cleaning up and adding the logo.svg

examples/ideal_gas_law/first_principles.ipynb

@@ -0,0 +1,100 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Ideal gas law from first principles\n",
+    "\n",
+    "The partition function, $Q$ for $N$ particles in a 3-dimensional box is, \n",
+    "\n",
+    "$$\n",
+    "Q = \\frac{V^N}{N!\\Lambda^{3N}},\n",
+    "$$\n",
+    "\n",
+    "where, $V$ is the volume of the system and $\\Lambda$ is the de Broglie thermal wavelength, \n",
+    "\n",
+    "$$\n",
+    "\\Lambda = \\bigg(\\frac{\\beta h^2}{2\\pi m}\\bigg)^{1/2}.\n",
+    "$$\n",
+    "\n",
+    "The most useful way to work with the partition function is in its logarithmic form, \n",
+    "\n",
+    "$$\n",
+    "\\ln Q = \\ln \\bigg(\\frac{V^N}{N!\\Lambda^{3N}}\\bigg) = \\ln V^N - \\ln N! - \\ln\\Lambda^{3N}.\n",
+    "$$\n",
+    "\n",
+    "Using Stirling's approximation,\n",
+    "\n",
+    "$$\n",
+    "\\ln Q = N\\ln V - (N\\ln N - N) - N\\ln\\Lambda^3,\n",
+    "$$\n",
+    "\n",
+    "and substituting the de Broglie thermal wavelength,\n",
+    "\n",
+    "$$\n",
+    "\\ln Q = N\\ln V - (N\\ln N - N) - \\frac{3}{2}N\\Bigg[\\ln \\beta + \\ln \\bigg(\\frac{\\beta h^2}{2\\pi m}\\bigg) \\Bigg].\n",
+    "$$\n",
+    "\n",
+    "The partition function can give information about thermodynamics of the system, \n",
+    "\n",
+    "$$\n",
+    "p = k_bT\\bigg(\\frac{\\partial\\ln Q}{\\partial V}\\bigg)_T,\n",
+    "$$\n",
+    "\n",
+    "This means, using the version of the partition function outlined above we can obtain the ideal gas law, \n",
+    "\n",
+    "$$\n",
+    "p = k_BT\\frac{N}{V} = \\frac{Nk_BT}{V}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "author": "Ideal gas law from first principles",
+  "kernelspec": {
+   "display_name": "Python [conda root]",
+   "language": "python",
+   "name": "conda-root-py"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.6.2"
+  },
+  "latex_envs": {
+   "LaTeX_envs_menu_present": true,
+   "autoclose": false,
+   "autocomplete": true,
+   "bibliofile": "biblio.bib",
+   "cite_by": "apalike",
+   "current_citInitial": 1,
+   "eqLabelWithNumbers": true,
+   "eqNumInitial": 1,
+   "hotkeys": {
+    "equation": "Ctrl-E",
+    "itemize": "Ctrl-I"
+   },
+   "labels_anchors": false,
+   "latex_user_defs": false,
+   "report_style_numbering": false,
+   "user_envs_cfg": false
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}