Skip to main content
Chemistry LibreTexts

9.7: Activity of an Uncharged Species

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

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

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

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

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

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

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

    The activity \(a_i\) of uncharged species \(i\) (i.e., a substance) is defined by the relation \begin{gather} \s{ a_i \defn \exp\left(\frac{\mu_i-\mu_i\st}{RT}\right) } \tag{9.7.1} \cond{(uncharged species)} \end{gather} or \begin{gather} \s{ \mu_i = \mu_i\st + RT\ln a_i } \tag{9.7.2} \cond{(uncharged species)} \end{gather} where \(\mu_i\st\) is the standard chemical potential of the species. The activity of a species in a given phase is a dimensionless quantity whose value depends on the choice of the standard state and on the intensive properties of the phase: temperature, pressure, and composition.

    Some chemists define the activity by \(\mu_i = \mu_i\rf + RT\ln a_i\). The activity defined this way is not the same as the activity used in this e-book unless the phase is at the standard pressure.

    The quantity \(a_i\) is sometimes called the relative activity of \(i\), because it depends on the chemical potential relative to a standard chemical potential. An important application of the activity concept is the definition of equilibrium constants (Sec. 11.8.1).

    For convenience in later applications, we specify that the value of \(a_i\) is the same in phases that have the same temperature, pressure, and composition but are at different elevations in a gravitational field, or are at different electric potentials. Section 9.8 10.1 will describe a modification of the defining equation \(\mu_i = \mu_i\st + RT\ln a_i\) for a system with phases of different elevations, and Sec. 10.1 will describe the modification needed for a charged species.

    9.7.1 Standard states

    The standard states of different kinds of mixture components have the same definitions as those for reference states (Table 9.3), with the additional stipulation in each case that the pressure is equal to the standard pressure \(p\st\).

    When component \(i\) is in its standard state, its chemical potential is the standard chemical potential \(\mu\st_i\). It is important to note from Eq. 9.7.2 that when \(\mu_i\) equals \(\mu_i\st\), the logarithm of \(a_i\) is zero and the activity in the standard state is therefore unity.

    The following equations in the form of Eq. 9.7.2 show the notation used in this e-book for the standard chemical potentials and activities of various kinds of uncharged mixture components: \begin{equation} \tx{Substance \(i\) in a gas mixture} \qquad \mu_i = \mu_i\st\gas + RT\ln a_i\gas \tag{9.7.3} \end{equation} \begin{equation} \tx{Substance \(i\) in a liquid or solid mixture} \qquad \mu_i = \mu_i\st + RT\ln a_i \tag{9.7.4} \end{equation} \begin{equation} \tx{Solvent A of a solution} \qquad \mu\A = \mu\A\st + RT\ln a\A \tag{9.7.5} \end{equation} \begin{equation} \tx{Solute B, mole fraction basis} \qquad \mu\B = \mu\xbB\st + RT\ln a\xbB \tag{9.7.6} \end{equation} \begin{equation} \tx{Solute B, concentration basis} \qquad \mu\B = \mu\cbB\st + RT\ln a\cbB \tag{9.7.7} \end{equation} \begin{equation} \tx{Solute B, molality basis} \qquad \mu\B = \mu\mbB\st + RT\ln a\mbB \tag{9.7.8} \end{equation}

    9.7.2 Activities and composition

    We need to be able to relate the activity of component \(i\) to the mixture composition. We can do this by finding the relation between the chemical potential of component \(i\) in its reference state and in its standard state, both at the same temperature. These two chemical potentials, \(\mu_i\rf\) and \(\mu_i\st\), are equal only if the mixture is at the standard pressure \(p\st\).

    It will be useful to define the following dimensionless quantity: \begin{equation} \G_i \defn \exp\left(\frac{\mu_i\rf-\mu_i\st}{RT}\right) \tag{9.7.9} \end{equation} The symbol \(\G_i\) for this quantity was introduced by Pitzer and Brewer (Thermodynamics, 2nd edition, McGraw-Hill, New York, 1961, p. 249). They called it the activity in a reference state. To see why, compare the definition of activity given by \(\mu_i = \mu_i\st + RT\ln a_i\) with a rearrangement of Eq. 9.7.9: \(\mu_i\rf = \mu_i\st + RT\ln\G_i\).

    At a given temperature, the difference \(\mu_i\rf-\mu_i\st\) depends only on the pressure \(p\) of the mixture, and is zero when \(p\) is equal to \(p\st\). Thus \(\G_i\) is a function of \(p\) with a value of 1 when \(p\) is equal to \(p\st\). This e-book will call \(\G_i\) the pressure factor of species \(i\).

    To understand how activity is related to composition, let us take as an example the activity \(a\mbB\) of solute B based on molality. From Eqs. 9.5.18 and 9.7.8, we have \begin{equation} \begin{split} \mu\B & = \mu\mbB\rf + RT\ln\left(\g\mbB\frac{m\B}{m\st}\right) \cr & = \mu\mbB\st + RT\ln a\mbB \end{split} \tag{9.7.10} \end{equation} The activity is then given by \begin{equation} \begin{split} \ln a\mbB & = \frac{\mu\mbB\rf-\mu\mbB\st}{RT} + \ln\left(\g\mbB\frac{m\B}{m\st}\right)\cr & = \ln\G\mbB + \ln\left(\g\mbB\frac{m\B}{m\st}\right) \end{split} \tag{9.7.11} \end{equation} \begin{equation} a\mbB = \G\mbB \g\mbB\frac{m\B}{m\st}\hspace{2.28cm} \tag{9.7.12} \end{equation} The activity of a constituent of a condensed-phase mixture is in general equal to the product of the pressure factor, the activity coefficient, and the composition variable divided by the standard composition.

    We are now able to write explicit formulas for \(\G_i\) for each kind of mixture component. They are collected in Table 9.6.

    Considering a constituent of a condensed-phase mixture, by how much is the pressure factor likely to differ from unity? If we use the values \(p\st=1\br\) and \(T=300\K\), and assume the molar volume of pure \(i\) is \(V_i^*=100\units{cm\(^3\) mol\(^{-1}\)}\) at all pressures, we find that \(\G_i\) is \(0.996\) in the limit of zero pressure, unity at \(1\br\), \(1.004\) at \(2\br\), \(1.04\) at \(10\br\), and \(1.49\) at \(100\br\). For a solution with \(V\B^{\infty}=100\units{cm\(^3\) mol\(^{-1}\)}\), we obtain the same values as these for \(\G\xbB\), \(\G\mbB\), and \(\G\cbB\). These values demonstrate that it is only at high pressures that the pressure factor differs appreciably from unity. For this reason, it is common to see expressions for activity in which this factor is omitted: \(a_i=\g_i x_i\), \(a\mbB=\g\mbB m\B/m\st\), and so on.

    In principle, we can specify any convenient value for the standard pressure \(p\st\). For a chemist making measurements at high pressures, it would be convenient to specify a value of \(p\st\) within the range of the experimental pressures, for example \(p\st=1\units{kbar}\), in order that the value of each pressure factor be close to unity.

    This page titled 9.7: Activity of an Uncharged Species is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Howard DeVoe via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.