update equation

Author: yoelcortes

docs/tutorial/Thermodynamics.ipynb

@@ -19,7 +19,7 @@
     "\n",
     "In BioSTEAM, mixture properties are estimated as a molar-weighted average of pure-component properties at a reference pressure (1 atm) plus the contribution of non-ideal interactions (i.e., excess properties) estimated by equations of state. For example, the heat capacity of a mixture is computed as:\n",
     "\n",
-    "$$ C_p^\\mathrm{mixture}(z, T, P) = \\sum_i z_i \\cdot C_{p, i}^\\mathrm{pure}(T, P_\\mathrm{ref}) + C_p^\\mathrm{excess}(z, T, P)$$\n",
+    "$$ C_p^\\mathrm{mixture}(z, T, P) = \\sum_i z_i \\cdot (C_{p, i}^\\mathrm{pure}(T, P_\\mathrm{ref})  - C_{p, i}^\\mathrm{pure\\ excess}(T, P_{ref})) + C_p^\\mathrm{mixture\\ excess}(z, T, P)$$\n",
     "\n",
     "- $C_p$ - Heat capacity [J/K/mol].\n",
     "- $z$ - Molar fraction [-].\n",
@@ -63,13 +63,13 @@
     "\n",
     "$$ P =\\frac{RT}{V} $$\n",
     "\n",
-    "At high pressures and/or low temperatures, molecular interactions become significant and may impact mass and energy balances within a process. In such cases, the Peng Robinson is commonly used to model hydrocarbon mixtures:\n",
+    "At high pressures and/or low temperatures, molecular interactions become significant and may impact mass and energy balances within a process. In such cases, the Peng Robinson model is commonly used to model hydrocarbon mixtures:\n",
     "\n",
     "$$ P = \\frac{RT}{V-b}-\\frac{a\\alpha(T)}{V(V+b)+b(V-b)} $$\n",
     "\n",
     "- $\\alpha$, $a$, $b$ - functions of chemical properties including the critical temperature, critical pressure, and the accentric factor.\n",
     "\n",
-    "For highly polar mixtures with multiple functional groups, an equation of state may not be enough to model chemical interactions within the liquid. In such cases, an activity model like the UNIFAC group contribution method may improve phase equilibrium calculations.   \n",
+    "For highly polar mixtures with multiple functional groups, an equation of state may not be enough to model chemical interactions within the liquid. In such cases, an activity model like the UNIFAC group contribution method may improve phase equilibrium calculations, particularly for liquid-liquid equilibrium.   \n",
     "\n",
     "$$ \\gamma = \\mathrm{UNIFAC}(x, T, P)$$\n",
     "\n",