# Chapter 5: Thermodynamic Potentials

$$\newcommand{\tx}[1]{\text{#1}} % text in math mode$$
$$\newcommand{\subs}[1]{_{\text{#1}}} % subscript text$$
$$\newcommand{\sups}[1]{^{\text{#1}}} % superscript text$$
$$\newcommand{\st}{^\circ} % standard state symbol$$
$$\newcommand{\id}{^{\text{id}}} % ideal$$
$$\newcommand{\rf}{^{\text{ref}}} % reference state$$
$$\newcommand{\units}[1]{\mbox{\thinspace#1}}$$
$$\newcommand{\K}{\units{K}} % kelvins$$
$$\newcommand{\degC}{^\circ\text{C}} % degrees Celsius$$
$$\newcommand{\br}{\units{bar}} % bar (\bar is already defined)$$
$$\newcommand{\Pa}{\units{Pa}}$$
$$\newcommand{\mol}{\units{mol}} % mole$$
$$\newcommand{\V}{\units{V}} % volts$$
$$\newcommand{\timesten}[1]{\mbox{\,\times\,10^{#1}}}$$
$$\newcommand{\per}{^{-1}} % minus one power$$
$$\newcommand{\m}{_{\text{m}}} % subscript m for molar quantity$$
$$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$$
$$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$$
$$\newcommand{\kT}{\kappa_T} % isothermal compressibility$$
$$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$$
$$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$$
$$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$$
$$\newcommand{\C}{_{\text{C}}} % subscript C$$
$$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$$
$$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$$
$$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$$
$$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$$
$$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$$
$$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$$
$$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$$
$$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$$
$$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$$
$$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$$
$$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$$
$$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$$
$$\newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces$$
$$\newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces$$
$$\newcommand{\ra}{\rightarrow} % right arrow (can be used in text mode)$$
$$\newcommand{\eq}{\subs{eq}} % equilibrium state$$
$$\newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation$$
$$\newcommand{\sys}{\subs{sys}} % system property$$
$$\newcommand{\sur}{\sups{sur}} % surroundings$$
$$\renewcommand{\in}{\sups{int}} % internal$$
$$\newcommand{\lab}{\subs{lab}} % lab frame$$
$$\newcommand{\cm}{\subs{cm}} % center of mass$$
$$\newcommand{\rev}{\subs{rev}} % reversible$$
$$\newcommand{\irr}{\subs{irr}} % irreversible$$
$$\newcommand{\fric}{\subs{fric}} % friction$$
$$\newcommand{\diss}{\subs{diss}} % dissipation$$
$$\newcommand{\el}{\subs{el}} % electrical$$
$$\newcommand{\cell}{\subs{cell}} % cell$$
$$\newcommand{\As}{A\subs{s}} % surface area$$
$$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$$
$$\newcommand{\allni}{\{n_i \}} % set of all n_i$$
$$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$$
$$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$$
$$\newcommand{\dil}{\tx{(dil)}}$$
$$\newcommand{\sln}{\tx{(sln)}}$$
$$\newcommand{\mix}{\tx{(mix)}}$$
$$\newcommand{\rxn}{\tx{(rxn)}}$$
$$\newcommand{\expt}{\tx{(expt)}}$$
$$\newcommand{\solid}{\tx{(s)}}$$
$$\newcommand{\liquid}{\tx{(l)}}$$
$$\newcommand{\gas}{\tx{(g)}}$$
$$\newcommand{\pha}{\alpha} % phase alpha$$
$$\newcommand{\phb}{\beta} % phase beta$$
$$\newcommand{\phg}{\gamma} % phase gamma$$
$$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$$
$$\newcommand{\bph}{^{\beta}} % beta phase superscript$$
$$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$$
$$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$$
$$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$$
$$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$$
$$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$$
$$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$$
$$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$$

$$\newcommand{\upOmega}{\Omega}$$

$$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$$
$$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$$
$$\newcommand{\df}{\dif\hspace{0.05em} f} % df$$

$$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$$
$$\newcommand{\dq}{\dBar q} % heat differential$$
$$\newcommand{\dw}{\dBar w} % work differential$$
$$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$$
$$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$$
$$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$$
$$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$$
$$\newcommand{\Del}{\Delta}$$
$$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$$
$$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$$
$$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$$
$$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$$
$$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$$
$$\newcommand{\dotprod}{\small\bullet}$$
$$\newcommand{\fug}{f} % fugacity$$
$$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$$
$$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$$
$$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$$
$$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$$
$$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$$
$$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$$
$$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$$

$$\newcommand{\D}{\displaystyle} % for a line in built-up$$
$$\newcommand{\s}{\smash[b]} % use in equations with conditions of validity$$
$$\newcommand{\cond}[1]{\\[-2.5pt]{}\tag*{#1}}$$
$$\newcommand{\nextcond}[1]{\\[-5pt]{}\tag*{#1}}$$
$$\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$$
$$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$$

$$\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$$
$$\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$$
$$\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$$

This chapter begins with a discussion of mathematical properties of the total differential of a dependent variable. Three extensive state functions with dimensions of energy are introduced: enthalpy, Helmholtz energy, and Gibbs energy. These functions, together with internal energy, are called thermodynamic potentials. (The term thermodynamic potential should not be confused with the chemical potential, $$\mu$$, to be introduced in Sec. 5.2.) Some formal mathematical manipulations of the four thermodynamic potentials are described that lead to expressions for heat capacities, surface work, and criteria for spontaneity in closed systems.