%% %% This is file `thermodynamics-examples.tex', %% generated with the docstrip utility. %% %% The original source files were: %% %% thermodynamics.dtx (with options: `example') %% %% This is a generated file. %% %% Copyright (C) 2022-2023 by Karl D. Hammond %% %% Karl D. Hammond, %% Department of Chemical Engineering %% University of Missouri %% Contact: hammondkd@missouri.edu %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3 %% of this license or (at your option) any later version. %% The latest version of this license is in %% http://www.latex-project.org/lppl.txt %% and version 1.3 or later is part of all distributions of LaTeX %% version 2005/12/01 or later. %%^^A = C_i \gammamol_i \Henrymol_i \documentclass{article} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb} \usepackage{thermodynamics} \title{Examples to Accompany the \textsf{thermodynamics} Package} \author{Karl D. Hammond} \date{} \begin{document} \NewThermodynamicProperty{B}{B} \maketitle\noindent The combined laws: \begin{align*} d\Et &= d\Ut + d\left(\frac12 m v^2\right) - d(m\phi) \\ d\Ut &= \dbar\Qt + \dbar\Wt + \Um d\Nt = \dbar\Qt - P d\Vt + \Hm d\Nt \\ &= \Partial*{\Ut}{\St}{\Vt,\allNs} d\St + \Partial*{\Ut}{\Vt}{\St,\allNs} d\Vt + \sumall_i \Partial*{\Ut}{\Nt_i}{\Vt,\St,\allNsbut{i}} d\Nt_i \\ &= T d\St - P d\Vt + \sumall_i \mu_i d\Nt_i \end{align*} With surfaces present: \begin{gather*} d\Ut = T d\St - P d\Vt + \sigma d\At + \sumall_i \mu_i d\Nt_i \\ d\Ht = T d\St + \Vt dP + \sigma d\At + \sumall_i \mu_i d\Nt_i \\ d\Ft = -\St dT - P d\Vt + \sigma d\At + \sumall_i \mu_i d\Nt_i \\ d\Gt = -\St dT + \Vt dP + \sigma d\At + \sumall_i \mu_i d\Nt_i \\ d\Lt = -\St dT - P d\Vt + \sigma d\At - \sumall_i \Nt_i d\mu_i \\ \Bt = \Ut + P\Vt - T\St - \sigma\At \\ d\Bt = -\St dT + \Vt dP - \At d\sigma + \sumall_i \mu_i d\Nt_i \\ \mu_i = \Bpm_i = \Gpm_i + \sigma \Apm_i \end{gather*} Some Maxwell reciprocity relations: \begin{gather*} \Partial*{\Vt}{T}{P,\allNs} = \PartialMixSecond*{\Gt}{T}{P}{\allNs} = \PartialMixSecond*{\Gt}{P}{T}{\allNs} = -\Partial{\St}{P}{T,\allNs} \\ \Partial*{\Gpm_i}{T}{P,\allXs} = \PartialMixSecond*{\Gt}{T}{\Nt_i}{P,\allNsbut{i}} = \PartialMixSecond*{\Gt}{\Nt_i}{T}{P,\allNsbut{i}} = -\Partial*{\St}{\Nt_i}{T,P,\allNsbut{i}} = -\Spm_i \end{gather*} Temporary changes of derivative delimiters: \begin{gather*} \begin{thermobrackets} \Partial*{\Vt}{T}{P,\allNs} \end{thermobrackets} \begin{thermobar} = \PartialMixSecond*{\Gt}{T}{P}{\allNs} \end{thermobar} \begin{thermomolesrange} = \PartialMixSecond*{\Gt}{P}{T}{\allNs} \end{thermomolesrange} \begin{thermobraces} = -\Partial*{\St}{P}{T,\allNs} \begin{thermoNOsubscripts} \begin{thermomolesrange} = -\Partial{\St}{P}{T,\allNs} \end{thermomolesrange} \end{thermoNOsubscripts} \end{thermobraces} \end{gather*} Inline derivatives: \begin{gather*} \mu_i = \Partialinline{\Gt}{\Nt_i}{T,P,\allNsbut{i}} = \Partialinline{\Ft}{\Nt_i}{T,\Vt,\allNsbut{i}} = \Partialinline{\Ht}{\Nt_i}{\St,P,\allNsbut{i}} = \Partialinline{\Ut}{\Nt_i}{\St,\Vt,\allNsbut{i}} \end{gather*} The heat capacities: \begin{gather*} \cV = T \Partial*{\Sm}{T}{\Vm,\allXs} = \Partial*{\Um}{T}{\Vm,\allXs} = -T\PartialSecond{\Fm}{T}{\Vm,\allXs} \\ \cP^\IGM = T \Partial*{\Sm^\IGM}{T}{P,\allYs} = \Partial*{\Hm^\IGM}{T}{P,\allYs} \begin{thermobrackets} = -T\PartialSecond{\Gm^\IGM}{T}{P,\allYs} \end{thermobrackets} \\ \cVt = T \Partial*{\St}{T}{\Vt,\allNs} = \Partial*{\Ut}{T}{\Vt,\allNs} \begin{thermobraces} = -T\PartialSecond{\Ft}{T}{\Vm,\allNs} \end{thermobraces} \\ \begin{thermobar} \cPt = T \Partial*{\St}{T}{P,\allNs} = \Partial*{\Ht}{T}{P,\allNs} = -T\PartialSecond{\Gt}{T}{P,\allNs} \end{thermobar} \\ \cVs = T \Partial*{\Ss}{T}{\Vs,\allWs} = \Partial*{\Us}{T}{\Vs,\allWs} \begin{thermoplain} = -T\PartialSecond{\Fs}{T}{\Vs,\allWs} \end{thermoplain} \\ \cPs = T \Partialbigg*{\Ss}{T}{P,\allWs} = \Partialbigg*{\Hs}{T}{P,\allWs} = -T\PartialSecondbigg{\Gs}{T}{P,\allWs} \\ \begin{split} \cPpm_i &= \Partial*{\cPt}{\Nt_i}{T,P,\allNsbut{i}} = T \PartialMixSecond*{\St}{\Nt_i}{T}{P,\allNsbut{i}} = T \PartialMixSecond*{\St}{T}{\Nt_i}{P,\allNsbut{i}} \\ &= T \Partial*{\Spm_i}{T}{P,\allXs} = \Partial*{\Hpm_i}{T}{P,\allXs} = \PartialMixSecond*{\Ht}{T}{\Nt_i}{P,\allNsbut{i}} = \PartialMixSecond*{\Ht}{\Nt_i}{T}{P,\allNsbut{i}} \\ &= -T\PartialSecond*{\Gpm_i}{T}{P,\allXs} = -T\Partial{{}^3 \Gt}{T^2\partial \Nt_i}{P,\allNsbut{i}} \end{split} \end{gather*} Other measurable quantities: \begin{align*} \alphaS &= \frac{1}{\Vm} \Partial{\Vm}{T}{\Sm} & \alphaP &= \frac{1}{\Vm} \Partial{\Vm}{T}{P} \\ \kappaS &= -\frac{1}{\Vm} \Partial{\Vm}{P}{\Sm} & \kappaT &= -\frac{1}{\Vm} \Partial{\Vm}{P}{T} \end{align*} The chemical potential, fugacity, and activity: \[ \mu_i = \Gpm_i = \Gm_i^\std + RT \ln a_i = \Gm_i^\std + RT\ln\left(\frac{\fmix_i}{\fstd_i}\right) \] Equilibrium in a chemical reaction: \[ \sumall_i \nu_i \mu_i = 0 \Rightarrow \exp\left(\frac{-\Delta\Gm^\std}{RT}\right) = K = \prodall_i a_i^{\nu_i} \] Partial molar quantities: \begin{align*} \Hpm{i} &= \Partial*{\Ht}{\Nt_i}{T,P,\allNsbut{i}} = \Hm + \Nt \Partial*{\Hm}{\Nt_i}{T,P,\allNsbut{i}} \\ &= \Hm + \Partial*{\Hm}{x_i}{T,P,\allXsbut{i}} - \sumallbutlast_j x_j \Partial*{\Hm}{x_j}{T,P,\allXsbut{j}} = \Partial*{\Hm}{x_i}{T,P,\allXsbut{i}} + \Hpm_\ncomponents \end{align*} \[ \Vpm_i = \Partial{\Vt}{\Nt_i}{T,P,\allNsbut{i}} \] Fugacity and related properties: \begin{gather*} \Gpm_i = \mu_i = \Gm_i^\std(T) + RT\ln a_i = \Gm_i^\std(T) + RT\ln\left(\frac{\fmix_i}{\fstd_i}\right) \\ a_i = \frac{\fmix_i}{\fstd_i} = x_i \gamma_i \exp\left(\frac{1}{RT} \int_{\Pstd}^P \Vm_i(T,p)\,dp\right) \approx x_i \gamma_i \\ \begin{split} \fmix_i &= x_i \phimix_i P = x_i \gamma_i \fpure_i = x_i \gamma_i \phipure_i P = x_i \gammarat_i \Henryrat_i = C_i \gammamol_i \Henrymol_i = x_i \gamma_i \fsat_i \exp\left(\frac{1}{RT} \int_{\Psat_i}^P \Vm_i(T,p)\,dp\right) \\ &= x_i \gamma_i \Psat_i \phisat_i \exp\left(\frac{1}{RT} \int_{\Psat_i}^P \Vm_i(T,p)\,dp\right) \approx x_i \gamma_i \Psat_i \end{split} \end{gather*} Chemical Equilibria: \begin{gather*} \Deltarxn\Hm^\std = \sumall_i \nu_i \Deltaf\Hm_i^\std \\ \Deltarxn\Gm^\std = \sumall_i \nu_i \Deltaf\Gm_i^\std = \sumall_i \nu_i \mu_i^\std \\ \Deltarxn\cP^\std = \sumall_i \nu_i \cP_i^\std \\ \mu_i = \mu_i^\std + RT\ln a_i \\ a_i = \begin{cases} \displaystyle \frac{y_i \phimix_i P}{\Pstd} \approx \frac{y_i P}{\Pstd} & \text{(gases)} \\ \rule{0pt}{5ex}% \displaystyle x_i \gamma_i \exp\left(\frac{1}{RT} \int_{\Pstd}^P \Vm_i(T,p)\,dp\right) \approx x_i \gamma_i \exp\left(\frac{\Vm_i(P-\Pstd)}{RT}\right) \approx x_i \gamma_i \approx 1 & \text{(solids, solvents)} \\ \displaystyle \rule{0pt}{5ex}% \frac{C_i \gammamol_i}{C_i^\std} \exp\left(\frac{1}{RT} \int_{\Pstd}^P \Vpm_i^\infty(T,p,\allXs)\,dp\right) \approx \frac{C_i \gammamol_i}{C_i^\std} \approx \frac{C_i}{C_i^\std} & \text{(solutes)} \end{cases} \end{gather*} Phase change properties: \begin{gather*} \Deltafus\Sm = \Sm^L - \Sm^S \\ \Deltasub\Vm = \Vm^V - \Vm^S \\ \Deltavap\Gm = \Gm^V - \Gm^L \end{gather*} Specific properties: \newcommand*{\Btilde}[2][]{\widetilde{B}_{#2}^{#1}} \[ \Btilde{j} \equiv \Partial{\Bt}{m_j}{T,\Vs,\allMsbut[i]{j}} \] and thus \[ \Btilde{i} = \frac{\Bpm{i}}{M_i} + \Biggl(\Vs - \frac{\Vpm_i}{M_i}\Biggr) \Partial*{\Bs}{\Vs}{T,\allMs} = \frac{\Bpm{i}}{M_i} + \Biggl(\Vs - \frac{\Vpm_i}{M_i}\Biggr) \Partial{\Bs}{\Vs}{T,m,\allWs} \] and \[ \Bs = \sumall_i w_i \Btilde{i}. \] Excess and Residual (Departure) Properties: \begin{align*} \HR &= \Hm - \Hm^\IG & \FR &= \Fm - \Fm^\IGM \\ \SE &= \Sm - \Sm^\IS & \VRpm_k &= \Vpm_k - \Vpm_k^\IGM \end{align*} Jacobians: \begin{gather*} \Jacobian{\St,\Vt}{T,P} = \Jacobiandet{\St,\Vt}{T,P} \\ \Jacobian{f,g,h}{x,y,z} = \Jacobiandet[\displaystyle]{f,g,h}{x,y,z} \\ \Jacobian{h_1,\dots,h_k}{z_1,\dots,z_k} = \Jacobiandet{h_1,\dots,h_k}{z_1,\dots,z_k} \\ \cP = T\Partial{\Sm}{T}{P} = T\Jacobian{\Sm,P}{T,P} = -T\Jacobian{\Sm,P}{T,\Vm} \Jacobian{T,\Vm}{T,P} = -T\Jacobian{P,\Sm}{\Vm,\Sm} \Jacobian{\Vm,\Sm}{T,\Vm} \Jacobian{T,\Vm}{T,P} = \gamma \cV \end{gather*} \end{document} \endinput %% %% End of file `thermodynamics-examples.tex'.