Homework/HW9-soln.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
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    "id": "K1nU4iJ1x-fa"
   },
   "source": [
    "# Chem 30324, Spring 2020, Homework 9\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "IfKlbKuzx-fc"
   },
   "source": [
    "# Due April 10, 2020\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "_n6NS06yx-fe"
   },
   "source": [
    "## The two-state system.\n",
    "### Consider a closed system containing $N$ objects, each of which can be in one of two energy states, of energy either 0 or $\\varepsilon$. The total internal energy $U$ of the box is the sum of the energies of the individual objects."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "ZaEC3jfAx-ff"
   },
   "source": [
    "![Boltzmann](https://github.com/wmfschneider/CHE30324/blob/master/Homework/imgs/Boltzmann.png?raw=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "5RQZ72Rpx-fh"
   },
   "source": [
    "### 1. Write down all the possible microstates for a box in which $N = 4$ and the internal energy $U = 2\\varepsilon$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "yW3gslpyyTP6"
   },
   "source": [
    "$$N = 4$$\n",
    "$$q = 2$$\n",
    "To find the number of microstates, \n",
    "$$\\Omega = \\frac{N!}{(q!)(N-q)!} = 6$$\n",
    "\n",
    "Possible Microstates:\n",
    "$$(00\\varepsilon\\varepsilon)\\qquad(0\\varepsilon 0\\varepsilon)\\qquad(0\\varepsilon \\varepsilon 0)\\qquad(\\varepsilon 00\\varepsilon)\\qquad(\\varepsilon 0\\varepsilon 0)\\qquad(\\varepsilon \\varepsilon 00)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "9_0luRKUx-fi"
   },
   "source": [
    "### 2. What does the postulate of *equal a priori probabilities* say about the relative likelihood of occurance of any one of these microstates?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "KL8LUv6Y1SE8"
   },
   "source": [
    "The postulate of equal a priori probabilities states that each microstate has the same probability of occuring."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "eVC6ZGgsx-fk"
   },
   "source": [
    "### 3. What is the entropy of the box? (Thank you, Ludwig Boltzmann.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 34
    },
    "colab_type": "code",
    "id": "dFtlZ51J1n4T",
    "outputId": "d4c4e2ce-c10d-40e1-e5af-839507563170"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The entropy of the box is kln6 = 2.474e-23 J/K.\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "k = 1.380649e-23 #J/K\n",
    "omega = 6 #number of possible microstates\n",
    "S = k*np.log(omega)\n",
    "print(\"The entropy of the box is kln6 = {0:8.3e} J/K.\".format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "_fTy02Zfx-fl"
   },
   "source": [
    "### 4. Suppose two identical such boxes are brought into thermal contact and allowed to come to equilibrium. Calculate the change in internal energy $\\Delta U$ and in entropy $\\Delta S$ associated with this process."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 51
    },
    "colab_type": "code",
    "id": "kyWj8VV-3Q_r",
    "outputId": "ac146e0d-d8b6-4a53-fc1a-de3c1cfe738a"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The change in entropy is 9.180988685797155e-24 J/K\n",
      "The change in energy is 0.0 epsilon\n",
      "The energy of a closed system is constant (first law)\n",
      "The entropy change in any spontaneous process in a closed system is > 0\n"
     ]
    }
   ],
   "source": [
    "from math import factorial\n",
    "import numpy as np\n",
    "\n",
    "k = 1.380649e-23 #J/K\n",
    "epsilon = 1. # some amount of energy\n",
    "\n",
    "def Omega(N,q):\n",
    "    return factorial(N)/(factorial(q)*factorial(N-q))\n",
    "\n",
    "def S(N,q):\n",
    "    return k*np.log(Omega(N,q))\n",
    "\n",
    "def U(N,q):\n",
    "    return q*epsilon\n",
    "\n",
    "#2 identical Boxes, separated\n",
    "N = 4; q = 2\n",
    "\n",
    "Sseparated = S(N,q)\n",
    "Useparated = U(N,q)\n",
    "\n",
    "#2 Boxes, Combined\n",
    "N2 = N*2\n",
    "q2 = q*2\n",
    "\n",
    "Scombined = S(N2,q2)\n",
    "Ucombined = U(N2,q2)\n",
    "\n",
    "deltaS = Scombined - 2*Sseparated\n",
    "deltaU = Ucombined - 2*Useparated\n",
    "print('The change in entropy is',deltaS,\"J/K\")\n",
    "print('The change in energy is',deltaU,\"epsilon\")\n",
    "print('The energy of a closed system is constant (first law)')\n",
    "print('The entropy change in any spontaneous process in a closed system is > 0')\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "Vr1rypWux-fn"
   },
   "source": [
    "## The canonical ensemble.\n",
    "### The energy spectrum of some molecule is described by the diagram below.  A large number $N$ of these distinguishable molecules is in thermal equilibrium with a much larger reservoir of temperature $T$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "dXAEz5I2x-fp"
   },
   "source": [
    "![Energy States](https://github.com/wmfschneider/CHE30324/blob/master/Homework/imgs/EnergyStates.png?raw=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "okqDVK5Ux-fq"
   },
   "source": [
    "### 5. Write the partition function $q$ for one of the molecules in the system (a) in terms of $T$ and $\\varepsilon$, (b) in terms of $\\beta = 1/k_B T$ and $\\varepsilon$, and (c) in terms of a characteristic temperature $\\theta = \\varepsilon/k_B$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "GWbj98ys7dC6"
   },
   "source": [
    "$a) q = e^{\\frac{0}{k_bT}}+2e^{\\frac{-\\epsilon}{k_bT}}+3e^{\\frac{-2\\epsilon}{k_bT}} = 1+2e^{\\frac{-\\epsilon}{k_bT}}+3e^{\\frac{-2\\epsilon}{k_bT}}$\n",
    "\n",
    "$b) q = e^{0\\beta}+2e^{-\\epsilon\\beta}+3e^{-2\\epsilon\\beta} = 1 +2e^{-\\epsilon\\beta}+3e^{-2\\epsilon\\beta}$\n",
    "\n",
    "$c) q = e^{\\frac{0\\theta}{T}}+2e^{\\frac{-\\theta}{T}}+3e^{\\frac{-2\\theta}{T}} = 1+2e^{\\frac{-\\theta}{T}}+3e^{\\frac{-2\\theta}{T}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "e_pJxsTYx-fs"
   },
   "source": [
    "### 6. Plot the relative fractions of molecules of energy 0, $\\varepsilon$, and $2\\varepsilon$  vs. temperature.  Assume $\\theta = 300$ K. Be sure to indicate the probabilities in the limits of $T\\rightarrow 0$ and $T \\rightarrow \\infty$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 329
    },
    "colab_type": "code",
    "id": "o9MubiEFr35G",
    "outputId": "9c52bcf6-2e05-4eca-e30c-48a1429ce9f5"
   },
   "outputs": [
    {
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36rNnzyYnJ4fTTz+9yfyZM2cyevTohmOdd955h3yszojL+zjqje2fxZurd8Y6DGNMHIt0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\nDPtLImARBwCLOABYxAHAIg4AlgAKhGEEORxlxgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "tags": []
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "When T --> 0, there is a 100% probability of E = 0. As T --> oo, E = 0 has a 1/6 probability\n",
      "E = 1E has a 1/3 probability and E = 2E has a 1/2 probability.\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "theta = 300 #K\n",
    "T = np.linspace (1,3000,1000)\n",
    "Ptot = 1 + 2*np.exp(-theta/T) + 3*np.exp(-2*theta/T)\n",
    "P0 = 1*np.exp(-0*theta/T)/ Ptot\n",
    "P1 = 2*np.exp(-theta/T) / Ptot\n",
    "P2 = 3*np.exp(-2*theta/T) / Ptot\n",
    "\n",
    "plt.plot(T,P0, label = 'E = 0')\n",
    "plt.plot(T,P1, label = 'E = 1E')\n",
    "plt.plot(T,P2, label = 'E = 2E')\n",
    "plt.legend()\n",
    "plt.xlabel('Temperature (K)')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Relative probabilites at theta = 300 K')\n",
    "plt.ylim(0,1)\n",
    "plt.xlim(0,3000)\n",
    "\n",
    "plt.show()\n",
    "print('When T --> 0, there is a 100% probability of E = 0. As T --> oo, E = 0 has a 1/6 probability')\n",
    "print('E = 1E has a 1/3 probability and E = 2E has a 1/2 probability.')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "WgcYA_cTx-ft"
   },
   "source": [
    "### 7. Derive an expression for the internal energy $U$ per molecule by summing over the possible microstates weighted by their probabilities. Plot the average energy vs. temperature, assuming  $\\theta =300$ K."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "TLS61_vAzT61"
   },
   "source": [
    "$U = 0P(0) + \\varepsilon P(\\varepsilon) + 2\\varepsilon P(2\\varepsilon) = \\varepsilon\\frac{2e^{-\\beta\\varepsilon}}{q} + 2\\varepsilon\\frac{3e^{-2\\beta\\varepsilon}}{q} = \\frac{2\\varepsilon e^{-\\beta\\varepsilon} + 6\\varepsilon e^{-2\\beta\\varepsilon}}{q}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 295
    },
    "colab_type": "code",
    "id": "5PHMqMGG2_Gz",
    "outputId": "22765f95-38e6-4cdb-aab1-ca1a8749f30a"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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54fyZfOLCWYwcocQukg2U4GPq6Vf28e+/W8P2A81cOm88X7ziZKaPLUl1WCIyjJTgY6a2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\n1xW4pgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "epsilon = theta*k # as we needed epsilon in expression\n",
    "U = (2.*epsilon * np.exp(-theta/T) + 6.*epsilon * np.exp(-2.*theta/T))/ Ptot # average energy J per particle\n",
    "plt.plot(T,U)\n",
    "plt.xlabel('Temperature (K)')\n",
    "plt.ylabel('Average energy (J/particle)')\n",
    "plt.title('Average energy')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "Inmmuyckx-fv"
   },
   "source": [
    "### 8. Derive an expression for the internal energy $U$ per molecule by taking the appropriate derivative of the partition function from problem 5 (*Hint:* it is easier to work with the expressions in term of $\\beta$ than in $T$.) Does your result agree with that from Question 7?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "ZZ8ncoZy7AMI"
   },
   "source": [
    "$\\left<U\\right> = -\\left(\\frac{dlnq}{d\\beta}\\right) = -\\frac{1}{q}\\left(\\frac{dq}{d\\beta}\\right) = -\\frac{1}{q}\\frac{d(1 + 2e^{-\\beta\\varepsilon} + 3e^{-2\\beta\\varepsilon})}{d\\beta} = \\frac{2\\varepsilon e^{-\\beta\\varepsilon} + 6\\varepsilon e^{-2\\beta\\varepsilon}}{q}$ \n",
    "\n",
    "Same expression as problem 7."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "wbu5H19Xx-fw"
   },
   "source": [
    "### 9. Derive an expression for the Helmholtz energy $A$ per molecule from the partition function. Plot $A$ vs. temperature, assuming $\\theta\n",
    "= 300$ K."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "TxLO0A0E7KsD"
   },
   "source": [
    "$A = U - TS = U - T(\\frac{U}{T}+k_B\\ln q) = -k_BTlnq = -k_BT\\ln(1 + 2e^{-\\beta\\varepsilon} + 3e^{-2\\beta\\varepsilon}) = -k_BT\\ln(1 + 2e^{-300/T} + 3e^{-600/T})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 295
    },
    "colab_type": "code",
    "id": "aSH9EvG_7Nvc",
    "outputId": "12c75a36-5ad0-451d-a761-d51ebdd072de"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Hc/FJSxMXdj+AD6/pQdPaFbnmtelc9pKPN+BcqsrzIbCkNOAS4Nhw0UfAM2a2p4hj+x/+\nELjw7NlrPD3hex787wIqlSvFXSe34sTWdaMOyzlXBLw7aJethWu2cN2IGcxcuYl+bevx9/4tqV6x\nQCV8zrkEs8+1gCS9Ef78TtLMrFNRBuuKT7M6lXnrsm5cd3xzxsxazfFDxzPOxyJ2LiXk1hCsrpmt\nltQ4u/VmVuwtsvwOoGjN+WEz142YwdzVmzm9UwNu6duCyuVKRx2Wc66A9vkOwMwyu3643MyWxU5A\nnu0AXPJpUa8K715xBFccfSAjp62k97AJTFm8PuqwnHNFJJ5aQMdns6xPYQfiEkOZUmn8pdchjLi0\nG2VKpXHW019w5/tz2L6r2J/5O+eKWG7PAC6T9B1wSJby/yWAPwMo4To2rs4HV3fnj10b8++JS+j7\n8ERmrtyY947OuaSR2zOAqkB14B7gxphVW8xsQzHE9j/8GUA0JizM4C8jZpKxdQdXHn0QVx5zEKXT\n47l5dM4lgvw8A9gErADaZ3kGEMmHv4tOj2a1GXttTwa0rcdDHy9k4GOTWbhmS9RhOecKKNevcWFj\nr/mSGhVTPC5BVS1fmgfPaMfjZ3dg5U/bOOnhiTwz4XsfdMa5JJbjgDAxqgOzJX0F/DrWoJn1L7Ko\nXMLq07ounZrU4Ka3vuOuD+by0Zw1PDCorQ8641wSiqcriCOzW25mnxdJRLnwZwCJw8wYOW0ld4wK\nBp25pe+hnN7JB51xLhF5VxCuSKz8aRvXj5jBF99v4NhD9uOeU1uzX+VyUYflnItRkAFhukj6WtJW\nSTsl7ZG0uWjCdMmmQfUKvHJRF24Nu5nuNXQ8H363Ou8dnXORi6cu3yPAWcBCoDxwEfBoUQblkkta\nmrig+wF8cHUPGtaowOUvf8M1r33Lpm27og7NOZeLuCpzm9kiIN3M9pjZc0Dvog3LJaOD9qvEm5d1\n49rjmvPBzNX0Gjae8Qsyog7LOZeDeBLANkllgOmS7pN0bZz7uRRUOj2Na45rxtuXH0HlcqX447Nf\n8X/vfMe2nbujDs05l0U8H+TnhttdSVANtCFwalEG5ZJf6wZVGXVVdwb3OICXv1xOn4cmMG2ZtyF0\nLpHkmQDC3j/3Ak2At4AbwyIh53JVrnQ6N5/UglcHd2HPXmPQE1O4d/Q8duz2juWcSwTx1AI6CVgM\nDCd4ILxIkvcG6uLWpWlNxgzpyemdGvLE54sZ8Mgk5vzgFcmci1o8RUD/Ao42s6PM7EjgaGBoQU4q\n6X5J88LeRd+WVK0gx3OJr1LZUtx7ahuePb8T67buZMCjE3n8s8Xs8a4knItMPAlgS5Yin++BgvYE\n9hHQyszaAAuAmwp4PJckjjmkDv+9tifHHVqHf46Zx5lPTWH5+m1Rh+VcSoonAUyV9KGk8yWdB4wC\nvpY0UNLA/JzUzP5rZpnVQr4AGuTnOC451ahYhsfO7sCDp7dl3uot9HloPK9/vZxkapXuXEkQTwIo\nB6wBjgSOAjIIGoT1A/oWQgwXAKNzWinpYklTJU3NyPA65SWFJAZ2aMCYa3vSpkE1/vrmdwx+YSoZ\nW3ZEHZpzKSO3AWHOAv5rZvkaFFbSOGD/bFbdbGbvhtvcDHQCBlocX/+8L6CSae9e47nJS/nnmHlU\nLluKfwxsTa+W2f3pOOfyI6e+gHLrDroRMEJSaeBjgm/pX8XzQQ1gZsflEdD5BHcQx8Z7TFcypaWJ\nC7sfQI9mtbj29elc8uI0BnVswK39WlC5XOmow3OuxIqnO+jKwHEE3T8cBswFxgBjzWxNvk4q9QYe\nBI40s7jLdfwOoOTbuXsvwz9eyGOfLaJetfL8a1BbDm9aM+qwnEtqhdYdtKQWQB/gBDPrlc9gFgFl\ngczipS/M7NK89vMEkDqmLdvAn9+YwfIN2xjcoynXndCcsqXSow7LuaS0zwlAUodcjrcDWG5mxTow\nrCeA1PLzjt3c/eFcXvlyOYfsX5mhZ7Tj0LpVog7LuaSTnwTwaS7HK0XwjOBRM7uvcELMmyeA1PTJ\nvDXcMPI7Nv2ykz8ffzAX92xKepqPPOZcvAp9RDBJZYFvzaxFQYOLlyeA1LXh553c/PZ3jJ71I52b\nVOfB09v5OMTOxWmfRwST1D2PY5YFLiloYM7FI2vjsd7DxvPG1yu88ZhzBZBbQ7BTJU2WdKukkyQd\nJqmnpAskvQi8D2wvpjid+7Xx2OghPWjdoCo3vDmTwS9MY91WbzzmXH7kWgQkqQZB3/9HAHWBXwiq\ngX5gZhOLJcIYXgTkMu3dazw7aQn3jZ1P5bKluGdga07wxmPOZavQnwFEwROAy2rBmi0MeW06c1Zv\n5vRODbilrzcecy6rfX4G4FwyaF6nMu9ccQRXHH0gI6etpM9DE/hqiY885lw8PAG4pFemVBp/6XUI\nIy7tSprEGU9N4Z7Rc33kMefy4AnAlRgdG9dg9DU9OLNzQ578/HsGPDKJuat95DHnchLPkJDTJF0h\nqXpxBORcQVQsW4p7Brbh3+eFI489MoknP/eRx5zLTjx3AGcA9QgGgXlNUi9J3gzTJbRjD63D2CE9\nOPqQ2twzeh5nPfUFKzb4yGPOxcozAZjZIjO7GWgOvAI8CyyTdEdYTdS5hFSzUlmeOKcjDwxqy5zV\nm73xmHNZxPUMQFIbgsHh7wfeBAYBm4FPii405wpOEqd1bMCYIT1oVT9oPHbxi954zDmI8xkAMBT4\nGmhjZleb2Zdm9i+CAeKdS3gNqlfg1cFduPnEQ/l8fga9h43nozn5Gs7CuRIjngFhmppZQnzQe0Mw\nVxjm/7iFIa9PZ27YeOzWfi2pVDa3wfGcS275GRIy08nZPPPdBEwzs+mFEZxzxeng/SvzzhXdGDZu\nIU9+vpgp36/nwdPb0bmJP9JyqSWeZwCdgEuB+uF0CcHwkE9LuqEIY3OuyJQtlc5fex/CG5d0RYjT\nn/TGYy71xJMAGgAdzOw6M7sO6AjsB/QEzi/C2Jwrcp2a1ODDLI3H5v3ojcdcaognAexHMARkpl1A\nHTP7Jcty55JSpbDx2DN/7MS6rTvo//AknhrvjcdcyRdPAngZ+FLSbZJuAyYBr0iqCMzJz0kl3Slp\npqTpkv4rqV5+juNcYTquRR3GDunJUQfX5h8fzuOsp73xmCvZ8hoPQARFQHUIxgQAmGRmBaqKI6mK\nmW0O568GWpjZpXnt57WAXHEwM0ZOW8kdo4LvN7f3b8mpHerjDeBdsspXLSAzM0kfmllroNA+eTM/\n/EMVAb/XdglDEoM6NaRL05pc98YMrh8xg3Fz1vCPga2pUbFM1OE5V2jiKQL6RlLnwj6xpLslrQDO\nBm7NZbuLJU2VNDUjI6Oww3AuRw1rVODVi7twY59D+HjeGnoNG8+n89dGHZZzhSaehmDzgGbAUuBn\nQAQ3B23y2G8ckN0YfTeb2bsx290ElDOz2/IK1ouAXFTm/LCZIa9/y4I1Wzm3S2NuOvEQKpTxxmMu\nOeR7SEhJjbNbbmbLCimwRsCHZtYqr209Abgobd+1hwfGzueZiUtoWqsiD57RjnYNq0UdlnN5yveQ\nkOEHfUPgmHB+Wzz75RFMs5iXA4B5BTmec8WhXOl0/q9vC1656HC279rDqY9P5qFxC9m9Z2/UoTmX\nL/F0Bncb8FfgpnBRaeClAp73XkmzJM0ETgCuKeDxnCs23Q6qxeghPenXpi5Dxy3g1CemsGTdz1GH\n5dw+i+eb/ClAf4Lyf8zsB6ByQU5qZqeaWSsza2Nm/cxsVUGO51xxq1q+NMPObM/DZ7Vn6bqfOfGh\nCbz85TIfa8AllXgSwE4L/qoNIGwA5pwD+rWtx9ghPenYuDo3vz2LC/7zNWu3bI86LOfiEk8CeEPS\nk0A1SYOBccDTRRuWc8lj/6rleOGCw7i9XwsmL15P72ETGDv7x6jDci5PedYCApB0PEFZvYCxZvZR\nUQeWHa8F5BLdwjXBWAOzf9jMoI4NuLVfCyqXKx11WC7F5bsaaCLxBOCSwc7dexn+8UIe+2wR9aqV\nZ+gZPtaAi1a+q4FKGihpoaRNkjZL2iLJ+8t1LgdlSqVxfa+DeeOSrqQpGGvgn2PmsXO3Vxd1iSWe\nZwD3Af3NrKqZVTGzymZWpagDcy7ZZY41cEanhjz+2WJOfnQSC9ZsiTos534VTwJYY2ZzizwS50qg\nSmVLce+pbXjq3I6s2bydvg9P5N8Tl7DXxxpwCSCezkymSnodeIeYAWDM7K0ii8q5EuaElvvTvlF1\nbnxzJne+P4dP5q3hgUFtqVu1fNShuRQWzx1AFYLuH04A+oVT36IMyrmSqHblsjxzXifuGdiab5dv\npNfQ8bw73dtAuuh4LSDnIrB03c9c+8Z0vl2+kf5t63HngFZUreDVRV3RKEgtoOaSPpY0K3zdRtL/\nFUWQzqWKJrUqMuKSrlx3fHM+/G41vR8az6RF66IOy6WYeIqAniboCG4XgJnNBM4syqCcSwWl0tO4\n6thmvHV5N8qXSefsZ77k76PmsH3XnqhDcykingRQwcy+yrJsd1EE41wqatOgGh9c1YPzujbm2UlL\n6PfwRGat2hR1WC4FxJMA1kk6kN86gzsNWF2kUTmXYsqXSeeOAa14/oLD2PTLLk55bBKPfbaIPV5d\n1BWheBLAFcCTwCGSVgFDgEuLNCrnUtSRzWszdkhPjm9Rh/vGzOfMp6awYsO2qMNyJVTctYDCbqDT\nzCyypoxeC8ilCjPj7W9Xcdu7s9lrxm39WzKoYwMkRR2aS0L5rgWUycx+jvLD37lUIomBHRowekgP\nWtWvyg0jZ3LpS9NYv3VH3js7F6cCje3rnCtaDapX4JXBXfjbiYfw6bwMeg2bwKfz1kYdlishPAE4\nl+DS08TFPQ/k3SuPoFalMvzpP19z89vfsW2nV8ZzBRNPQ7AKkm6R9HT4upkk7wrCuWJ2aN0qvHPF\nEVzcsymvfLWck4ZP5NvlP0Udlkti8dwBPEfQCVzX8PUq4K7COLmk6ySZpFqFcTznSrpypdP524mH\n8spFXdi5ey+nPTGFoR8tYNceH2vA7bt4EsCBZnYfv7UE3kYwNGSBSGpI0MHc8oIey7lU0/XAmowe\n0oMBbevx0McLOe3xyXyfsTXqsFySiScB7JRUnt8agh1ITLfQBTAUuCHzuM65fVOlXGkePKMdj/6h\nA8s2bOPE4RN48YtlJFMHjy5a8SSA24AxQENJLwMfE3xw55ukAcAqM5sRx7YXS5oqaWpGRkZBTutc\niXRSm7qMHdKTzk1qcMs7s/jTf75m7ebtUYflkkBcDcEk1QS6EBT9fGFmeXZbKGkcsH82q24G/gac\nYGabJC0FOsVzTG8I5lzOzIwXv1jG3R/MpUKZdO4Z2JrerepGHZZLAAXpDlpAH6Cjmb0PVJB0WF77\nmdlxZtYq6wR8DxwAzAg//BsA30jKLlk45+IkiT92bcIHV/egQfUKXPrSN1w/YgZbtu+KOjSXoOIp\nAnqMoAbQWeHrLcCj+T2hmX1nZvuZWRMzawKsBDqY2Y/5PaZz7jcH7VeJty7vxlXHHMRb36yk97AJ\nfLVkQ9RhuQQUTwI43MyuALYDmNlPQJkijco5VyCl09O47oSDGXFpN0qlizOemsK9o+exY7ePNeB+\nE08C2CUpnd9qAdUGCq3ScXgn4EMhOVcEOjauzodX9+DMzg154vPFnPzoZOb/6F16uUA8CWA48Daw\nn6S7gYnAP4o0KudcoalYthT3DGzDM3/sRMaW7fR7ZCLPTPievT7WQMrLNQFISgOWEFT7vIdgIJiT\nzWxEMcTmnCtEx7Wow5ghPenZrDZ3fTCXc/79JT9s/CXqsFyE8qwGKulbM2tfTPHkyquBOldwZsYb\nU1dwx6g5pKeJu05uxYB29aMOyxWhgowH8LGkU+UjUThXIkjijM6NGH1ND5rXqcw1r03nqle/ZdM2\nry6aauJJAJcAI4AdkjZL2iJpcxHH5ZwrYo1rVuT1i7tw/QnNGf3danoNG8/EhV4fI5XkmAAkHRHO\n1jazNDMrY2ZVzKyymVUppvicc0WoVHoaVx7TjLcvP4KKZdM5599fcseo2Wzf5dVFU0FudwDDw5+T\niyMQ51x0WjeoygdX9+D8bk14btJS+j48kVmrNkUdlitiOT4ElvQFMBM4GXgt63ozu7poQ/tf/hDY\nuaI3fkEGfxk5g/Vbd3Lt8c259MgDSU/zR4DJLD8PgfsCnwC/ANOymZxzJVDP5rUZO6QnvVrtz/1j\n53PGk1NYvn5b1GG5IhBPNUAlWloAABTgSURBVNC28XTbXBz8DsC54mNmvDv9B255dxZ79xq39WvJ\noE4N8AqBySff1UAT5cPfOVe8JHFy+/qMGdKTNg2qccObM7nkxWms31oY40G5RBBPNVDnXAqrX608\nL190OP930qF8Nj+DXsPG8/HcNVGH5QqBJwDnXJ7S0sRFPZry3lVHUKtSWS58fio3vfUdP+/YHXVo\nrgDiGRBmj6R7Y1sCS/qmaMNyziWiQ/avwrtXHsElRzblta+Xc9LwCXyz/Keow3L5FM8dwOxwu/9K\nqhEu86dAzqWosqXSuanPobw2uAu79hiDnpjCg/+dz649hdZLvCsm8SSA3WZ2A/AMMEFSR8KxAZxz\nqevwpjUZM6QHJ7erz/BPFnHa45P5PmNr1GG5fRBPAhCAmb0OnAE8BzQtyqCcc8mhcrnS/Ov0tjx+\ndgeWbdjGicMn8OIXy8irerlLDPEkgKsyZ8xsFtADKPZWwM65xNWndV3GDunJYQfU5JZ3ZnHBf75m\n7ZbtUYfl8hDviGC/MrNNwJCiCcc5l6zqVCnH83/qzB39WzJ58Xp6D5vA2Nk/Rh2Wy0VuvYHuH5b3\nl5fUXlKHcDoKqFBsETrnkoYkzuvWhA+u7k69auW45MVp/GXEDLZ6ddGEVCqXdb2A84EGwIMxy7cA\nfyvISSXdDgwGMsJFfzOzDwtyTOdc4jhov8q8ddkRDP94IY99togvlqxn6Ont6NSkRt47u2ITT19A\np5rZm4V60iABbDWzB/ZlP+8LyLnkM3XpBq59YzqrfvqFy446kGuObU6ZUt4GtTjl1BdQjncAkv6c\n3XwmM3sw6zLnnMuqU5MajL6mJ38fNZtHP13M5wsyGHZGOw7ar3LUoaW83NJw5TymgrpS0kxJz0qq\nntNGki6WNFXS1IyMjJw2c84lsEplS3HfaW154pyO/LBxOycNn8jzk5d6ddGI5VkElO8DS+OA/bNZ\ndTPwBbCOoEHZnUBdM7sgr2N6EZBzyW/tlu3cMHImn83PoEezWjwwqC11qpSLOqwSLacioHieATQA\nHgYyxwieAFxjZisLKbAmwPtm1iqvbT0BOFcymBkvf7mcuz6YQ7nS6fzjlNac2Lpu1GGVWPkeD4Cg\n5e97QL1wGhUuK0gwse/0KcCsghzPOZdcJHFOl8Z8eHUPGteowOUvf8Of35jO5u27og4tpcSTAGqb\n2XNmtjuc/gPULuB575P0naSZwNHAtQU8nnMuCTWtXYmRl3Xj6mOb8c63q+gzbAJfLdkQdVgpI54E\nsF7SOZLSw+kcYH1BTmpm55pZazNrY2b9zWx1QY7nnEtepdPT+PPxzRl5WTdKpYsznprCvaPnsWP3\nnqhDK/HiSQAXAKcDPwKrgdOAPxVlUM651NOhUXU+vLoHZ3ZuyBOfL+bkRyezYM2WqMMq0YqsFlBR\n8IfAzqWGj+as4cY3Z7Jlx27+2vsQ/tStCWlpPgxJfu1zQ7CYHWsTdNvQJHb7eKptOudcfhzfog7t\nG/Xkxjdncuf7c/hk3hoeGNSWulXLRx1aiRJPEdC7QFVgHPBBzOScc0WmVqWyPP3HTtwzsDXfLNtI\nr6HjGTXjh6jDKlHyvAMAKpjZX4s8Euecy0ISZx3WiK5NazLk9elc9eq3jJu7hr8PaEXV8qWjDi/p\nxXMH8L6kE4s8Euecy0GTWhUZeWlXrj2uOe/PXE2fYeOZvHhd1GElvdzGA9giaTNwDUES+EXS5pjl\nzjlXbEqlp3HNcc1487JulC2dztnPfMndH8zx6qIFkGMCMLPKZlYl/JlmZuVjXlcpziCdcy5Tu4bV\n+ODq7px9eCOenrCEAY9MYt6P/p00P+LqlFtSG0n9JQ3MnIo6MOecy0mFMqW46+TWPHd+Z9Zt3Un/\nhyfx9Pjv2bs3eaq1J4I8E4CkZ4FngVOBfuHUt4jjcs65PB19yH6MHdKDow6uzd0fzuUPz3zBqo2/\nRB1W0oinN9A5ZtaimOLJlTcEc85lx8wYMXUld4yaTVqauHNAKwa0q4fkjcegYL2BTpGUEAnAOeey\nI4nTOzdk9DU9aV6n8q9VRjdt895FcxNPAniBIAnMD0fwyuzF0znnEkqjmhV445Ku/KXXwYyZ9SO9\nho1n0iKvLpqTeBLAv4Fzgd78Vv7fryiDcs65/EpPE1ccfRBvX34EFcsG1UX/PmoO23d5ddGs4kkA\nGWb2npktMbNlmVORR+accwXQukFV3r+qB+d1bcyzk5bQ7+GJzP5hU9RhJZR4EsC3kl6RdJZXA3XO\nJZPyZdK5Y0Arnr/gMDb9souTH53E458tZo9XFwXiSwDlgR3ACXg1UOdcEjqyeW3GDunJcYfW4Z9j\n5nHWU1+wYsO2qMOKnI8H4JxLGWbGW9+s4rb3ZgNwR/+WDOxQv8RXF93n8QAkPQzkmB3M7OpCis05\n54qFJE7t2IDDDqjBdW/M4LoRMxg3dw3/OKU11SuWiTq8Ypdbd9BF+lVb0lXAFcAe4AMzu6Eoz+ec\nc5ka1qjAqxd34anx3/PgR/OZtuwn7h/UliOb1446tGKVYwIws+djX0uqYGaFUmgm6WhgANDWzHZI\n2q8wjuucc/FKTxOXHXUgPZrV4trXp3Pes19xXtfG3NjnUMqXSY86vGIRT19AXSXNAeaFr9tKeqyA\n570MuNfMdgCY2doCHs855/KlVf2qjLqqOxcccQDPT1lG34cnMGtValQXjacW0DCgF7AewMxmAD0L\neN7mQA9JX0r6XFLnAh7POefyrVzpdG7t14KXLjycn3fs4eRHJ/Hop4tKfHXRuLqDNrMVWRbl2aRO\n0jhJs7KZBhAUPdUAugB/Ad5QDo/hJV0saaqkqRkZGfGE65xz+dK9WS3GDOlB71b7c//Y+Zz+5BSW\nry+51UXjSQArJHUDTFJpSdcDc/PaycyOM7NW2UzvAiuBtyzwFbAXqJXDcZ4ys05m1ql27dR6QOOc\nK37VKpTh4bPaM+yMdixYs4U+D43nja9XkExV5uMVTwK4lKC2Tn1gFdAufF0Q7wBHA0hqDpQBvMcm\n51xCkMTJ7eszZkhPWjeoyg1vzuSSF6exfuuOqEMrVJE0BJNUhmCQmXbATuB6M/skr/28IZhzrrjt\n3Ws8M/F7Hhi7gCrlS3P/aW04+pDkqriYU0OwHBNAIjYE8wTgnIvK3NWbGfLadOav2cI5XRrxtxMP\npUKZ3JpSJY78DAgzFZgWTv1j5jMn55xLGYfWrcK7Vx7B4B4H8PKXy+k7fCIzVmyMOqwCiasISNK3\nZta+GOLJld8BOOcSweTF67j+jRms2bKDq49pxhVHH0ip9LgqVUaiIENCQi5FQc45l2q6HViL0UN6\n0rdNXYaOW8CgJ6ewdN3PUYe1zxI3ZTnnXAKrWr40D53ZnuFntWfx2q2cOHwCr361PKmqi+aYACRt\nkbRZ0magTeZ85vJijNE55xJW/7b1GHttT9o1rMZNb33H4BemsS5JqovmmADMrLKZVQmnUjHzlc2s\nSnEG6Zxziaxu1fK8dOHh3NK3BeMXZtBr6HjGzVkTdVh58iIg55wrBGlp4sLuBzDqyu7sV6UcF70w\nlZvemsnPO3ZHHVqOPAE451whOnj/yrxzRTcuObIpr329gpOGT+Cb5T9FHVa2PAE451whK1sqnZv6\nHMprg7uwa48x6IkpPPjRAnbt2Rt1aL/jCcA554rI4U1rMnpIDwa0rcfwjxdy2uOT+T5ja9Rh/coT\ngHPOFaEq5Urz4BntePQPHVi2YRsnDp/AS18sS4jqop4AnHOuGJzUpi5jh/Skc5Ma/N87s7jgP1+z\ndsv2SGPyBOCcc8WkTpVyPP+nw7i9XwsmL15P72ETGDv7x8ji8QTgnHPFKC1NnH/EAXxwdXfqVi3H\nJS9O44aRM9gaQXVRTwDOOReBg/arzNuXH8HlRx3IyGkr6fPQeKYu3VCsMXgCcM65iJQplcYNvQ/h\n9Uu6AnD6k1O4f+w8du4unuqingCccy5inZvU4MOre3BqhwY8+uliBj4+iUVrtxT5eT0BOOdcAqhc\nrjT3D2rLE+d0YNVPv3DS8Ik8P3lpkVYX9QTgnHMJpHeroLpo1wNrctt7sznvua9Zs7loqot6AnDO\nuQSzX5VyPHd+Z+4c0JKvlqyn17DxTFm8vtDPE8mIxpJeBw4OX1YDNppZuyhicc65RCSJc7s2odtB\ntbj9vdk0qVWh0M8RSQIwszMy5yX9C9gURRzOOZfoDqxdiRcvPLxIjh1JAsgkScDpwDFRxuGcc6ko\n6mcAPYA1ZrYwpw0kXSxpqqSpGRkZxRiac86VbEV2ByBpHLB/NqtuNrN3w/mzgFdzO46ZPQU8BdCp\nU6fou89zzrkSosgSgJkdl9t6SaWAgUDHoorBOedczqIsAjoOmGdmKyOMwTnnUlaUCeBM8ij+cc45\nV3QiqwVkZudHdW7nnHPR1wJyzjkXESXCuJTxkpQBLMvn7rWAdYUYTpT8WhJPSbkO8GtJVAW5lsZm\nVjvrwqRKAAUhaaqZdYo6jsLg15J4Ssp1gF9LoiqKa/EiIOecS1GeAJxzLkWlUgJ4KuoACpFfS+Ip\nKdcBfi2JqtCvJWWeATjnnPu9VLoDcM45F8MTgHPOpaiUSACSekuaL2mRpBujjicvkpZK+k7SdElT\nw2U1JH0kaWH4s3q4XJKGh9c2U1KHiGN/VtJaSbNilu1z7JLOC7dfKOm8BLqW2yWtCt+b6ZJOjFl3\nU3gt8yX1ilke6d+fpIaSPpU0R9JsSdeEy5PufcnlWpLxfSkn6StJM8JruSNcfoCkL8O4XpdUJlxe\nNny9KFzfJK9rzJOZlegJSAcWA02BMsAMoEXUceUR81KgVpZl9wE3hvM3Av8M508ERgMCugBfRhx7\nT6ADMCu/sQM1gO/Dn9XD+eoJci23A9dns22L8G+rLHBA+DeXngh/f0BdoEM4XxlYEMabdO9LLteS\njO+LgErhfGngy/D3/QZwZrj8CeCycP5y4Ilw/kzg9dyuMZ4YUuEO4DBgkZl9b2Y7gdeAARHHlB8D\ngOfD+eeBk2OWv2CBL4BqkupGESCAmY0HNmRZvK+x9wI+MrMNZvYT8BHQu+ij/70criUnA4DXzGyH\nmS0BFhH87UX+92dmq83sm3B+CzAXqE8Svi+5XEtOEvl9MTPbGr4sHU5GMELiyHB51vcl8/0aCRwr\nSeR8jXlKhQRQH1gR83oluf/BJAID/itpmqSLw2V1zGx1OP8jUCecT4br29fYE/2argyLRp7NLDYh\nSa4lLDZoT/BtM6nflyzXAkn4vkhKlzQdWEuQUBcDG81sdzZx/RpzuH4TUJMCXEsqJIBk1N3MOgB9\ngCsk9YxdacF9X1LW303m2EOPAwcC7YDVwL+iDSd+kioBbwJDzGxz7Lpke1+yuZakfF/MbI+ZtQMa\nEHxrP6Q4z58KCWAV0DDmdYNwWcIys1Xhz7XA2wR/GGsyi3bCn2vDzZPh+vY19oS9JjNbE/7T7gWe\n5rdb7YS+FkmlCT4wXzazt8LFSfm+ZHctyfq+ZDKzjcCnQFeCIrfMrvpj4/o15nB9VWA9BbiWVEgA\nXwPNwifrZQgenrwXcUw5klRRUuXMeeAEYBZBzJm1Ls4DMsdVfg/4Y1hzowuwKea2PlHsa+xjgRMk\nVQ9v5U8Il0Uuy/OVUwjeGwiu5cywpsYBQDPgKxLg7y8sJ/43MNfMHoxZlXTvS07XkqTvS21J1cL5\n8sDxBM80PgVOCzfL+r5kvl+nAZ+Ed245XWPeivOpd1QTQa2GBQTlazdHHU8esTYleKI/A5idGS9B\nWd/HwEJgHFDDfqtJ8Gh4bd8BnSKO/1WCW/BdBGWRF+YnduACgodZi4A/JdC1vBjGOjP8x6sbs/3N\n4bXMB/okyt8f0J2geGcmMD2cTkzG9yWXa0nG96UN8G0Y8yzg1nB5U4IP8EXACKBsuLxc+HpRuL5p\nXteY1+RdQTjnXIpKhSIg55xz2fAE4JxzKcoTgHPOpShPAM45l6I8ATjnXIryBOCSgqSaMT09/pil\n58cyUccXS9JRkroV4fHLS/o87EagiX7fW+ngsAuR6pIekHRMUcXhkl+pvDdxLnpmtp6gmT+Sbge2\nmtkDUcUjqZT91l9LVkcBW4HJhXS8rC4A3jKzPUG7qF+PcS5wFXCMmf0k6WGCVrGfxBuHSy1+B+CS\nlqSO4TfhaZLGxnRr8JmkoZKmSporqbOktxT0YX9XuE0TSfMkvRxuM1JShTiOO0zBGA3XSOqnoF/2\nbyWNk1Qn7KDsUuDa8O6kh6T/SDotJu6t4c+jJE2Q9B4wJ/xGf7+krxV0anZJDpd+Nr+1Ds085ukE\nXTqfYGbrAMxsGVBT0v6F9Ct3JYwnAJesBDwMnGZmHYFngbtj1u80s04E/am/C1wBtALOl1Qz3OZg\n4DEzOxTYDFwe9jOT23HLmFknM/sXMBHoYmbtCboTvsHMlobnHGpm7cxsQh7X0QG4xsyaE7Q03mRm\nnYHOwOCwaf9vFx0UdzUNz5OpMfAIwYf/j1mO/w1wRB4xuBTlRUAuWZUl+ED/KCwGSSfotiFTZr8u\n3wGzLewfSdL3BB1nbQRWmNmkcLuXgKuBMXkc9/WY+QbA6+EdQhlgST6u4ysL+nCHoG+dNjF3C1UJ\n+nWJPW6tMPZYGQTjFpwODM2ybi1QLx9xuRTgCcAlKxF8sHfNYf2O8OfemPnM15l/91n7QbE4jvtz\nzPzDwINm9p6kowhGpcrObsK7bUlpBMkiu+MJuMrMcutg7ReCPmFibSPo12aCpLVm9nLMunLhPs79\nDy8CcslqB1BbUlcIugiW1HIfj9Eoc3/gDwRFOvP34bhV+a3b3djxcbcQDFeYaSnQMZzvTzDyU3bG\nApeFxVBIaq6gR9hfWTASV7qkclmWryUYnesf+v2YsM35rWdM537HE4BLVnsJusT9p6QZBL1C7mvV\ny/kEA+7MJRjj9nELhgeM97i3AyMkTQPWxSwfBZyS+RCYoCbOkeHxuvL7b/2xngHmAN+EVTufJPu7\n9P8S9Ir5O2FRUn/gWUmHhYnkIGBqTr8Al9q8N1CXksLaOu+bWauIQ9lnkjoA15rZuXlsdwrBAOq3\nFE9kLtn4HYBzScaCQdE/lZSex6alSJKhEV00/A7AOedSlN8BOOdcivIE4JxzKcoTgHPOpShPAM45\nl6I8ATjnXIr6f95PGtU586elAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ptot = 1. + 2.*np.exp(-theta/T) + 3.*np.exp(-2.*theta/T) # partition function\n",
    "A = -k*T*np.log(Ptot) # J/particle\n",
    "plt.plot(T,A)\n",
    "plt.xlabel('Temperature (K)')\n",
    "plt.ylabel('Helmholtz free energy (J/particle)')\n",
    "plt.title('Helmholtz free energy')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "QYMyQORvx-fz"
   },
   "source": [
    "### 10. Derive an expression for the entropy $S$ per molecules and plot vs. temperature, again assuming $\\theta = 300$ K."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "K32kxVmh7ZLw"
   },
   "source": [
    "$S = \\frac{U-A}{T}= \\frac{U}{T} + k_B\\ln q = \\frac{2\\varepsilon e^{-\\beta\\varepsilon} + 6\\varepsilon e^{-2\\beta\\varepsilon}}{(1 + 2e^{-\\beta\\varepsilon} + 3e^{-2\\beta\\varepsilon})T} + k_B\\ln(1 + 2e^{-\\beta\\varepsilon} + 3e^{-2\\beta\\varepsilon})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 295
    },
    "colab_type": "code",
    "id": "aDHcGo1X7b_i",
    "outputId": "6fc2ebb4-1e2e-4a31-8853-327c6baad2fe"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "S = (U-A)/T\n",
    "\n",
    "plt.plot(T,S)\n",
    "plt.xlabel('Temperature (K)')\n",
    "plt.ylabel('Entropy (J/K)')\n",
    "plt.title('Three-state system entropy')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "38CO030Lx-f1"
   },
   "source": [
    "### 11. In class we took the First Law as a postulate and demonstrated the Second Law. Look at your results for Problems 6 and 10. Can you use them to rationalize the Third Law? Explain your answer."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "hZvGKNyi7k6-"
   },
   "source": [
    "Yes. The entropy at absolute zero is equal to zero. From problem 6, we can get when T = 0 K, the molecule has the probability of 1 to be in the 0 energy state. $S(T = 0 K) = k_Bln(1) = 0$. From number 10, we can also get $S \\rightarrow 0$ when $T\\rightarrow 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "w5Ac5b7i7lid"
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "colab": {
   "name": "PChem HW 9 Solutions",
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}