SherbyNet - /page/wikibook/entry.mu

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Activity coefficient

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In <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a>, an activity coefficient is a factor used to account for deviation of a <a href="Mixture" title="Mixture">mixture</a> of <a href="Chemical_substance" title="Chemical substance">chemical substances</a> from ideal behaviour.<a href="#cite_note-1">[1]</a> In an <a href="Ideal_mixture" class="mw-redirect" title="Ideal mixture">ideal mixture</a>, the microscopic interactions between each pair of <a href="Chemical_species" title="Chemical species">chemical species</a> are the same (or macroscopically equivalent, the <a href="Enthalpy_change_of_solution" title="Enthalpy change of solution">enthalpy change of solution</a> and volume variation in mixing is zero) and, as a result, properties of the mixtures can be expressed directly in terms of simple <a href="Concentration" title="Concentration">concentrations</a> or <a href="Partial_pressure" title="Partial pressure">partial pressures</a> of the substances present e.g. <a href="Raoult's_law" title="Raoult's law">Raoult's law</a>. Deviations from ideality are accommodated by modifying the concentration by an activity coefficient. Analogously, expressions involving gases can be adjusted for non-ideality by scaling partial pressures by a <a href="Fugacity" title="Fugacity">fugacity</a> coefficient.

The concept of activity coefficient is closely linked to that of <a href="Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity in chemistry</a>.

<div class="mw-heading mw-heading2"><h2 id="Thermodynamic_definition">Thermodynamic definition</h2></div>

The <a href="Chemical_potential" title="Chemical potential">chemical potential</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mu _{\mathrm {B} }}</annotation>

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</math><img src="./7b31896b7b9936e15a733f25a5f1d20200201cca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.798ex; height:2.176ex;" alt="{\displaystyle \mu _{\mathrm {B} }}" loading="lazy">, of a substance B in an <a href="Ideal_mixture" class="mw-redirect" title="Ideal mixture">ideal mixture</a> of liquids or an <a href="Ideal_solution" title="Ideal solution">ideal solution</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+RT\ln x_{\mathrm {B} }\,}</annotation>

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</math><img src="./36286fe3e906945ad59ab5fee1f537b87720f9ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.875ex; height:3.176ex;" alt="{\displaystyle \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+RT\ln x_{\mathrm {B} }\,}" loading="lazy">,</dd></dl>

where μ<s>o</s> B is the chemical potential of a pure substance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} }</annotation>

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</math><img src="./93003d072991ba424a73ed1e081afe55c124b6ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle \mathrm {B} }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {B} }}</annotation>

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</math><img src="./ad6a9136d2d12bf85c92f95826f812b0839f4a87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.726ex; height:2.009ex;" alt="{\displaystyle x_{\mathrm {B} }}" loading="lazy"> is the <a href="Mole_fraction" title="Mole fraction">mole fraction</a> of the substance in the mixture.

This is generalised to include non-ideal behavior by writing

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<annotation encoding="application/x-tex">{\displaystyle \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+RT\ln a_{\mathrm {B} }\,}</annotation>

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</math><img src="./6345acbacfa90e2aca32f3deacd685c47e718643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.775ex; height:3.176ex;" alt="{\displaystyle \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+RT\ln a_{\mathrm {B} }\,}" loading="lazy"></dd></dl>

when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {B} }}</annotation>

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</math><img src="./f460e717dd02276783e773a0cdfb1eaee2a03fc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.626ex; height:2.009ex;" alt="{\displaystyle a_{\mathrm {B} }}" loading="lazy"> is the activity of the substance in the mixture,

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<annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {B} }=x_{\mathrm {B} }\gamma _{\mathrm {B} }}</annotation>

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</math><img src="./45ba36653cdbc9bbe94f6a2b5d9b43384ee5f67e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.05ex; height:2.176ex;" alt="{\displaystyle a_{\mathrm {B} }=x_{\mathrm {B} }\gamma _{\mathrm {B} }}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {B} }}</annotation>

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</math><img src="./57727b121efc343e6f430ebb55c485f3f2ca06ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.6ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {B} }}" loading="lazy"> is the activity coefficient, which may itself depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {B} }}</annotation>

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</math><img src="./ad6a9136d2d12bf85c92f95826f812b0839f4a87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.726ex; height:2.009ex;" alt="{\displaystyle x_{\mathrm {B} }}" loading="lazy">. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {B} }}</annotation>

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</math><img src="./57727b121efc343e6f430ebb55c485f3f2ca06ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.6ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {B} }}" loading="lazy"> approaches 1, the substance behaves as if it were ideal. For instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {B} }}</annotation>

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</math><img src="./57727b121efc343e6f430ebb55c485f3f2ca06ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.6ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {B} }}" loading="lazy"> ≈ 1, then <a href="Raoult's_law" title="Raoult's law">Raoult's law</a> is accurate. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {B} }}</annotation>

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</math><img src="./57727b121efc343e6f430ebb55c485f3f2ca06ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.6ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {B} }}" loading="lazy"> > 1 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {B} }}</annotation>

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</math><img src="./57727b121efc343e6f430ebb55c485f3f2ca06ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.6ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {B} }}" loading="lazy"> < 1, substance B shows positive and negative deviation from Raoult's law, respectively. A positive deviation implies that substance B is more volatile.

In many cases, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {B} }}</annotation>

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</math><img src="./ad6a9136d2d12bf85c92f95826f812b0839f4a87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.726ex; height:2.009ex;" alt="{\displaystyle x_{\mathrm {B} }}" loading="lazy"> goes to zero, the activity coefficient of substance B approaches a constant; this relationship is <a href="Henry's_law" title="Henry's law">Henry's law</a> for the solvent. These relationships are related to each other through the <a href="Gibbs%E2%80%93Duhem_equation" title="Gibbs–Duhem equation">Gibbs–Duhem equation</a>.<a href="#cite_note-2">[2]</a>

Note that in general activity coefficients are dimensionless.

In detail: <a href="Raoult's_law" title="Raoult's law">Raoult's law</a> states that the partial pressure of component B is related to its vapor pressure (saturation pressure) and its mole fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {B} }}">

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</math><img src="./ad6a9136d2d12bf85c92f95826f812b0839f4a87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.726ex; height:2.009ex;" alt="{\displaystyle x_{\mathrm {B} }}" loading="lazy"> in the liquid phase,

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<annotation encoding="application/x-tex">{\displaystyle p_{\mathrm {B} }=x_{\mathrm {B} }\gamma _{\mathrm {B} }p_{\mathrm {B} }^{\sigma }\;,}</annotation>

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</math><img src="./7fd68a3fccad9f76ef4a5b25adce483b7acc54d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:14.936ex; height:2.843ex;" alt="{\displaystyle p_{\mathrm {B} }=x_{\mathrm {B} }\gamma _{\mathrm {B} }p_{\mathrm {B} }^{\sigma }\;,}" loading="lazy"></dd></dl>

with the convention

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x_{\mathrm {B} }\to 1}\gamma _{\mathrm {B} }=1\;.}</annotation>

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</math><img src="./b3a8f08809dd27967988061c0d81ddc66cb7a8d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.054ex; height:4.176ex;" alt="{\displaystyle \lim _{x_{\mathrm {B} }\to 1}\gamma _{\mathrm {B} }=1\;.}" loading="lazy">

In other words: Pure liquids represent the ideal case.

At infinite dilution, the activity coefficient approaches its limiting value, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {B} }}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {B} }}</annotation>

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</math><img src="./57727b121efc343e6f430ebb55c485f3f2ca06ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.6ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {B} }}" loading="lazy">∞. Comparison with <a href="Henry's_law" title="Henry's law">Henry's law</a>,

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<annotation encoding="application/x-tex">{\displaystyle p_{\mathrm {B} }=K_{\mathrm {H,B} }x_{\mathrm {B} }\quad {\text{for}}\quad x_{\mathrm {B} }\to 0\;,}</annotation>

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</math><img src="./6482cb3b0b41206fc53e60218379a20c7626c06b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:29.763ex; height:2.843ex;" alt="{\displaystyle p_{\mathrm {B} }=K_{\mathrm {H,B} }x_{\mathrm {B} }\quad {\text{for}}\quad x_{\mathrm {B} }\to 0\;,}" loading="lazy"></dd></dl>

immediately gives

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<annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {H,B} }=p_{\mathrm {B} }^{\sigma }\gamma _{\mathrm {B} }^{\infty }\;.}</annotation>

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</math><img src="./0612db3181859b3df25d504033ec515dfbab8e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.17ex; height:2.843ex;" alt="{\displaystyle K_{\mathrm {H,B} }=p_{\mathrm {B} }^{\sigma }\gamma _{\mathrm {B} }^{\infty }\;.}" loading="lazy"></dd></dl>

In other words: The compound shows nonideal behavior in the dilute case.

The above definition of the activity coefficient is impractical if the compound does not exist as a pure liquid. This is often the case for electrolytes or biochemical compounds. In such cases, a different definition is used that considers infinite dilution as the ideal state:

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {B} }^{\dagger }\equiv \gamma _{\mathrm {B} }/\gamma _{\mathrm {B} }^{\infty }}</annotation>

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</math><img src="./b1ba46e67bfc8281ca64ddf9d2ef39959382d7d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.616ex; height:3.509ex;" alt="{\displaystyle \gamma _{\mathrm {B} }^{\dagger }\equiv \gamma _{\mathrm {B} }/\gamma _{\mathrm {B} }^{\infty }}" loading="lazy"></dd></dl>

with

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x_{\mathrm {B} }\to 0}\gamma _{\mathrm {B} }^{\dagger }=1\;,}</annotation>

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</math><img src="./91b5b6ba236b3dd1d254ec61f8f163144161b62b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.054ex; height:4.843ex;" alt="{\displaystyle \lim _{x_{\mathrm {B} }\to 0}\gamma _{\mathrm {B} }^{\dagger }=1\;,}" loading="lazy">

and

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<annotation encoding="application/x-tex">{\displaystyle \mu _{\mathrm {B} }=\underbrace {\mu _{\mathrm {B} }^{\ominus }+RT\ln \gamma _{\mathrm {B} }^{\infty }} _{\mu _{\mathrm {B} }^{\ominus \dagger }}+RT\ln \left(x_{\mathrm {B} }\gamma _{\mathrm {B} }^{\dagger }\right)}</annotation>

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</math><img src="./ecf43485ad2f2dddb722427e1513beba0b2967ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:37.586ex; height:8.676ex;" alt="{\displaystyle \mu _{\mathrm {B} }=\underbrace {\mu _{\mathrm {B} }^{\ominus }+RT\ln \gamma _{\mathrm {B} }^{\infty }} _{\mu _{\mathrm {B} }^{\ominus \dagger }}+RT\ln \left(x_{\mathrm {B} }\gamma _{\mathrm {B} }^{\dagger }\right)}" loading="lazy"></dd></dl>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{\dagger }}">

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</math><img src="./598d18ac9976abfe7b86f07414365168a487d612.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0.962ex; height:2.509ex;" alt="{\displaystyle ^{\dagger }}" loading="lazy"> symbol has been used here to distinguish between the two kinds of activity coefficients. Usually it is omitted, as it is clear from the context which kind is meant. But there are cases where both kinds of activity coefficients are needed and may even appear in the same equation, e.g., for solutions of salts in (water + alcohol) mixtures. This is sometimes a source of errors.

Modifying mole fractions or concentrations by activity coefficients gives the effective activities of the components, and hence allows expressions such as <a href="Raoult's_law" title="Raoult's law">Raoult's law</a> and <a href="Equilibrium_constant" title="Equilibrium constant">equilibrium constants</a> to be applied to both ideal and non-ideal mixtures.

<div class="mw-heading mw-heading3"><h3 id="Ionic_solutions">Ionic solutions</h3></div>

Knowledge of activity coefficients is particularly important in the context of <a href="Electrochemistry" title="Electrochemistry">electrochemistry</a> since the behaviour of <a href="Electrolyte" title="Electrolyte">electrolyte</a> solutions is often far from ideal, even starting at low densities due to the effects of the <a href="Ionic_atmosphere" title="Ionic atmosphere">ionic atmosphere</a>. Additionally, they are particularly important in the context of <a href="Soil_chemistry" title="Soil chemistry">soil chemistry</a> due to the low volumes of solvent and, consequently, the high concentration of <a href="Electrolytes" class="mw-redirect" title="Electrolytes">electrolytes</a>.<a href="#cite_note-3">[3]</a>

For solution of substances which ionize in solution the activity coefficients of the cation and anion cannot be experimentally determined independently of each other because solution properties depend on both ions. Single ion activity coefficients must be linked to the activity coefficient of the dissolved electrolyte as if undissociated. In this case a mean stoichiometric activity coefficient of the dissolved electrolyte, γ±, is used. It is called stoichiometric because it expresses both the deviation from the ideality of the solution and the incomplete ionic dissociation of the ionic compound which occurs especially with the increase of its concentration.

For a 1:1 electrolyte, such as <a href="Sodium_chloride" title="Sodium chloride">NaCl</a> it is given by the following:

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\pm }={\sqrt {\gamma _{+}\gamma _{-}}}}</annotation>

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</math><img src="./3b68582e04e92cb981047aadac6e9a86462ff145.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.18ex; height:3.009ex;" alt="{\displaystyle \gamma _{\pm }={\sqrt {\gamma _{+}\gamma _{-}}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {+} }}">

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</math><img src="./0206d44562d7fb40dfe40a14c3806b24eba6131c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.715ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {+} }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {-} }}">

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</math><img src="./464da06e2c2c1be230cc8481d203bf2aa6c39513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.715ex; height:2.176ex;" alt="{\displaystyle \gamma _{\mathrm {-} }}" loading="lazy"> are the activity coefficients of the cation and anion respectively.

More generally, the mean activity coefficient of a compound of formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mathrm {p} }B_{\mathrm {q} }}">

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<annotation encoding="application/x-tex">{\displaystyle A_{\mathrm {p} }B_{\mathrm {q} }}</annotation>

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</math><img src="./870d29985f7580899c0caa698b8bb868f0ae5e8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.765ex; height:2.843ex;" alt="{\displaystyle A_{\mathrm {p} }B_{\mathrm {q} }}" loading="lazy"> is given by<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\pm }={\sqrt[{p+q}]{\gamma _{\mathrm {A} }^{p}\gamma _{\mathrm {B} }^{q}}}.}</annotation>

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</math><img src="./ebf812d2d894885ab43d46779e4fa8bdbd6cae2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.866ex; height:4.843ex;" alt="{\displaystyle \gamma _{\pm }={\sqrt[{p+q}]{\gamma _{\mathrm {A} }^{p}\gamma _{\mathrm {B} }^{q}}}.}" loading="lazy"></dd></dl>

The prevailing view that single ion activity coefficients are unmeasurable independently, or perhaps even physically meaningless, has its roots in the work of Guggenheim in the late 1920s.<a href="#cite_note-Guggenheim1928-5">[5]</a> In this view, the partitioning of the physical <a href="Electrochemical_potential" title="Electrochemical potential">electrochemical potentials</a> into an activity contribution and a <a href="Galvani_potential" title="Galvani potential">Galvani potential</a> contribution is arbitrary, thus nonidealities in ion activities can be remapped to nonidealities in Galvani potential and vice versa. Nevertheless, certain products of activities (such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\pm }}">

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</math><img src="./16ffca6a32533af26551a9361897688d6f0dca50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.715ex; height:2.176ex;" alt="{\displaystyle \gamma _{\pm }}" loading="lazy">) reflect a charge-neutral stoichiometry that is anyway insensitive to this partitioning, so these products are physically meaningful even if the single-ion activities are not.<a href="#cite_note-Guggenheim1928-5">[5]</a> However, chemists have never been able to give up the idea of single ion activities, and by implication single ion activity coefficients. For example, <a href="PH" title="PH">pH</a> is defined as the negative logarithm of the hydrogen ion activity. If the prevailing view on the physical meaning and measurability of single ion activities is correct then defining pH as the negative logarithm of the hydrogen ion activity places the quantity squarely in the unmeasurable category. Recognizing this logical difficulty, <a href="International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">International Union of Pure and Applied Chemistry</a> (IUPAC) states that the activity-based definition of pH is a notional definition only.<a href="#cite_note-6">[6]</a> Despite the prevailing negative view on the measurability of single ion coefficients, the concept of single ion activities continues to be discussed in the literature.<a href="#cite_note-Rockwood2015-7">[7]</a><a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Experimental_determination_of_activity_coefficients">Experimental determination of activity coefficients</h2></div>

Activity coefficients may be determined experimentally by making measurements on non-ideal mixtures. Use may be made of <a href="Raoult's_law" title="Raoult's law">Raoult's law</a> or <a href="Henry's_law" title="Henry's law">Henry's law</a> to provide a value for an ideal mixture against which the experimental value may be compared to obtain the activity coefficient. Other <a href="Colligative" class="mw-redirect" title="Colligative">colligative</a> properties, such as <a href="Osmotic_pressure" title="Osmotic pressure">osmotic pressure</a> may also be used.

<div class="mw-heading mw-heading3"><h3 id="Radiochemical_methods">Radiochemical methods</h3></div>

Activity coefficients can be determined by <a href="Radiochemistry" title="Radiochemistry">radiochemical</a> methods.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="At_infinite_dilution">At infinite dilution</h3></div>

Activity coefficients for binary mixtures are often reported at the infinite dilution of each component. Because activity coefficient models simplify at infinite dilution, such empirical values can be used to estimate interaction energies. Examples are given for water:

<caption>Binary solutions with water<a href="#cite_note-10">[10]</a>

</caption>

<th>X

</th>

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<td><a href="Ethanol" title="Ethanol">Ethanol</a></td>

<td>3.2800 (298.15)

</td></tr>

<tr>

<td><a href="Acetone" title="Acetone">Acetone</a></td>

<td>6.0200 (307.85)

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Theoretical_calculation_of_activity_coefficients">Theoretical calculation of activity coefficients</h2></div>

Activity coefficients of electrolyte solutions may be calculated theoretically, using the <a href="Debye%E2%80%93H%C3%BCckel_equation" class="mw-redirect" title="Debye–Hückel equation">Debye–Hückel equation</a> or extensions such as the <a href="Davies_equation" title="Davies equation">Davies equation</a>,<a href="#cite_note-King1964-11">[11]</a> <a href="Pitzer_equations" title="Pitzer equations">Pitzer equations</a><a href="#cite_note-davies-12">[12]</a> or TCPC model.<a href="#cite_note-GeWang2007-13">[13]</a><a href="#cite_note-GeZhang2008-14">[14]</a><a href="#cite_note-15">[15]</a><a href="#cite_note-GeWang2009-16">[16]</a> <a href="Specific_ion_interaction_theory" title="Specific ion interaction theory">Specific ion interaction theory</a> (SIT)<a href="#cite_note-17">[17]</a> may also be used.

For non-electrolyte solutions correlative methods such as <a href="UNIQUAC" title="UNIQUAC">UNIQUAC</a>, <a href="Non-random_two-liquid_model" title="Non-random two-liquid model">NRTL</a>, <a href="MOSCED" title="MOSCED">MOSCED</a> or <a href="UNIFAC" title="UNIFAC">UNIFAC</a> may be employed, provided fitted component-specific or model parameters are available. COSMO-RS is a theoretical method which is less dependent on model parameters as required information is obtained from <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> calculations specific to each molecule (sigma profiles) combined with a statistical thermodynamics treatment of surface segments.<a href="#cite_note-Klamt-18">[18]</a>

For uncharged species, the activity coefficient γ0 mostly follows a <a href="Salting-out" class="mw-redirect" title="Salting-out">salting-out</a> model:<a href="#cite_note-Butler-19">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle \log _{10}(\gamma _{0})=bI}</annotation>

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</math><img src="./ff34e2f88b334ca2ba7cdf84310887428e01a364.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.184ex; height:2.843ex;" alt="{\displaystyle \log _{10}(\gamma _{0})=bI}" loading="lazy"></dd></dl>

This simple model predicts activities of many species (dissolved undissociated gases such as CO2, H2S, NH3, undissociated acids and bases) to high <a href="Ionic_strength" title="Ionic strength">ionic strengths</a> (up to 5 mol/kg). The value of the constant b for CO2 is 0.11 at 10 °C and 0.20 at 330 °C.<a href="#cite_note-EllisGolding1963-20">[20]</a>

For <a href="Water" title="Water">water</a> as solvent, the activity aw can be calculated using:<a href="#cite_note-Butler-19">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle \ln(a_{\mathrm {w} })={\frac {-\nu b}{55.51}}\varphi }</annotation>

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</math><img src="./c11be070f16aeb088700d8e23b02fafc40356513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.149ex; height:5.343ex;" alt="{\displaystyle \ln(a_{\mathrm {w} })={\frac {-\nu b}{55.51}}\varphi }" loading="lazy"></dd></dl>

where ν is the number of ions produced from the dissociation of one molecule of the dissolved salt, b is the molality of the salt dissolved in water, φ is the <a href="Osmotic_coefficient" title="Osmotic coefficient">osmotic coefficient</a> of water, and the constant 55.51 represents the <a href="Molality" title="Molality">molality</a> of water. In the above equation, the activity of a solvent (here water) is represented as inversely proportional to the number of particles of salt versus that of the solvent.

<div class="mw-heading mw-heading3"><h3 id="Link_to_ionic_diameter">Link to ionic diameter</h3></div>

The ionic activity coefficient is connected to the <a href="Ionic_radius" title="Ionic radius">ionic diameter</a> by the formula obtained from <a href="Debye%E2%80%93H%C3%BCckel_theory" title="Debye–Hückel theory">Debye–Hückel theory</a> of <a href="Electrolyte" title="Electrolyte">electrolytes</a>:

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<annotation encoding="application/x-tex">{\displaystyle \log(\gamma _{i})=-{\frac {Az_{i}^{2}{\sqrt {I}}}{1+Ba{\sqrt {I}}}}}</annotation>

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</math><img src="./53f7ae92f6671128b91c297e96f1f280ff9faed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.632ex; height:7.176ex;" alt="{\displaystyle \log(\gamma _{i})=-{\frac {Az_{i}^{2}{\sqrt {I}}}{1+Ba{\sqrt {I}}}}}" loading="lazy"></dd></dl>

where A and B are constants, zi is the valence number of the ion, and I is <a href="Ionic_strength" title="Ionic strength">ionic strength</a>.

<div class="mw-heading mw-heading3"><h3 id="Concentrated_ionic_solutions">Concentrated ionic solutions</h3></div>

Ionic activity coefficients can be calculated theoretically, for example by using the <a href="Debye%E2%80%93H%C3%BCckel_equation" class="mw-redirect" title="Debye–Hückel equation">Debye–Hückel equation</a>. The theoretical equation can be tested by combining the calculated single-ion activity coefficients to give mean values which can be compared to experimental values.

<div class="mw-heading mw-heading4"><h4 id="Stokes–Robinson_model">Stokes–Robinson model</h4></div>

For concentrated ionic solutions the hydration of ions must be taken into consideration, as done by Stokes and Robinson in their hydration model from 1948.<a href="#cite_note-21">[21]</a> The activity coefficient of the electrolyte is split into electric and statistical components by E. Glueckauf who modifies the Robinson–Stokes model.

The statistical part includes <a href="Solvation_shell" title="Solvation shell">hydration index number</a> h, the number of ions from the dissociation and the ratio r between the <a href="Apparent_molar_property" title="Apparent molar property">apparent molar volume</a> of the electrolyte and the molar volume of water and molality b.

Concentrated solution statistical part of the activity coefficient is:

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<annotation encoding="application/x-tex">{\displaystyle \ln \gamma _{s}={\frac {h-\nu }{\nu }}\ln \left(1+{\frac {br}{55.5}}\right)-{\frac {h}{\nu }}\ln \left(1-{\frac {br}{55.5}}\right)+{\frac {br(r+h-\nu )}{55.5\left(1+{\frac {br}{55.5}}\right)}}}</annotation>

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</math><img src="./4f5e26cd000cd076ee827aab12cb3d39061870ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:67.073ex; height:8.343ex;" alt="{\displaystyle \ln \gamma _{s}={\frac {h-\nu }{\nu }}\ln \left(1+{\frac {br}{55.5}}\right)-{\frac {h}{\nu }}\ln \left(1-{\frac {br}{55.5}}\right)+{\frac {br(r+h-\nu )}{55.5\left(1+{\frac {br}{55.5}}\right)}}}" loading="lazy"><a href="#cite_note-Glueckauf1955-22">[22]</a><a href="#cite_note-Glueckauf1957-23">[23]</a><a href="#cite_note-Kortüm1960-24">[24]</a></dd></dl>

The Stokes–Robinson model has been analyzed and improved by other investigators.<a href="#cite_note-Miller1956-25">[25]</a><a href="#cite_note-26">[26]</a> The problem with this widely accepted idea that electrolyte activity coefficients are driven at higher concentrations by changes in hydration is that water activities are completely dependent on the concentration of the ions themselves, as imposed by a thermodynamic relationship called the Gibbs-Duhem equation. This means that the activity coefficients and the corresponding water activities are linked together fundamentally, regardless of molecular-level hypotheses. Due to this high correlation, such hypotheses are not independent enough to be satisfactorily tested.

<div class="mw-heading mw-heading4"><h4 id="Ion_trios">Ion trios</h4></div>

The rise in activity coefficients found with most aqueous strong electrolyte systems can be explained by increasing electrostatic repulsions between ions of the same charge which are forced together as the available space between them decreases. In this way, the initial attractions between cations and anions at the low concentrations described by Debye and Hueckel are progressively overcome. It has been proposed<a href="#cite_note-27">[27]</a> that these electrostatic repulsions take place predominantly through the formation of so-called ion trios in which two ions of like charge interact, on average and at distance, with the same counterion as well as with each other. This model accurately reproduces the experimental patterns of activity and osmotic coefficients exhibited by numerous 3-ion aqueous electrolyte mixtures.

<div class="mw-heading mw-heading2"><h2 id="Dependence_on_state_parameters">Dependence on state parameters</h2></div>

The derivative of an activity coefficient with respect to temperature is related to <a href="Excess_molar_quantity" class="mw-redirect" title="Excess molar quantity">excess molar enthalpy</a> by

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<annotation encoding="application/x-tex">{\displaystyle {\bar {H}}_{i}^{\mathsf {E}}=-RT^{2}{\frac {\partial }{\partial T}}\ln(\gamma _{i})}</annotation>

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</math><img src="./148858061a99877d95739eb429a64b3a1487eeaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.652ex; height:5.509ex;" alt="{\displaystyle {\bar {H}}_{i}^{\mathsf {E}}=-RT^{2}{\frac {\partial }{\partial T}}\ln(\gamma _{i})}" loading="lazy"></dd></dl>

Similarly, the derivative of an activity coefficient with respect to pressure can be related to excess molar volume.

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<annotation encoding="application/x-tex">{\displaystyle {\bar {V}}_{i}^{\mathsf {E}}=RT{\frac {\partial }{\partial P}}\ln(\gamma _{i})}</annotation>

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</math><img src="./7e8acd03372f5051c1e6514b5df7b75ecf110c94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.539ex; height:5.509ex;" alt="{\displaystyle {\bar {V}}_{i}^{\mathsf {E}}=RT{\frac {\partial }{\partial P}}\ln(\gamma _{i})}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Application_to_chemical_equilibrium">Application to chemical equilibrium</h2></div>

At equilibrium, the sum of the chemical potentials of the reactants is equal to the sum of the chemical potentials of the products. The <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> change for the reactions, ΔrG, is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{\mathrm {A} }+\beta _{\mathrm {B} }=\sigma _{\mathrm {S} }+\tau _{\mathrm {T} },}</annotation>

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</math><img src="./0b13b2fb376a347da6f75626ab397a33b1a61be3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.999ex; height:2.509ex;" alt="{\displaystyle \alpha _{\mathrm {A} }+\beta _{\mathrm {B} }=\sigma _{\mathrm {S} }+\tau _{\mathrm {T} },}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{\mathrm {r} }G=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }-(\alpha \mu _{\mathrm {A} }+\beta \mu _{\mathrm {B} })=0\,}</annotation>

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</math><img src="./ef21f380899dea94f79583a3d244cd14c9a776c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.1ex; height:2.843ex;" alt="{\displaystyle \Delta _{\mathrm {r} }G=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }-(\alpha \mu _{\mathrm {A} }+\beta \mu _{\mathrm {B} })=0\,}" loading="lazy"></dd></dl>

Substitute in the expressions for the chemical potential of each reactant:

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{\mathrm {r} }G=\sigma \mu _{S}^{\ominus }+\sigma RT\ln a_{\mathrm {S} }+\tau \mu _{\mathrm {T} }^{\ominus }+\tau RT\ln a_{\mathrm {T} }-(\alpha \mu _{\mathrm {A} }^{\ominus }+\alpha RT\ln a_{\mathrm {A} }+\beta \mu _{\mathrm {B} }^{\ominus }+\beta RT\ln a_{\mathrm {B} })=0}</annotation>

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</math><img src="./3b868588131ecbdc146076db7cef8d1ea5e663ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:90.844ex; height:3.176ex;" alt="{\displaystyle \Delta _{\mathrm {r} }G=\sigma \mu _{S}^{\ominus }+\sigma RT\ln a_{\mathrm {S} }+\tau \mu _{\mathrm {T} }^{\ominus }+\tau RT\ln a_{\mathrm {T} }-(\alpha \mu _{\mathrm {A} }^{\ominus }+\alpha RT\ln a_{\mathrm {A} }+\beta \mu _{\mathrm {B} }^{\ominus }+\beta RT\ln a_{\mathrm {B} })=0}" loading="lazy"></dd></dl>

Upon rearrangement this expression becomes

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{\mathrm {r} }G=\left(\sigma \mu _{\mathrm {S} }^{\ominus }+\tau \mu _{\mathrm {T} }^{\ominus }-\alpha \mu _{\mathrm {A} }^{\ominus }-\beta \mu _{\mathrm {B} }^{\ominus }\right)+RT\ln {\frac {a_{\mathrm {S} }^{\sigma }a_{\mathrm {T} }^{\tau }}{a_{\mathrm {A} }^{\alpha }a_{\mathrm {B} }^{\beta }}}=0}</annotation>

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</math><img src="./e51e36c4af3cd0091812e2d213f1e6fdbd5598a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:54.762ex; height:7.176ex;" alt="{\displaystyle \Delta _{\mathrm {r} }G=\left(\sigma \mu _{\mathrm {S} }^{\ominus }+\tau \mu _{\mathrm {T} }^{\ominus }-\alpha \mu _{\mathrm {A} }^{\ominus }-\beta \mu _{\mathrm {B} }^{\ominus }\right)+RT\ln {\frac {a_{\mathrm {S} }^{\sigma }a_{\mathrm {T} }^{\tau }}{a_{\mathrm {A} }^{\alpha }a_{\mathrm {B} }^{\beta }}}=0}" loading="lazy"></dd></dl>

The sum

σμ<s>o</s> S + τμ<s>o</s> T − αμ<s>o</s> A − βμ<s>o</s> B is the standard free energy change for the reaction, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\mathrm {r} }G^{\ominus }}">

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</math><img src="./8ce1a0a95bd7500bb8dd352492794dbbd1c5a071.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.15ex; height:2.843ex;" alt="{\displaystyle \Delta _{\mathrm {r} }G^{\ominus }}" loading="lazy">.

Therefore,

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K}</annotation>

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</math><img src="./cf2d942868fdcf65d484f3cc0a2fe1b5cd398241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.334ex; height:2.843ex;" alt="{\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K}" loading="lazy"></dd></dl>

where K is the <a href="Equilibrium_constant" title="Equilibrium constant">equilibrium constant</a>. Note that activities and equilibrium constants are dimensionless numbers.

This derivation serves two purposes. It shows the relationship between standard free energy change and equilibrium constant. It also shows that an equilibrium constant is defined as a quotient of activities. In practical terms this is inconvenient. When each activity is replaced by the product of a concentration and an activity coefficient, the equilibrium constant is defined as

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }}}\times {\frac {\gamma _{\mathrm {S} }^{\sigma }\gamma _{\mathrm {T} }^{\tau }}{\gamma _{\mathrm {A} }^{\alpha }\gamma _{\mathrm {B} }^{\beta }}}}</annotation>

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</math><img src="./95d7e43431f4648654306641285e434365d7af57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.381ex; height:7.176ex;" alt="{\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }}}\times {\frac {\gamma _{\mathrm {S} }^{\sigma }\gamma _{\mathrm {T} }^{\tau }}{\gamma _{\mathrm {A} }^{\alpha }\gamma _{\mathrm {B} }^{\beta }}}}" loading="lazy"></dd></dl>

where [S] denotes the <a href="Concentration" title="Concentration">concentration</a> of S, etc. In practice equilibrium constants are <a href="Determination_of_equilibrium_constants" title="Determination of equilibrium constants">determined</a> in a medium such that the quotient of activity coefficients is constant and can be ignored, leading to the usual expression

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }}}}</annotation>

</semantics>

</math><img src="./59e684b2a458ef9a608ef9a8d8f9e55473da1e86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.435ex; height:6.509ex;" alt="{\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }}}}" loading="lazy"></dd></dl>

which applies under the conditions that the activity quotient has a particular (constant) value.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFNesbitt1982" class="citation journal cs1">Nesbitt, H. Wayne (1982). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF00649040">"The stokes and robinson hydration theory: A modification with application to concentrated electrolyte solutions"</a>. Journal of Solution Chemistry. 11 (6): 415–422. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00649040">10.1007/BF00649040</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:94189765">94189765</a>.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFMayMay2024" class="citation journal cs1">May, Peter M.; May, Eric (2024). <a rel="nofollow" class="external text" href="https://pubs.acs.org/doi/10.1021/acsomega.4c07525">"Ion Trios: Cause of Ion Specific Interactions in Aqueous Solutions and Path to a Better pH Definition"</a>. ACS Omega. 9 (46): 46373–46386. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facsomega.4c07525">10.1021/acsomega.4c07525</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11579776">11579776</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://aiomfac.lab.mcgill.ca/">AIOMFAC online-model</a> An interactive group-contribution model for the calculation of activity coefficients in organic–inorganic mixtures.</li>

<li><a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/article/pii/0013468676850256?np=y">Electrochimica Acta</a> Single-ion activity coefficients</li></ul>

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