Replace spaces with underscores in module directory names

Rename module directories (module 6 → module_6, module 7 → module_7) to
fix Google Colab "Open in Colab" link compatibility. Update Colab badge
URL in Peng_Robinson_EOS.ipynb accordingly.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

module_6/Peng_Robinson_EOS.ipynb

@@ -4,48 +4,7 @@
    "cell_type": "markdown",
    "id": "3c29e0e8",
    "metadata": {},
-   "source": [
+   "source": "[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/url/https%3A%2F%2Flem.che.udel.edu%2Fgit%2Ffurst%2Fthermohub%2Fraw%2Fbranch%2Fmain%2Fmodule_6%2FPeng_Robinson_EOS.ipynb)\n\n# Generalized Peng-Robinson Equation of State\n\nRoutines to calcualte the Generalized Peng-Robinson Equation of State\n\nSIS is Stanley I. Sandler, *Chemical, Biochemcial and Engineering Thermodynamics*, 5th ed.\n\nEric Furst  \nNovember 2025\n\nThe Generalized Peng-Robinson equation of state is\n\n$$ P = \\frac{RT}{\\underline{V}-b} - \\frac{a(T)}{\\underline{V}(\\underline{V}+b) + b(\\underline{V}-b)} \\tag{Eq. 6.4-2}$$\n\nwith\n\n$$ b = 0.07780 \\frac{RT_c}{P_c} \\tag{Eq. 6.7-2}$$\n$$ a(T) = a(T_c)\\alpha(T) = 0.45724 \\frac{R^2T_c^2}{P_c}\\alpha(T) \\tag{Eq. 6.7-1}$$\n$$ \\sqrt{\\alpha} = 1 + \\kappa \\left ( 1- \\sqrt{\\frac{T}{T_c}} \\right ) \\tag{Eq. 6.7-3}$$\n$$ \\kappa = 0.37464 + 1.54226\\omega − 0.26992\\omega^2 \\tag{Eq. 6.7-4}$$\n\nThe acentric factor $\\omega$ and the crticial temperatures and pressures are given in SIS table 6.6-1.\n\nCalculating the pressure $P$ given $\\underline{V}$ and $T$ is straightforward, but to calculate the molar volume given $P$ and $T$, we need to solve the cubic equation of state of the form\n\n$$ Z^3 + \\alpha Z^2 + \\beta Z + \\gamma = 0 \\tag{Eq. 6.4-4}$$\n\nwhere $Z$ is the compressibility factor\n\n$$ Z = \\frac{P \\underbar{V{}}}{RT} $$\n\nFor the Peng-Robinson EOS (see SIS Table 6.4-3),\n\n$$ \\alpha = -1 + B $$\n$$ \\beta = A - 3B^2 -2B $$\n$$ \\gamma = -AB + B^2 + B^3 $$\n\nand \n\n$$ A = \\frac{aP}{(RT)^2} $$\n$$ B = \\frac{bP}{RT} $$"
-    "# Generalized Peng-Robinson Equation of State\n",
-    "\n",
-    "Routines to calcualte the Generalized Peng-Robinson Equation of State\n",
-    "\n",
-    "SIS is Stanley I. Sandler, *Chemical, Biochemcial and Engineering Thermodynamics*, 5th ed.\n",
-    "\n",
-    "Eric Furst  \n",
-    "November 2025\n",
-    "\n",
-    "The Generalized Peng-Robinson equation of state is\n",
-    "\n",
-    "$$ P = \\frac{RT}{\\underline{V}-b} - \\frac{a(T)}{\\underline{V}(\\underline{V}+b) + b(\\underline{V}-b)} \\tag{Eq. 6.4-2}$$\n",
-    "\n",
-    "with\n",
-    "\n",
-    "$$ b = 0.07780 \\frac{RT_c}{P_c} \\tag{Eq. 6.7-2}$$\n",
-    "$$ a(T) = a(T_c)\\alpha(T) = 0.45724 \\frac{R^2T_c^2}{P_c}\\alpha(T) \\tag{Eq. 6.7-1}$$\n",
-    "$$ \\sqrt{\\alpha} = 1 + \\kappa \\left ( 1- \\sqrt{\\frac{T}{T_c}} \\right ) \\tag{Eq. 6.7-3}$$\n",
-    "$$ \\kappa = 0.37464 + 1.54226\\omega − 0.26992\\omega^2 \\tag{Eq. 6.7-4}$$\n",
-    "\n",
-    "The acentric factor $\\omega$ and the crticial temperatures and pressures are given in SIS table 6.6-1.\n",
-    "\n",
-    "Calculating the pressure $P$ given $\\underline{V}$ and $T$ is straightforward, but to calculate the molar volume given $P$ and $T$, we need to solve the cubic equation of state of the form\n",
-    "\n",
-    "$$ Z^3 + \\alpha Z^2 + \\beta Z + \\gamma = 0 \\tag{Eq. 6.4-4}$$\n",
-    "\n",
-    "where $Z$ is the compressibility factor\n",
-    "\n",
-    "$$ Z = \\frac{P \\underbar{V{}}}{RT} $$\n",
-    "\n",
-    "For the Peng-Robinson EOS (see SIS Table 6.4-3),\n",
-    "\n",
-    "$$ \\alpha = -1 + B $$\n",
-    "$$ \\beta = A - 3B^2 -2B $$\n",
-    "$$ \\gamma = -AB + B^2 + B^3 $$\n",
-    "\n",
-    "and \n",
-    "\n",
-    "$$ A = \\frac{aP}{(RT)^2} $$\n",
-    "$$ B = \\frac{bP}{RT} $$"
-   ]
   },
   {
    "cell_type": "code",
@@ -184,4 +143,4 @@
  },
  "nbformat": 4,
  "nbformat_minor": 5
-}
+}