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Chapter 12: Equilibrium Conditions in Multicomponent Systems

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 \( \newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p\)
 \( \newcommand{\kT}{\kappa_T} % isothermal compressibility\)
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 \( \newcommand{\bd}{_{\text{b}}}  % subscript b for boundary or boiling point\)
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 \( \newcommand{\f}{_{\text{f}}}  % subscript f for freezing point\)
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 \( \newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)\)
 \( \newcommand{\xbB}{_{x,\text{B}}}       % x basis, B\)
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 \( \newcommand{\cbB}{_{c,\text{B}}}       % c basis, B\)
 \( \newcommand{\mbB}{_{m,\text{B}}}       % m basis, B\)
 \( \newcommand{\kHi}{k_{\text{H},i}}      % Henry's law constant, x basis, i\)
 \( \newcommand{\kHB}{k_{\text{H,B}}}      % Henry's law constant, x basis, B\)
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 \( \newcommand{\pha}{\alpha}        % phase alpha\)
 \( \newcommand{\phb}{\beta}         % phase beta\)
 \( \newcommand{\phg}{\gamma}        % phase gamma\)
 \( \newcommand{\aph}{^{\alpha}}     % alpha phase superscript\)
 \( \newcommand{\bph}{^{\beta}}      % beta phase superscript\)
 \( \newcommand{\gph}{^{\gamma}}     % gamma phase superscript\)
 \( \newcommand{\aphp}{^{\alpha'}}   % alpha prime phase superscript\)
 \( \newcommand{\bphp}{^{\beta'}}    % beta prime phase superscript\)
 \( \newcommand{\gphp}{^{\gamma'}}   % gamma prime phase superscript\)
 \( \newcommand{\apht}{\small\aph} % alpha phase tiny superscript\)
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 \( \newcommand{\gpht}{\small\gph} % gamma phase tiny superscript\)

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

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 \( \newcommand{\Dif}{\mathop{}\!\mathrm{D}}   % roman D in math mode, preceded by space\)
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 \(\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential \)
 \( \newcommand{\dq}{\dBar q} % heat differential\)
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 \( \newcommand{\dQ}{\dBar Q} % infinitesimal charge\)
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 \( \newcommand{\fug}{f} % fugacity\)
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 \( \newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)\)
 \( \newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential\)
 \( \newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential\)
 \( \newcommand{\Ej}{E\subs{j}} % liquid junction potential\)
 \( \newcommand{\mue}{\mu\subs{e}} % electron chemical potential\)
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 \( \newcommand{\Rsix}{8.31447\units{J$\,$K$\per\,$mol$\per$}} % gas constant value - 6 sig figs\)

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This chapter applies equilibrium theory to a variety of chemical systems of more than one component.  Two different approaches will be used as appropriate: one based on the relation \(\mu_i\aph=\mu_i\bph\) for transfer equilibrium, the other based on \(\sum_i\!\nu_i\mu_i=0\) or \(K=\prod_i a_i^{\nu_i}\) for reaction equilibrium.

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