9: Mixtures

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

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

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

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

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

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