02-elementary_propagation_i.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Elementary Propagation\n",
    "S Singleton, J Mills, R Estler"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Background Reading\n",
    "- *An Introduction to Error Analysis*, Taylor, University Science Books, 1997, Ch 3.\n",
    "- Wikipedia [Propagation of Uncertainty](https://en.wikipedia.org/wiki/Propagation_of_uncertainty#Simplification)\n",
    "- \"Error propagation: A functional approach,\" I.F. Hughes, T.P.A. Hase, *J. Chem. Ed.*, 89, 2012, p821,  [DOI: 10.1021/ed2004627](http://pubs.acs.org/doi/abs/10.1021/ed2004627)\n",
    "- *A Student's Guide to Data and Error Analysis*, Berendsen, Cambridge U. Press, 2011, Ch 3.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task\n",
    "Apply the general uncertainty propagation equation to the measurement situation listed\n",
    "below to develop specific explicit uncertainty expressions.\n",
    "\n",
    "For example\n",
    "<center>Absorbance = absorptivity $\\times$ path length $\\times$ molarity</center>\n",
    "\n",
    "$$A = a \\times b \\times M$$\n",
    "\n",
    "The uncertainty in $A$ is $\\delta A$, and so on.\n",
    "\n",
    "\n",
    "$$\n",
    "  \\delta A = \\sqrt{\\left( \\frac{\\partial A}{\\partial a}\\right) ^2\n",
    "    _{bM} \\delta a^2 + \\left(\\frac{\\partial A}{\\partial b}\n",
    "    \\right)^2_{aM} \\delta b^2 + \\left(\\frac{\\partial A}{\\partial\n",
    "        M}\\right)^2_{ab} \\delta\n",
    "    M^2 }\n",
    "$$ \n",
    "\n",
    "Take the derivative of $A$ with respect to each variable and simplify:\n",
    "\n",
    "$$\n",
    "  = \\sqrt{(b^2 M^2 \\delta a^2 + a^2 M^2 \\delta b^2 + a^2 b^2 \\delta\n",
    "  M^2)} $$\n",
    "\n",
    "This is a symbolic manipulation, so we use `sympy`. It's safe to `import *` because we won't be doing any numerical work in this notebook. The uncertainty terms (e.g, $\\delta a$) could be included, but I chose not to. In this case, I would manually include these terms in a solution (sometimes the computer isn't the best tool for everything?)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# execute next line if running python2\n",
    "#from __future__ import division, print_function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAOBAMAAACr0JNIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqtUdu+7iTJEIs0Q\nmd0e9zMQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVQYGTWOsUvDUBCHv9dUS1/zSii4uKhQ\ncGwGce3DSYeiLm7agKMIHR2D/4CTe3QQnBRXFSJSN0kXJxcpjg6VVmyloNek3nC/7z7u4IDKrw/k\nv/alSz3tZVlrSR6PsgFuUtBzi5KV3r/tp6AurkG1g6ktjlMwZgsur+zUFt5SMHk5tgsh6OX1D0rV\nhi/+zBnixDWh02i2R3LHj/C87vPIDrjbmIB6zLfYDgOsHoAZU7a84mZ214nUEGYCkpB7ivKb9rh9\nICd/JC3qxJ+TG1TEi53skfgcONGItgdl2Aw5sVCyqltgiUPINd9ZYbXZjXGrR+cbPK95/AFQCEGg\n54PGfAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$M a b$$"
      ],
      "text/plain": [
       "M⋅a⋅b"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('a b M')\n",
    "\n",
    "A = a * b * M\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAaBAMAAADfznSWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\sqrt{M^{2} a^{2} + M^{2} b^{2} + a^{2} b^{2}}$$"
      ],
      "text/plain": [
       "   _______________________\n",
       "  ╱  2  2    2  2    2  2 \n",
       "╲╱  M ⋅a  + M ⋅b  + a ⋅b  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dA_da2 = diff(A, a)**2\n",
    "dA_db2 = diff(A, b)**2\n",
    "dA_dM2 = diff(A, M)**2\n",
    "\n",
    "sum_squares = dA_da2 + dA_db2 + dA_dM2\n",
    "sum_squares\n",
    "\n",
    "sqrt(sum_squares)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAaBAMAAADfznSWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\sqrt{M^{2} a^{2} + M^{2} b^{2} + a^{2} b^{2}}$$"
      ],
      "text/plain": [
       "   _______________________\n",
       "  ╱  2  2    2  2    2  2 \n",
       "╲╱  M ⋅a  + M ⋅b  + a ⋅b  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(sum(diff(A,x)**2 for x in ['a', 'b', 'M']))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a\n",
      "M⋅b\n",
      "b\n",
      "M⋅a\n",
      "M\n",
      "a⋅b\n"
     ]
    }
   ],
   "source": [
    "A = a*b*M\n",
    "for x in ['a', 'b', 'M']:\n",
    "    print(x)\n",
    "    pprint(diff(A, x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Your Turn\n",
    "Derive uncertainty relationships for the following expressions. Constants are noted; all other labels are variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Root-mean-square speed of a gas-phase molecule\n",
    "\n",
    "$$ \n",
    " v_{\\text{rms}} = \\sqrt{\\frac{3 RT}{M}}   \\qquad  R \\text{ constant}\n",
    " $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAAAyBAMAAABBvE9uAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\sqrt{\\frac{3 R}{4 M T} + \\frac{3 R T}{4 M^{3}}}$$"
      ],
      "text/plain": [
       "     _______________\n",
       "    ╱  3⋅R    3⋅R⋅T \n",
       "   ╱  ───── + ───── \n",
       "  ╱   4⋅M⋅T       3 \n",
       "╲╱             4⋅M  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('R T M')\n",
    "v_rms = sqrt(3*R*T/M)\n",
    "\n",
    "delta_v2 = sum(diff(v_rms, x)**2 for x in ['T', 'M'])\n",
    "sqrt(delta_v2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAA/BAMAAACvAWH7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\sqrt{3}}{2} \\sqrt{\\frac{R}{M} \\left(\\frac{1}{T} + \\frac{T}{M^{2}}\\right)}$$"
      ],
      "text/plain": [
       "             ____________\n",
       "            ╱   ⎛1   T ⎞ \n",
       "           ╱  R⋅⎜─ + ──⎟ \n",
       "          ╱     ⎜T    2⎟ \n",
       "  ___    ╱      ⎝    M ⎠ \n",
       "╲╱ 3 ⋅  ╱     ────────── \n",
       "      ╲╱          M      \n",
       "─────────────────────────\n",
       "            2            "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " # underscore is an alias for \"output from last line\"; use with caution\n",
    "_.simplify() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### van der Waals [equation of state](https://en.wikipedia.org/wiki/Van_der_Waals_equation) for real gases\n",
    " \n",
    "$$P = \\frac{RT}{V-b} - \\frac{a}{V^2} \\qquad  R \\text{ constant}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAqBAMAAACkSaOPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqZBGUTEi/gclabGf/TaArwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{R T}{V - b} - \\frac{a}{V^{2}}$$"
      ],
      "text/plain": [
       " R⋅T    a \n",
       "───── - ──\n",
       "V - b    2\n",
       "        V "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('T V b a R')\n",
    "\n",
    "P = R*T/(V-b) - (a/V**2)\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPYAAAAqBAMAAABywLuvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqu7IjJ23e9UmRCJ\nzUQBtxRlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADx0lEQVRYCe1YTWgTQRR+m58m2W26oSB4srGH\n4kFpFZSiiIt6KkJzE70Y1OpB0KBgDyqteFAUIXjz1LWC4KkBQXqRhIoVSqEiaCkSmovgQWhta9XW\nGt8kmd3Z+Wm2e+4cdt/7vu99bzudSXYC4I72amfnfjdVRf33OqsH9v1V0Q38i4Lve35QyiwD9PdI\nGRacBrgP8IyFxPjEuogRxEjDTEVCRVcBQj8lhBdKQ2QNYJcX5LJXo4reoQLEhzkxSVuwb2xFQnig\naAq0PwBXPKCQtCh6x3PyPy88DDBiCzYc0IaTg8YVDuZSVe/kury3mYbETc5DmuLDNxuq3lgXW4fI\n3oFbXodzXRd+eBFF1m8rCBfepPdCCb5Z4UVXS6JrBdhpeSF51pGX4wy6Se8bYNwFndtPvQCmzdQr\nw+6CkqKEujd21dchlqXK+n0Jt3faC8kz3N7Nhrr3eYB4DxQzHoco/rObL3MsIdu72VD2TthwsliC\nEfDMHdneM1mjmSvUtnczlbL3WYCLxTwcTVisRXIRoCN7hoXkcVjxucGqVb0jt6dHc8msdr2VVevz\nSynQD2VZTBq/nd/okhIMqO/5tZtJ3TBZrVZzRte741ddbDviZsCwOCBIGtAkkQ3SjKsJaBKwzNs8\noEnAsu3eZAYUk4d7XT7wU85TJlchSoTG7CCOOxkMTY+w5uH0lnF1hXhtLZdnH5fLtshsAQluopiu\nLfRGaUCTgGXk2aYOFxqPKJpMTlCuIZHdxDKZSoa1WqEPDVww0VP8S5PMQSiTiaRYzIrSl33BRC9p\ni3ggGNrIQ3Tooafc6M3R3ClrJqQFcLk6B3D6tx2vRP41QMeEcgA6mZMiuTxxKutBMk2BthSNmgip\nDIwNDBNpcqEvGY6Jy41nUFA7HhElO8w8m9Vj38IHqA+RGv5lFCHKvf5MBORVqYU8BDsWCmxWj30L\n51E/QGqm6oXs1eFqay2EC+IUS5P4PQ9g7lvYbUGkhAWajRduUA6ieJ6EVlwQNqeAj5de8pB/IR6w\ndVL9Hb4KJg1uPBMhxy9tFTR+yo2Ngmnxdb6FHVnYgcXRuadHeA98Ea5xMUt7hFzbmjjluEDNEl/n\nW7hg16Y8Xv+a8do0uOjkhIWEsQxpvOF4Q76LBj9hhJsPzw7c8C0s9tRWOVdeTznul1bhZUkbFvI8\nCH6FZk79gwjHDfUJXcwUjIm7zK8wtmILjhTguGPiijAreEQWhl9heFl8bmrGcWMVSjj3UCpsO4kT\n+BXq9GPcqXQDjmt3GRoZL2SnKb9CLU19xDvL/QfSKFaLQl9GngAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{R}{V - b} + \\frac{T}{V - b} - \\frac{1}{V^{2}} + \\frac{2 a}{V^{3}}$$"
      ],
      "text/plain": [
       "  R       T     1    2⋅a\n",
       "───── + ───── - ── + ───\n",
       "V - b   V - b    2     3\n",
       "                V     V "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "state_vars = [T, V, b, a, R]\n",
    "\n",
    "partials = sum(diff(P,v) for v in state_vars)\n",
    "partials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATUAAAA/BAMAAABz3ULlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\sqrt{\\left(\\frac{R}{V - b} + \\frac{T}{V - b} - \\frac{1}{V^{2}} + \\frac{2 a}{V^{3}}\\right)^{2}}$$"
      ],
      "text/plain": [
       "      _____________________________\n",
       "     ╱                           2 \n",
       "    ╱  ⎛  R       T     1    2⋅a⎞  \n",
       "   ╱   ⎜───── + ───── - ── + ───⎟  \n",
       "  ╱    ⎜V - b   V - b    2     3⎟  \n",
       "╲╱     ⎝                V     V ⎠  "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deltaP = sqrt(partials**2)\n",
    "deltaP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compressibility of an ideal gas\n",
    "$$Z = \\frac{PV}{RT} \\qquad  R \\text{ constant}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOwAAAAyBAMAAACtySp4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiRDdIrtUdpkyzUSr\nZu9x33h7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAE6UlEQVRYCb2YT4gbVRzHv5PN5s+m2aSoUOhh\nx6sg7kHwprl6SvBSKbUbFXttqErxIBkKhaUiLRQ8iJKgUGERkuLJk3tQCiWSHBSUiluoXhR0W3fd\n2qYdf2/m/Z15b3Yuu++Q+f1+7/O+v5l5b+bNL4BoK+EhNpEUWFXmIVrFziEmU6lKPWUfolW5kkx2\n7FU/GUr4BmE4CTB2bcRCkvRaR4bJmOkbhOGYHPesxDNJdKFV3UnGTN8gDMfkuGclXpdoebo7OXkB\ny+PiQxkzDZ14O7wGXH0cuPFM4rhSbgyB0z5QuqdipqURhQfUVdpkPw48k/iMBvI2aAJnaWEvOBe3\nTvxFo46wkU48ixiykXHrj4F+ABwTgdRRJ0bU+x4jnHgGUQvYyLit0WHaRT3gfvqgE9MWinR7MvAM\notxV4jRbxfvAdXytYqalEytjVFivG88gFn0pXL0LfNVE9dqZV2TMNAxiq4efqTsDzyC0t0V5fnT9\nHWA5DLfNbNIziG+C6B5n4BnEJ1ITi44HwUE814nWsepNWRnE+wpurCrbbhlEY/WWnZLRDOJbCWEQ\nKNtuGcTCzn58BqFdIXsos5tBLO762TTchL7Jj/aTgUFUHu6TFW6i3pNjL4UnpG03TKK+aadU1E2k\nN3k16gAt7bE9wCwp6dQmnyKAQssStIVyg1CbvE0njuX+yMsFvsxEj7uzyZ5caozOBX7BSG2TZ661\n5VJjI/OApTkjh+xnn5ZHLZLIAz77IqH6Ju9Mnkctd9ruZR/wusQ3XKXXdiQm752TExulOD8nyAS3\nKKW2yccpUr/exsbnf2xsBKmOZCAv2G7SR19ysM0XF2HrM2K5QPbNq23yJOBdvKBkVOGSVlN9jO+I\nQSlQFxT2Et1pbZOnsU/iplCAVrik1YwqSX2dp0BdUNjef4C2yVPC5zFoibxa4ZJS0/oIf1p+C6VA\nXVDYpUeykq9P55PvZvgT7THav0/CUzfvaqWNpvZLOJm8FldJxcenqGYaNXHOklYTFFwkTmdZmENu\n8o0OirsUO+vjKHAH+FArbWozcQsAgto0l3RjvSaWV1GZVd+QaTVQCQouEmdCdyA3+UGP6oErwAvA\nZvSJfste2lT3qO6hyaGyhz4b2wHqY68g06qzgxIUXCTOiKkvN/k+pXypBS9AdYb6v8A5e2lTZimp\n+qWy59PoyS/7P1jTKkHBMfGo9cfysT1N943KkJ/odUmXQ6c/tpc2i7fpXtE1ksKYnTdqhZ41rRLk\nXCQepR305CZP28KbPdSaNbpqNmXUrKVNYxMlmgjeR2sApfX1+zPGm80QZJwQp3dxR2zy1fn5NRr7\n8Zm3fAJoylylzeDGu7T+ednDyjRqRcvcGoIxx8WBpdsfROOA8j3UfwP+DkMWYC9rRyXU9/FrS/Sx\nNUDtx3k3Ouo/hmDMcXGgsjPkKJuy+NRZgKbM1WjKGoHoZGvA0QzBBFd/NOSjaMqKNBu8RVMhHPNI\nUHtThOI1IDzjaAgmuOI/AWcHs2ghxx6fMkOGO1WaWFrIvEVrQDjm0RBMcmGXw/SUeXvwY49PGe8y\nDuyxXekVeGyra3TqjiGY5NZ8jtJ/G/W9ajf2MsrcpW1acL2P+Cg5nPvawRBMck9w8FL4PXDxfOw9\nNXpwQxPQzcpoPkPlRC+KeSfDL2NLR2JbF0xz19MDDjryP1MhXqnobHi5AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{\\frac{P^{2}}{R^{2} T^{2}} + \\frac{P^{2} V^{2}}{R^{2} T^{4}} + \\frac{V^{2}}{R^{2} T^{2}}}$$"
      ],
      "text/plain": [
       "      _______________________\n",
       "     ╱    2     2  2      2  \n",
       "    ╱    P     P ⋅V      V   \n",
       "   ╱   ───── + ───── + ───── \n",
       "  ╱     2  2    2  4    2  2 \n",
       "╲╱     R ⋅T    R ⋅T    R ⋅T  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var(\"P V T R\")\n",
    "Z = P*V/(R*T)\n",
    "\n",
    "deltaZ2 = sum(diff(Z,x)**2 for x in [P,V,T])\n",
    "sqrt(deltaZ2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rotational partition function\n",
    " \n",
    "$$q_r  = \\frac{2IkT}{\\sigma h^2} \\qquad k,\\: \\sigma,\\: h\n",
    "  \\text{ constant}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWsAAAA/BAMAAADUAPQeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLdP/AQGAqMq+qpJhAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( \\sqrt{\\frac{4 I^{2} k^{2}}{h^{4} \\sigma^{2}} + \\frac{4 T^{2} k^{2}}{h^{4} \\sigma^{2}}}, \\quad 2 \\sqrt{\\frac{k^{2} \\left(I^{2} + T^{2}\\right)}{h^{4} \\sigma^{2}}}\\right )$$"
      ],
      "text/plain": [
       "⎛      ___________________          ______________⎞\n",
       "⎜     ╱    2  2      2  2          ╱  2 ⎛ 2    2⎞ ⎟\n",
       "⎜    ╱  4⋅I ⋅k    4⋅T ⋅k          ╱  k ⋅⎝I  + T ⎠ ⎟\n",
       "⎜   ╱   ─────── + ─────── , 2⋅   ╱   ──────────── ⎟\n",
       "⎜  ╱      4  2      4  2        ╱        4  2     ⎟\n",
       "⎝╲╱      h ⋅σ      h ⋅σ       ╲╱        h ⋅σ      ⎠"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('I k T sigma h')\n",
    "\n",
    "q_r = 2*I*k*T/(sigma*h**2)\n",
    "deltaq_r2 = sum(diff(q_r, x)**2 for x in [I,T])\n",
    "sqrt(deltaq_r2), sqrt(deltaq_r2).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Translational partition function\n",
    " \n",
    "$$ q_{tr} = \\left( \\frac{2 \\pi mkT}{h^2}\\right) ^{3/2} \\qquad\n",
    "  k, h \\text{ constant}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAA6BAMAAABYEUazAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 \\sqrt{2} \\pi^{\\frac{3}{2}} \\left(\\frac{T k}{h^{2}} m\\right)^{\\frac{3}{2}}$$"
      ],
      "text/plain": [
       "                    3/2\n",
       "    ___  3/2 ⎛T⋅k⋅m⎞   \n",
       "2⋅╲╱ 2 ⋅π   ⋅⎜─────⎟   \n",
       "             ⎜   2 ⎟   \n",
       "             ⎝  h  ⎠   "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# exponent is a rational number\n",
    "var('m k T h')\n",
    "\n",
    "q_tr = ( 2*pi*m*k*T/h**2 )**(S(3)/2)\n",
    "q_tr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQoAAAAyBAMAAABWu4iOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{\\frac{18 m}{h^{6}} \\pi^{3} T^{3} k^{3} + \\frac{18 T}{h^{6}} \\pi^{3} k^{3} m^{3}}$$"
      ],
      "text/plain": [
       "      _______________________________\n",
       "     ╱     3  3  3         3    3  3 \n",
       "    ╱  18⋅π ⋅T ⋅k ⋅m   18⋅π ⋅T⋅k ⋅m  \n",
       "   ╱   ───────────── + ───────────── \n",
       "  ╱           6               6      \n",
       "╲╱           h               h       "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deltaq_tr = sqrt(sum(diff(q_tr,x)**2 for x in [m,T]))\n",
    "deltaq_tr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANsAAAAVBAMAAAA9T4a3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny9nPNZY+n0r4f8BR+A0r/3X6FfA6CAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( p, \\quad v, \\quad r, \\quad t, \\quad a, \\quad b\\right )$$"
      ],
      "text/plain": [
       "(p, v, r, t, a, b)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('p,v,r,t,a,b')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAmCAMAAADjhQVFAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n71l8wFPRKQrKUsdK8g/fpBMsysrp7wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( \\frac{p v}{r t}, \\quad \\frac{v}{r t}\\right )$$"
      ],
      "text/plain": [
       "⎛p⋅v   v ⎞\n",
       "⎜───, ───⎟\n",
       "⎝r⋅t  r⋅t⎠"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# compressibility of ideal gas\n",
    "z = p*v/(r*t)\n",
    "z, diff(z, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " v \n",
      "───\n",
      "r⋅t\n",
      "\n",
      "\n",
      " p \n",
      "───\n",
      "r⋅t\n",
      "\n",
      "\n",
      "-p⋅v \n",
      "─────\n",
      "    2\n",
      " r⋅t \n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for x in [p, v, t]:\n",
    "    w = diff(z, x)\n",
    "    pprint(w), print('\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJIAAAAyCAMAAACAuXJSAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n8lEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sqrt{\\frac{p}{r t} - \\frac{p v}{r t^{2}} + \\frac{v}{r t}}$$"
      ],
      "text/plain": [
       "     __________________\n",
       "    ╱  p    p⋅v     v  \n",
       "   ╱  ─── - ──── + ─── \n",
       "  ╱   r⋅t      2   r⋅t \n",
       "╲╱          r⋅t        "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(sum([diff(z,x) for x in [p,v,t]]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAArBAMAAABGPk3oAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$- \\frac{a}{v^{2}} + \\frac{r t}{- b + v}$$"
      ],
      "text/plain": [
       "  a     r⋅t  \n",
       "- ── + ──────\n",
       "   2   -b + v\n",
       "  v          "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = r*t/(v-b) - (a/v**2)\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAesAAABMCAMAAAB6Q29xAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{\\frac{r^{2} t^{2}}{\\left(- b + v\\right)^{4}} + \\frac{r^{2}}{\\left(- b + v\\right)^{2}} + \\left(\\frac{2 a}{v^{3}} - \\frac{r t}{\\left(- b + v\\right)^{2}}\\right)^{2} + \\frac{1}{v^{4}}}$$"
      ],
      "text/plain": [
       "      _________________________________________________\n",
       "     ╱    2  2          2                       2      \n",
       "    ╱    r ⋅t          r       ⎛2⋅a      r⋅t   ⎞    1  \n",
       "   ╱   ───────── + ───────── + ⎜─── - ─────────⎟  + ── \n",
       "  ╱            4           2   ⎜  3           2⎟     4 \n",
       "╲╱     (-b + v)    (-b + v)    ⎝ v    (-b + v) ⎠    v  "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = [diff(P, x)**2 for x in (a,b,v,t)]\n",
    "sqrt(sum(w))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAyBAMAAABmEs6GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\sqrt{\\frac{3 r}{4 m t} + \\frac{3 r t}{4 m^{3}}}$$"
      ],
      "text/plain": [
       "     _______________\n",
       "    ╱  3⋅r    3⋅r⋅t \n",
       "   ╱  ───── + ───── \n",
       "  ╱   4⋅m⋅t       3 \n",
       "╲╱             4⋅m  "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m=var('m')\n",
    "vrms = sqrt(3*r*t/m)\n",
    "\n",
    "w = sum([diff(vrms, x)**2 for x in (t,m)])\n",
    "w**(S(1)/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "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": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}