Partial pressure

{title2=$\mu_{\rm m}=\mu_{\rm m}^\circ+RT\ln(p/p^\circ)$}

For a pure <ideal gas> at <temperature> $T$ and <pressure> $p$,
$$
\mu_{\rm m}(T,p)=\mu_{\rm m}^\circ(T)+RT\ln\frac{p}{p^\circ},
$$
where $R$ is the <molar gas constant> and $p^\circ$ specifies the reference <pressure> of the <standard state>. The ratio inside the <natural logarithm> is <dimensionless>. The <chemical potential> difference therefore depends on a ratio of <pressures>, not an absolute difference. See https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/e763a06aa3b5de09cd6bfea6280f219c_5_60_lecture14.pdf[MIT 5.60, Lecture 14, page 4].

For a <chemical species> in an ideal <gas> <mixture>, replace $p$ by its <partial pressure>. More generally write $\mu_{{\rm m},i}=\mu_{{\rm m},i}^\circ+RT\ln a_i$ using its <thermodynamic activity> $a_i$ and a consistent <standard state>. For an ideal dilute <chemical solution>, $a_i=c_i/c^\circ$ with reference <concentration> $c^\circ$; the <thermodynamic activity> section describes departures from this idealization.

As a computed comparison, take identical <chemical species> at $T=300\,\mathrm K$, with $p_A=2p^\circ$ and $p_B=p^\circ$. Stipulate $R=8.314462618\,\mathrm{J\,mol^{-1}\,K^{-1}}$. Then
$$
\mu_{{\rm m},B}-\mu_{{\rm m},A}
=RT\ln\frac{p_B}{p_A}
=-RT\ln2
\simeq-1728.944\,\mathrm{J\,mol^{-1}}.
$$
Halving the <pressure> gives the same decrement whether starting from $2p^\circ$ or $20p^\circ$, provided the <ideal gas> approximation remains valid. This comparison uses different <pressures>; it is not the common-<pressure> reservoir setup of <chemical potential equality for particle exchange>. The fixed-<volume (thermodynamics)> example in <chemical potential and Helmholtz free energy> supplies an exchange experiment with unequal initial <pressures>.

\Image[/chemical-potential-and-pressure.png]
{title=<Ideal gas> <chemical potential> difference versus <pressure> ratio}
{description=At a common <temperature>, the <chemical potential> difference is $RT\ln(p_B/p_A)$. The arrows give the initial transfer direction through an open route between rigid compartments; the decreasing <thermodynamic potential> there is <Helmholtz free energy>. The marked comparison uses $300\,\mathrm K$ and the stipulated <molar gas constant>.}


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 Incoming links: Chemical potential of an ideal gas