Question

Consider the below physical situations:

(i) Closed rigid container of volume V that is isolated from the rest of the universe with uniform temperature T and pressure P filled and mixed with two species of ideal gases 1 and 2 with particle counts N1,N2. The total number of particles N=N1+N2 is fixed.

(ii) Closed rigid container of volume V that is isolated from the rest of the universe with uniform temperature T and pressure P divided into two components of equal volume 1 and 2, separated by a partition that allows particle exchange. Both components are filled with ideal gas of the same kind with particle counts in both components N1,N2. The total number of particles N=N1+N2 is fixed.

In both physical situations (i) and (ii), it seems like the chemical potentials of 1 and 2 are)μ_1=μ^∘_1+k_BTln(N_1/N)
μ_2=μ^∘_2+k_BTln(N_2/N)

Is that true?

Answer 1

The chemical potentials μ_1 and μ_2 for two species of ideal gases in a closed container, whether mixed or separated by a partition, are accurately represented by the equations given, depending on the molar fraction of the gases.

Step-by-step explanation:

The chemical potentials for species 1 and 2 in a mixture of ideal gases can be determined using the fact that each gas behaves independently. Considering a closed rigid container of volume V filled with two species of ideal gases with particle counts N1 and N2, and total number of particles N=N1+N2 being fixed, the chemical potential for species 1 (μ_1) and species 2 (μ_2) are given by:

μ_1 = μ°_1 + k_B T ln(N1/N)

μ_2 = μ°_2 + k_B T ln(N2/N)

This formula indeed reflects the fact that the chemical potential depends on the molar fraction of the gas in the mixture, which is in agreement with the equation provided.

When the container is divided into two components of equal volume and each is filled with the same ideal gas at the same temperature, and pressure P, allowing for particle exchange, the scenario does not change the expression of the chemical potentials. Since both components are ideal gases and the partition allows for particle exchange, the chemical potentials remain as is, assuming both halves reach equilibrium.

These relationships arise because ideal gases do not interact, and the only factor affecting pressure (and hence chemical potential) is the number of moles present in a given volume at a given temperature, according to the ideal gas law.