{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculus approach to data of H&H section 4.2.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Repeat the calculation of the uncertainty $\\alpha_P$ in problem 3 using the \"calculus approximation\" of the uncertainties."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "P(V_m,T) = \\frac{RT}{V_m-b} - \\frac{a}{V_m^2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\alpha_P = \\sqrt{\\left(\\alpha_P^T\\right)^2+\\left(\\alpha_P^V\\right)^2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\alpha_P^T = \\left(\\frac{\\partial P}{\\partial T}\\right)\\alpha_T = \\frac{R}{V_m-b}\\alpha_T\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\alpha_P^V = \\left(\\frac{\\partial P}{\\partial V}\\right)\\alpha_V \n",
    "= \\left[RT\\left(\\frac{-1}{(V_m-b)^2}\\right) - \\frac{a(-2)}{V_m^3}\\right]\\alpha_V \n",
    "= \\left[\\frac{-RT}{(V_m-b)^2} + \\frac{2a}{V_m^3}\\right]\\alpha_V\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = 8.3145\n",
    "Tbest = 298.0\n",
    "alpha_T = 0.2\n",
    "Vmbest = 2.000e-4\n",
    "alpha_V = 0.003e-4\n",
    "a = 1.408e-1\n",
    "b = 3.913e-5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10336.918008329707, -18162.58562642741)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha_PT = (R/(Vmbest-b))*alpha_T\n",
    "alpha_PV = ((-1*R*Tbest/(Vmbest-b)**2)+2*a/Vmbest**3)*alpha_V\n",
    "alpha_PT, alpha_PV"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'2.089812e+04'"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha_P = np.sqrt(alpha_PV**2 + alpha_PT**2)\n",
    "format(alpha_P,'e')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.020898119306488768"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# And divide by 10^6\n",
    "alpha_P/1e6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, the uncertainty in the pressue is 0.02 MPa. Basically the same thing that we got with the other approaches to Section 4.2.2. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Version information\n",
    "`version_information` is from J.R. Johansson (jrjohansson at gmail.com); see <a href='http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb'>Introduction to scientific computing with Python</a> for more information and instructions for package installation.\n",
    "\n",
    "`version_information` is installed on the linux network at Bucknell"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext version_information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.7.7 64bit [GCC 7.3.0]"
        },
        {
         "module": "IPython",
         "version": "7.16.1"
        },
        {
         "module": "OS",
         "version": "Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core"
        },
        {
         "module": "numpy",
         "version": "1.18.5"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.7.7 64bit [GCC 7.3.0]</td></tr><tr><td>IPython</td><td>7.16.1</td></tr><tr><td>OS</td><td>Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core</td></tr><tr><td>numpy</td><td>1.18.5</td></tr><tr><td colspan='2'>Fri Aug 07 11:02:07 2020 EDT</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.7.7 64bit [GCC 7.3.0] \\\\ \\hline\n",
       "IPython & 7.16.1 \\\\ \\hline\n",
       "OS & Linux 3.10.0 1062.9.1.el7.x86\\_64 x86\\_64 with centos 7.7.1908 Core \\\\ \\hline\n",
       "numpy & 1.18.5 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Fri Aug 07 11:02:07 2020 EDT} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.7.7 64bit [GCC 7.3.0]\n",
       "IPython 7.16.1\n",
       "OS Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core\n",
       "numpy 1.18.5\n",
       "Fri Aug 07 11:02:07 2020 EDT"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%version_information numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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",
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