correct thermodynamic description; previous equation superscript should have been residual (not excess) property

Author: yoelcortes

docs/tutorial/Thermodynamics.ipynb

@@ -19,8 +19,8 @@
     "\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",
-    "$$ C_p^\\mathrm{mixture}(z, T, P) = \\sum_i z_i \\cdot C_{p, i}^\\mathrm{pure\\ ideal}(T, P_\\mathrm{ref}) + C_p^\\mathrm{mixture\\ excess}(z, T, P)$$\n",
-    "$$ C_p^\\mathrm{mixture}(z, T, P) = \\sum_i z_i \\cdot \\bigg (C_{p, i}^\\mathrm{pure}(T, P_\\mathrm{ref})  - C_{p, i}^\\mathrm{pure\\ excess}(T, P_{ref}) \\bigg ) + C_p^\\mathrm{mixture\\ excess}(z, T, P)$$\n",
+    "$$ C_p^\\mathrm{mixture}(z, T, P) = \\sum_i z_i \\cdot C_{p, i}^\\mathrm{pure}(T, P_\\mathrm{ref}) + C_p^\\mathrm{mixture\\ excess}(z, T, P)$$\n",
+    "$$ C_p^\\mathrm{mixture\\ excess}(z, T, P) = C_{p}^\\mathrm{mixture\\ residual}(T, P) - \\sum_i z_i \\cdot C_{p, i}^\\mathrm{pure\\ residual}(T, P) $$\n",
     "\n",
     "- $C_p$ - Heat capacity [J/K/mol].\n",
     "- $z$ - Molar fraction [-].\n",
@@ -29,7 +29,7 @@
     "- $P_{\\mathrm{ref}}$ - Reference pressure [Pa].\n",
     "- $i$ - Arbitrary chemical.\n",
     "\n",
-    "BioSTEAM uses the most rigorous correlations available within the database to estimate pure component properties. While thermodynamically inconsistent, it provides more accurate estimates than using equations of state to estimate pure component properties. For thermodynamic consistency, pure component properties need to have the same equation of state. \n",
+    "BioSTEAM uses the most rigorous correlations available within the database to estimate pure component properties. While thermodynamically inconsistent, it provides more accurate estimates than using equations of state to estimate properties of both pure components and mixtures. \n",
     "\n",
     "The equation above only applies to mixture properties on a molar basis (e.g., enthalpy, heat capacity). Some mixture properties cannot be calculated through equations of state (e.g., conductivity) and are estimated by mixing rules such as a simple molar-weighted average:\n",
     "\n",