# Chemical potential of a mixture of ideal gases

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.38.

The chemical potential is defined in terms of the entropy as

$\displaystyle \mu\equiv-T\left(\frac{\partial S}{\partial N}\right)_{U,V} \ \ \ \ \ (1)$

This definition leads to a general thermodynamic identity

$\displaystyle dU=TdS-PdV+\mu dN \ \ \ \ \ (2)$

For a mixture of ideal gases, each species ${i}$ constitutes a molar fraction ${x_{i}}$ of the total number ${N_{total}}$ of molecules, so each species has its own chemical potential defined as

$\displaystyle \mu_{i}=-T\left(\frac{\partial S}{\partial N_{i}}\right)_{U,V,N_{j\ne i}} \ \ \ \ \ (3)$

where all ${N_{j}}$ with ${j\ne i}$ are held constant in the derivative.

Also, for an ideal gas, each species contributes its own portion of the overall entropy, independently of the other species. We can see this by noting that if we have a mixture of, say, 2 gases, then for each configuration of the gas ${A}$ molecules there is a multiplicity of ${\Omega_{B}}$ of the gas ${B}$ molecules and since for an ideal gas, the molecules don’t interact, the total multiplicity of the mixture is ${\Omega_{total}=\Omega_{A}\Omega_{B}}$, so the entropy is the sum of the entropies for the separate species: ${S_{total}=S_{A}+S_{B}}$.

Since ideal gas molecules don’t interact, species ${i}$ contributes a fraction ${x_{i}}$ of the total pressure, or in other words, its partial pressure is

$\displaystyle P_{i}=x_{i}P \ \ \ \ \ (4)$

We can therefore write the thermodynamic identity for a mixture of ideal gases as

$\displaystyle dU=T\sum_{i}dS_{i}-\left(\sum_{i}P_{i}\right)dV+\sum_{i}\mu_{i}dN_{i} \ \ \ \ \ (5)$

Since ${dS_{j\ne i}=0}$ in 3 (because only the number ${N_{i}}$ of species ${i}$ is changing, and no properties of any of the other species are changing), we can write the chemical potential of species ${i}$ as

$\displaystyle \mu_{i}=-T\left(\frac{\partial S_{i}}{\partial N_{i}}\right)_{U,V,N_{j\ne i}} \ \ \ \ \ (6)$

But this is the definition of chemical potential in a system containing only species ${i}$ at partial pressure ${P_{i}}$ in volume ${V}$. Thus, for a mixture of ideal gases, the chemical potential of each species is independent of the other species. In a mixture of real gases, however, this is probably not the case, since interactions between the species means the total entropy isn’t a simple sum of the entropies of the individual species.