 Solution

ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2021/section-b/4/c/solution

μ_{α} = (\partial F / \partial N_{α})_{T, V, N_{α^{'} \neq = α}}

. For the combined noninteracting mixture, add the free energies from (a) and (b). Since the thermal de Broglie wavelengths are independent of particle number at fixed temperature, differentiation gives

μ_{CO} = k_{B} T ln (n_{CO} λ_{CO}^{3}), μ_{O_{2}} = k_{B} T ln (n_{O_{2}} λ_{O_{2}}^{3}),

μ_{C O_{2}} = k_{B} T ln (n_{c} λ_{c}^{3}) - \frac{ϵ}{2} .

Each logarithm has a dimensionless argument, and each chemical potential has units of energy per molecule. The binding offset must be kept consistently when comparing species.

 New to topics? Read the docs here!