Question

How do you define Chemical potential of a species i in U, H, G and A. and how they are
related to each other?

Answer 1

Chemical potential is energy required to add or release a particle from a given species.

So In tyermodynamic relations change in internal energy of a recation we write as

dU = TdS – PdV + SuidN,

So chemical potential gor i th particle change on constant Volume ans entroy we write as

au li Ni S.V.Ni

Now gibbs free energy

G=U + PV - TS

so,

dG = dU + PDV + V DP - TIS - SDT

now substituting for dU

we get

dG = -SDT +VdP + MidN,

so,we define for constant T and P

$\mu_i =\frac{\partial G}{\partial N_i}_{T,P,N_{i\neq j}}$

similarly

H = U + PV

we write

$dH=dU+PdV+VdP$

Substitute for dU

$dH=TdS+VdP+\Sigma _i^N\mu_i dN_i$

so,

$\mu_i =\frac{\partial H}{\partial N_i}_{S,P,N_{i\neq j}}$

simlarly

A= U-TS

So,

dA= dU-TdS-SdT

so,

$dA=-PdV-SdT+\Sigma_i^N\mu_idN_i$

so,

$\mu_i =\frac{\partial A}{\partial N_i}_{V,T,N_{i\neq j}}$