Excess chemical potential

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The difference between the chemical potential of a given species and that of an ideal gas under the same conditions.[1]

chemical potential of pure fluid can be estimated by the Particle 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}}{\Lambda^{dN}N!}\int_{0}^{1}\ldots\int_{0}^{1}ds^{N}exp[-\beta U(s^{N};L)] s is a scaled coordinate.

F(N,V,T)= -k_{B}TlnQ=-k_{B}Tln(\frac{V^{N}}{\Lambda^{dN}N!})-k_{B}Tln{\int ds^{N}exp[-\beta U(s^{N};L)]}=F_{id}(N,V,T)+F_{ex}(N,V,T)

combine the above equation with the definition of chemical potential: \mu_{a}= (\frac{\partial G}{\partial Na})_{PTN}

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

\mu= -k_{B}Tln(Q_{N+1}/Q_{N})=-k_{B}Tln(\frac{V/\Lambda^{d}}{N+1}) - k_{B}Tln{\frac{\int ds^{N+1}exp[-\beta U(s^{N+1})]}{\int ds^{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 μex, 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

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

and

\mu_{ex}= -k_{B}T\int ln ds_{N+1}<exp(-\beta\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 Windom Insertion particle method, or other ensembles such as NPT, NVE.

[edit] References

  1. ^ Frenkel, Daan; Smit, Berend [2001]. Understanding Molecular Simulation : from algorithms to applications. San Diego, California: Academic Press. ISBN 0-12-267351-4.