5.5: Open Systems

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

\( \newcommand{\upOmega}{\Omega}\)

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

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

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

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

An open system of one substance in one phase, with expansion work only, has three independent variables. The total differential of \(U\) is given by Eq. 5.2.5: \begin{equation} \dif U = T \dif S - p \dif V + \mu \dif n \tag{5.5.1} \end{equation} In this open system the natural variables of \(U\) are \(S\), \(V\), and \(n\). Substituting this expression for \(\dif U\) into the expressions for \(\dif H\), \(\dif A\), and \(\dif G\) given by Eqs. 5.3.4–5.3.6, we obtain the following total differentials: \begin{equation} \dif H = T \dif S + V \difp + \mu \dif n \tag{5.5.2} \end{equation} \begin{equation} \dif A = -S \dif T - p \dif V + \mu \dif n \tag{5.5.3} \end{equation} \begin{equation} \dif G = -S \dif T + V \difp + \mu \dif n \tag{5.5.4} \end{equation} Note that these are the same as the four Gibbs equations (Eqs. 5.4.1–5.4.4) with the addition of a term \(\mu \dif n\) to allow for a change in the amount of substance.

Identification of the coefficient of the last term on the right side of each of these equations shows that the chemical potential can be equated to four different partial derivatives: \begin{equation} \mu = \Pd{U}{n}{S,V} = \Pd{H}{n}{S,p} = \Pd{A}{n}{T,V} = \Pd{G}{n}{T,p} \tag{5.5.5} \end{equation} All four of these partial derivatives must have the same value for a given state of the system; the value, of course, depends on what that state is.

The last partial derivative on the right side of Eq. 5.5.5, \(\pd{G}{n}{T,p}\), is especially interesting because it is the rate at which the Gibbs energy increases with the amount of substance added to a system whose intensive properties remain constant. Thus, \(\mu\) is revealed to be equal to \(G\m\), the molar Gibbs energy of the substance.

Suppose the system contains several substances or species in a single phase (a mixture) whose amounts can be varied independently. We again assume the only work is expansion work. Then, making use of Eq. 5.2.6, we find the total differentials of the thermodynamic potentials are given by

\begin{equation} \dif U = T \dif S - p \dif V + \sum_i\mu_i\dif n_i \tag{5.5.6} \end{equation} \begin{equation} \dif H = T \dif S + V \difp + \sum_i\mu_i\dif n_i \tag{5.5.7} \end{equation} \begin{equation} \dif A = -S \dif T - p \dif V + \sum_i\mu_i\dif n_i \tag{5.5.8} \end{equation} \begin{equation} \dif G = -S \dif T + V \difp + \sum_i\mu_i\dif n_i \tag{5.5.9} \end{equation} The independent variables on the right side of each of these equations are the natural variables of the corresponding thermodynamic potential. Section F.4 shows that all of the information contained in an algebraic expression for a state function is preserved in a Legendre transform of the function. What this means for the thermodynamic potentials is that an expression for any one of them, as a function of its natural variables, can be converted to an expression for each of the other thermodynamic potentials as a function of its natural variables.

Willard Gibbs, after whom the Gibbs energy is named, called Eqs. 5.5.6–5.5.9 the fundamental equations of thermodynamics, because from any single one of them not only the other thermodynamic potentials but also all thermal, mechanical, and chemical properties of the system can be deduced (J. Willard Gibbs, in Henry Andrews Bumstead and Ralph Gibbs Van Name, editors, The Scientific Papers of J. Willard Gibbs, Vol. I, Ox Bow Press, Woodbridge, Connecticut, 1993, p. 86). Problem 5.4 illustrates this useful application of the total differential of a thermodynamic potential.

In Eqs. 5.5.6–5.5.9, the coefficient \(\mu_i\) is the chemical potential of species \(i\). The equations show that \(\mu_i\) can be equated to four different partial derivatives, similar to the equalities shown in Eq. 5.5.5 for a pure substance: \begin{equation} \mu_i = \Pd{U}{n_i}{S,V,n_{j \ne i}} = \Pd{H}{n_i}{S,p,n_{j \ne i}} = \Pd{A}{n_i}{T,V,n_{j \ne i}} = \Pd{G}{n_i}{T,p,n_{j \ne i}} \tag{5.5.10} \end{equation} The partial derivative \(\pd{G}{n_i}{T,P,n_{j \ne i}}\) is called the partial molar Gibbs energy of species \(i\), another name for the chemical potential as will be discussed in Sec. 9.2.6.