\( \newcommand{\tx}[1]{\text{#1}} % text in math mode\) \( \newcommand{\subs}[1]{_{\text{#1}}} % subscript text\) \( \newcommand{\sups}[1]{^{\text{#1}}} % superscript text\) \( \newcommand{\st}{^\circ} % standard state symbol\) \( \newcommand{\id}{^{\text{id}}} % ideal\) \( \newcommand{\rf}{^{\text{ref}}} % reference state\) \( \newcommand{\units}[1]{\mbox{$\thinspace$#1}}\) \( \newcommand{\K}{\units{K}} % kelvins\) \( \newcommand{\degC}{^\circ\text{C}} % degrees Celsius\) \( \newcommand{\br}{\units{bar}} % bar (\bar is already defined)\) \( \newcommand{\Pa}{\units{Pa}}\) \( \newcommand{\mol}{\units{mol}} % mole\) \( \newcommand{\V}{\units{V}} % volts\) \( \newcommand{\timesten}[1]{\mbox{$\,\times\,10^{#1}$}}\) \( \newcommand{\per}{^{-1}} % minus one power\) \( \newcommand{\m}{_{\text{m}}} % subscript m for molar quantity\) \( \newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V\) \( \newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p\) \( \newcommand{\kT}{\kappa_T} % isothermal compressibility\) \( \newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A\) \( \newcommand{\B}{_{\text{B}}} % subscript B for solute or state B\) \( \newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point\) \( \newcommand{\C}{_{\text{C}}} % subscript C\) \( \newcommand{\f}{_{\text{f}}} % subscript f for freezing point\) \( \newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)\) \( \newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)\) \( \newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)\) \( \newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)\) \( \newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)\) \( \newcommand{\xbB}{_{x,\text{B}}} % x basis, B\) \( \newcommand{\xbC}{_{x,\text{C}}} % x basis, C\) \( \newcommand{\cbB}{_{c,\text{B}}} % c basis, B\) \( \newcommand{\mbB}{_{m,\text{B}}} % m basis, B\) \( \newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i\) \( \newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B\) \( \newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces\) \( \newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces\) \( \newcommand{\ra}{\rightarrow} % right arrow (can be used in text mode)\) \( \newcommand{\eq}{\subs{eq}} % equilibrium state\) \( \newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation\) \( \newcommand{\sys}{\subs{sys}} % system property\) \( \newcommand{\sur}{\sups{sur}} % surroundings\) \( \renewcommand{\in}{\sups{int}} % internal\) \( \newcommand{\lab}{\subs{lab}} % lab frame\) \( \newcommand{\cm}{\subs{cm}} % center of mass\) \( \newcommand{\rev}{\subs{rev}} % reversible\) \( \newcommand{\irr}{\subs{irr}} % irreversible\) \( \newcommand{\fric}{\subs{fric}} % friction\) \( \newcommand{\diss}{\subs{diss}} % dissipation\) \( \newcommand{\el}{\subs{el}} % electrical\) \( \newcommand{\cell}{\subs{cell}} % cell\) \( \newcommand{\As}{A\subs{s}} % surface area\) \( \newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)\) \( \newcommand{\allni}{\{n_i \}} % set of all n_i\) \( \newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}\) \( \newcommand{\solmB}{\tx{(sol,$\,$$m\B$)}}\) \( \newcommand{\dil}{\tx{(dil)}}\) \( \newcommand{\sln}{\tx{(sln)}}\) \( \newcommand{\mix}{\tx{(mix)}}\) \( \newcommand{\rxn}{\tx{(rxn)}}\) \( \newcommand{\expt}{\tx{(expt)}}\) \( \newcommand{\solid}{\tx{(s)}}\) \( \newcommand{\liquid}{\tx{(l)}}\) \( \newcommand{\gas}{\tx{(g)}}\) \( \newcommand{\pha}{\alpha} % phase alpha\) \( \newcommand{\phb}{\beta} % phase beta\) \( \newcommand{\phg}{\gamma} % phase gamma\) \( \newcommand{\aph}{^{\alpha}} % alpha phase superscript\) \( \newcommand{\bph}{^{\beta}} % beta phase superscript\) \( \newcommand{\gph}{^{\gamma}} % gamma phase superscript\) \( \newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript\) \( \newcommand{\bphp}{^{\beta'}} % beta prime phase superscript\) \( \newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript\) \( \newcommand{\apht}{\small\aph} % alpha phase tiny superscript\) \( \newcommand{\bpht}{\small\bph} % beta phase tiny superscript\) \( \newcommand{\gpht}{\small\gph} % gamma phase tiny superscript\) \( \newcommand{\upOmega}{\Omega}\) \( \newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space\) \( \newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space\) \( \newcommand{\df}{\dif\hspace{0.05em} f} % df\) \(\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential \) \( \newcommand{\dq}{\dBar q} % heat differential\) \( \newcommand{\dw}{\dBar w} % work differential\) \( \newcommand{\dQ}{\dBar Q} % infinitesimal charge\) \( \newcommand{\dx}{\dif\hspace{0.05em} x} % dx\) \( \newcommand{\dt}{\dif\hspace{0.05em} t} % dt\) \( \newcommand{\difp}{\dif\hspace{0.05em} p} % dp\) \( \newcommand{\Del}{\Delta}\) \( \newcommand{\Delsub}[1]{\Delta_{\text{#1}}}\) \( \newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line\) \( \newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up\) \( \newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}\) \( \newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}\) \( \newcommand{\dotprod}{\small\bullet}\) \( \newcommand{\fug}{f} % fugacity\) \( \newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general\) \( \newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)\) \( \newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential\) \( \newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential\) \( \newcommand{\Ej}{E\subs{j}} % liquid junction potential\) \( \newcommand{\mue}{\mu\subs{e}} % electron chemical potential\) \( \newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol\) \( \newcommand{\D}{\displaystyle} % for a line in built-up\) \( \newcommand{\s}{\smash[b]} % use in equations with conditions of validity\) \( \newcommand{\cond}[1]{\\[-2.5pt]{}\tag*{#1}}\) \( \newcommand{\nextcond}[1]{\\[-5pt]{}\tag*{#1}}\) \( \newcommand{\R}{8.3145\units{J$\,$K$\per\,$mol$\per$}} % gas constant value\) \( \newcommand{\Rsix}{8.31447\units{J$\,$K$\per\,$mol$\per$}} % gas constant value - 6 sig figs\) \( \newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt} \) \( \newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt} \) \( \newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt} \) Recall that \(\g_{\pm}\) is the mean ionic activity coefficient of a strong electrolyte, or the stoichiometric activity coefficient of an electrolyte that does not dissociate completely. The general procedure described in this section for evaluating \(\g_{\pm}\) requires knowledge of the osmotic coefficient \(\phi_m\) as a function of molality. \(\phi_m\) is commonly evaluated by the isopiestic method (Sec. 9.6.4) or from measurements of freezing-point depression (Sec. 12.2). The osmotic coefficient of a binary solution of an electrolyte is defined by \begin{gather} \s{ \phi_m \defn \frac{\mu\A^*-\mu\A}{RTM\A\nu m\B} } \tag{10.6.1} \cond{(binary electrolyte solution)} \end{gather} That is, for an electrolyte the sum \(\sum_{i\neq \tx{A}}m_i\) appearing in the definition of \(\phi_m\) for a nonelectrolyte solution (Eq. 9.6.11) is replaced by \(\nu m\B\), the sum of the ion molalities assuming complete dissociation. It will now be shown that \(\phi_m\) defined this way can be used to evaluate \(\g_{\pm}\). The derivation is like that described in Sec. 9.6.3 for a binary solution of a nonelectrolyte. Solving Eq. 10.6.1 for \(\mu\A\) and taking the differential of \(\mu\A\) at constant \(T\) and \(p\), we obtain \begin{equation} \dif\mu\A = -RTM\A\nu(\phi_m\dif m\B + m\B\dif\phi_m) \tag{10.6.2} \end{equation} From Eq. 10.3.9, we obtain \begin{equation} \dif\mu\B = RT\nu\left(\dif\ln\g_{\pm} + \frac{\dif m\B}{m\B}\right) \tag{10.6.3} \end{equation} Substitution of these expressions in the Gibbs–Duhem equation \(n\A \dif\mu\A + n\B \dif\mu\B = 0\), together with the substitution \(n\A M\A = n\B/m\B\), yields \begin{equation} \dif\ln\g_{\pm} = \dif\phi_m + \frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.4} \end{equation} Then integration from \(m\B = 0\) to any desired molality \(m'\B\) gives the result \begin{equation} \ln\g_{\pm}(m'\B) = \phi_m (m'\B) - 1 + \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.5} \end{equation} The right side of this equation is the same expression as derived for \(\ln\g\mbB\) for a nonelectrolyte (Eq. 9.6.20). The integrand of the integral on the right side of Eq. 10.6.5 approaches \(-\infty\) as \(m\B\) approaches zero, making it difficult to evaluate the integral by numerical integration starting at \(m\B = 0\). (This difficulty does not exist when the solute is a nonelectrolyte.) Instead, we can split the integral into two parts \begin{equation} \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.6} \end{equation} where the integration limit \(m''\B\) is a low molality at which the value of \(\phi_m\) is available and at which \(\g_{\pm}\) can either be measured or estimated from the Debye–Hückel equation. We next rewrite Eq. 10.6.5 with \(m\B'\) replaced with \(m\B''\): \begin{equation} \ln\g_{\pm}(m''\B) = \phi_m (m''\B) - 1 + \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.7} \end{equation} By eliminating the integral with an upper limit of \(m\B''\) from Eqs. 10.6.6 and 10.6.7, we obtain \begin{equation} \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \ln\g_{\pm}(m''\B) - \phi_m (m''\B) + 1 + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.8} \end{equation} Equation 10.6.5 becomes \begin{equation} \ln\g_{\pm}(m'\B) = \phi_m(m'\B) - \phi_m(m''\B) + \ln\g_{\pm}(m''\B) + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.9} \end{equation} The integral on the right side of this equation can easily be evaluated by numerical integration.