Stochastic ordering

The excess chemical potential is defined as the difference between the chemical potential of a given species and that of an ideal gas under the same conditions (in particular, at the same pressure, temperature, and composition).

Chemical potential of pure fluid can be estimated by the Widom insertion method.

For a system of diameter $L$ and volume $V$, at constant temperature :$T$, the classical partition function $Q(N,V,T)=\frac{V^N}{{\mathrm{\Lambda }}^{dN}N!}\underset{0}{\overset{1}{\int }}\dots \underset{0}{\overset{1}{\int }}d{s}^N\exp [-\beta U(s^N;L)]$ s is a scaled coordinate.

$F(N,V,T)=-k_{B}T\ln Q=-k_{B}T\ln \left(\frac{V^N}{{\mathrm{\Lambda }}^{dN}N!}\right)-k_{B}T\ln \int d{s}^N\exp [-\beta U(s^N;L)]=$

$=F_{id}(N,V,T)+F_{ex}(N,V,T)$

combining the above equation with the definition of chemical potential: ${\mu }_a={\left(\frac{\partial G}{\partial N_a}\right)}_{PTN}$

we get the chemical potential of a sufficient large system is :

$\mu =-k_{B}T\ln (Q_{N+1}/Q_N)=-k_{B}T\ln \left(\frac{V/{\mathrm{\Lambda }}^d}{N+1}\right)-k_{B}T\ln \frac{\int d{s}^{N+1}\exp [-\beta U(s^{N+1})]}{\int d{s}^N\exp [-\beta U(s^N)]}={\mu }_{id}(\rho )+{\mu }_{ex}$

wherein the chemical potential of ideal gas can be evaluated analytically. Now let's focus on ${\mu }_{ex}$, since the potential energy of an N+1 particle system can be separated into the potential energy of an N particle system and the potential of the excess particle interacts with the N particle system, there is

$\mathrm{\Delta }U\equiv U(s^{N+1})-U(s^N)$

and

${\mu }_{ex}=-k_{B}T\ln \int d{s}_{N+1}⟨\exp (-\beta \mathrm{\Delta }U){⟩}_N$

thus far we converted the excess chemical potential into an ensemble average, and the integral in the above equation can be sampled by brute force Monte Carlo method.

The calculating of excess chemical potential is not only limited to homogeneous system, it has also been extended to inhomogeneous system by Widom insertion method, or other ensembles such as NPT, NVE.

See also

Apparent molar property

References

• Note: the equations and presentation in this article are drawn from Excess Chemical Potential via the Widom Method

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