperlimit))`\n",
        "Find an expression for the average speed of molecules moving in three-dimensions."
      ],
      "metadata": {
        "id": "jXNMC--y7MqQ"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "wYwkJp6fahG8"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Use `subs` to find a value for the average speed for He and for Ar at 298 K."
      ],
      "metadata": {
        "id": "0yqjVbeO7sTY"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "CL64Z1747x1-"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Do the same for the $x$-component of the velocity. What are the limits in this case?"
      ],
      "metadata": {
        "id": "xiuG87pq8WXn"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "ZXCwgoq88e8S"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Do it again for only positive values of velocity. (or negative ones)."
      ],
      "metadata": {
        "id": "wjNJYV8K8gcG"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "gc0szV8i8lf1"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "A similar technique can get the average squared speed in 1D and 3D."
      ],
      "metadata": {
        "id": "2153ZdX021tm"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "MJFqrkJPaqY_"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Find an expression for the average cubed speed. Not that anyone cares about this but we can do it just the same."
      ],
      "metadata": {
        "id": "Ln_-w6Tb72Ne"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "y7oNQBrCaud7"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The average kinetic energy expression should work just the same. Find this expression too for 1D and 3D."
      ],
      "metadata": {
        "id": "sUzbypN68At8"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "g0tOBk16aw-n"
      },
      "outputs": [],
      "source": [
        "      # average kinetic energy"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "D0JzNXNXMbrb"
      },
      "source": [
        "# Collision Frequncy\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "cs,N1,N2,V,mu=sp.symbols('sigma,N_1,N_2,V,mu')"
      ],
      "metadata": {
        "id": "HGaNQEvC9jK4"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Define `v_avg` to be the average speed in three dimensions."
      ],
      "metadata": {
        "id": "gsAb-lPM-5w-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        ""
      ],
      "metadata": {
        "id": "ZcHJOkvd_DNv"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The collision frequency is the cross-sectional area multiplied by the average speed multiplied by the density of molecules."
      ],
      "metadata": {
        "id": "FpbAd7_r9To3"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "tO6hRZ22MulI"
      },
      "outputs": [],
      "source": [
        "z11=cs*v_avg*N1/V\n",
        "z11"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "This is appropriate for a single molecule of type 1 colliding with other type 1 molecules. Find the collision frequency of an Ar atom at 298 in one mole of Ar in a 24 L container. The $\\sigma$ of Ar is 140 pm."
      ],
      "metadata": {
        "id": "cvCFYRa0BK0K"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "p9rVVvwWQBUc"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The collision frequency of each molecule type 1 with a second type of molecule would be just the same except multiplied by the concentration of molecule type 2."
      ],
      "metadata": {
        "id": "MBYMTBvSDwed"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "JOBQLG1Mo9xe",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 60
        },
        "outputId": "c372a924-9f52-482b-dc24-1f159bb93703"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "             _____  \n",
              "2⋅√2⋅N₂⋅√T⋅╲╱ k_B ⋅σ\n",
              "────────────────────\n",
              "      √π⋅V⋅√m       "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\frac{2 \\sqrt{2} N_{2} \\sqrt{T} \\sqrt{k_{B}} \\sigma}{\\sqrt{\\pi} V \\sqrt{m}}$"
          },
          "metadata": {},
          "execution_count": 85
        }
      ],
      "source": [
        "z12=cs*v_avg*N2/V\n",
        "Z2"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "In the case of two different molecules, the mass needs to be replaced by the reduced mass $\\dfrac{m_1m_2}{m_1+m_2}$\n",
        "\n",
        "For the case of an argon atom colliding with helium molecules,"
      ],
      "metadata": {
        "id": "2DSEFxruHZLe"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Zcwin6jMoqIA"
      },
      "outputs": [],
      "source": [
        "round(Z2.subs(N2,Avogadro).subs(V,.024).subs(T,298).subs(cs,pi*140e-12**2).subs(kb,Boltzmann).subs(m,(0.040*0.004)/(0.004+0.040)/Avogadro).evalf(),2)"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Calculate for an argon atom colliding with xenon atoms."
      ],
      "metadata": {
        "id": "djZLkGt9HxEZ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "4FE-wmyaH8wl"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LgzDGTuvA2pL"
      },
      "source": [
        "It's actually not the average velocity that we want in this application. We need the average *relative* velocity. The average relative velocity is the difference between the velocity vectors of the two molecules. $\\vec{v_{rel}}=\\vec{v_1}-\\vec{v_2}$ The magnitude of the relative velocity can be obtained from the dot product of the relative velocity with itself. $$v_{rel}=\\sqrt{\\vec{v_{rel}}\\cdot\\vec{v_{rel}}}=\\sqrt{\\left(\\vec{v_1}-\\vec{v_2}\\right)\\cdot\\left(\\vec{v_1}-\\vec{v_2}\\right)}=\\sqrt{\\vec{v_1}^2-2\\vec{v_1}\\cdot\\vec{v_2}+\\vec{v_2}^2}$$\n",
        "For random, uncorrelated velocity vectors, the $\\vec{v_1}\\cdot\\vec{v_2}=0$ so $$\\vec{v_{rel}}=\\sqrt{\\vec{v_1}^2+\\vec{v_2}^2}$$  Since the two velocity vectors are the same $$\\vec{v_{rel}}=\\sqrt{2\\vec{v_1}^2}=\\sqrt{2}\\left<v\\right>$$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "FSMuVSwxA1Do"
      },
      "outputs": [],
      "source": [
        "z11=cs*v_avg*sp.sqrt(2)*N1/V\n",
        "z11"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "aAoEO2n9KdRj",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 60
        },
        "outputId": "2a7a4e25-0f99-4016-b298-9444549a55a8"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "          _____  \n",
              "4⋅N₂⋅√T⋅╲╱ k_B ⋅σ\n",
              "─────────────────\n",
              "     √π⋅V⋅√m     "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\frac{4 N_{2} \\sqrt{T} \\sqrt{k_{B}} \\sigma}{\\sqrt{\\pi} V \\sqrt{m}}$"
          },
          "metadata": {},
          "execution_count": 89
        }
      ],
      "source": [
        "z12=(cs*v_avg*sp.sqrt(2)*N2/V).subs(m,mu)\n",
        "z12"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "If, instead of the collision frequency of a single molecule, we want the total number of collisions between all molecules, we just multiply by the concentration of molecule 1."
      ],
      "metadata": {
        "id": "g2GEbSC0IUrL"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "drzeJ9u3KyQc",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 60
        },
        "outputId": "e9909703-a8d8-4358-fe87-c7887b482816"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "    2      _____  \n",
              "2⋅N₁ ⋅√T⋅╲╱ k_B ⋅σ\n",
              "──────────────────\n",
              "         2        \n",
              "     √π⋅V ⋅√m     "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\frac{2 N_{1}^{2} \\sqrt{T} \\sqrt{k_{B}} \\sigma}{\\sqrt{\\pi} V^{2} \\sqrt{m}}$"
          },
          "metadata": {},
          "execution_count": 91
        }
      ],
      "source": [
        "Z11=N1/V*z11/2 #capital Z is for *all* molecules 1 and lower-case is for a single molecule 1\n",
        "Z11"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "o0Bbd4N_K5hB"
      },
      "outputs": [],
      "source": [
        "print(round(Z11.subs(N2,Avogadro).subs(N1,Avogadro).subs(V,.024).subs(T,298).subs(cs,pi*140e-12**2).subs(kb,Boltzmann).subs(m,0.040/Avogadro).evalf(),2),\"1/s\")"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "A similar expression could be developed for $Z_{12}$, the collision frequency for all molecules 1 with molecules of type 2."
      ],
      "metadata": {
        "id": "k1Nn0T8KIo8U"
      }
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "O6hmqC-feQcE"
      },
      "source": [
        "# Mean Free Path"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The mean free path, $\\lambda$, is the distance that molecules travel, on average, between collisions. We have collision frequency and velocity. The reciprocal of collision frequency is the time between collisions. Since distance is rate multiplied by time, we can find the average distance. "
      ],
      "metadata": {
        "id": "I9nu26waIz18"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "uiuj6QnUeYbh",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 59
        },
        "outputId": "3a8b59fc-bc51-4b46-fd13-391ac39e9abe"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "    √2⋅T⋅k_B\n",
              "λ = ────────\n",
              "     2⋅p₁⋅σ "
            ],
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\lambda = \\frac{\\sqrt{2} T k_{B}}{2 p_{1} \\sigma}$"
          },
          "metadata": {},
          "execution_count": 93
        }
      ],
      "source": [
        "mfp,p1=sp.symbols('lambda,p_1')\n",
        "expr2=sp.Eq(mfp,v_avg/Z1)\n",
        "expr2.subs(V/N1,kb*T/p1) #substituting the ideal gas law gives us an expression involving pressure and temperature rather than volume and temperature."
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Calculate the mean free path for an argon molecule at 1 bar (or 100000 Pa) and 300 K. "
      ],
      "metadata": {
        "id": "mAob3odFJc06"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "vqIPLRW_hF4i"
      },
      "outputs": [],
      "source": [
        ""
      ]
    }
  ],
  "metadata": {
    "colab": {
      "collapsed_sections": [],
      "name": "Copy of Maxwell-Boltzmann.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyPeGGShdt48MUmMWijsFXrV",
      "include_colab_link": true
    },
    "kernelspec": {
      "display_name": "Python 3",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}