Definition of chemical potential at constant T and P: The chemical potential μ_i of component i is the increase in a system's capacity to do non-expansion (useful) work per infinitesimal mole of i added at constant temperature and pressure. Which expression correctly defines μ_i?

Introduction / Context:
Chemical potential links composition changes to thermodynamic potentials. At constant temperature and pressure, it measures the system's capacity to perform non-expansion work when matter is added—central to phase equilibrium, reaction equilibria, and transport driving forces.

Given Data / Assumptions:

• Infinitesimal addition of component i.
• Temperature and pressure held constant.
• All other component amounts n_{j≠i} held fixed for the partial derivative.

Concept / Approach:
Gibbs free energy G is the appropriate potential for processes at constant T and P; its differential is dG = −S dT + V dP + Σ μ_i dn_i. Thus, at constant T and P, μ_i equals the partial derivative of G with respect to n_i. While U and A are legitimate thermodynamic potentials with their own natural variables, the definition in terms of 'capacity to do non-expansion work at constant T and P' points specifically to G, not U or A.

Step-by-Step Solution:
Write the fundamental relation: dG = −S dT + V dP + Σ μ_i dn_i.Hold T and P constant: dT = 0, dP = 0.Then dG at constant T,P reduces to Σ μ_i dn_i → μ_i = (∂G/∂n_i){T,P,n{j≠i}}.Therefore, choose the Gibbs free energy expression.

Verification / Alternative check:
Alternative potentials: dA = −S dT − P dV + Σ μ_i dn_i, so μ_i = (∂A/∂n_i){T,V}; however, the question explicitly fixes T and P and references non-expansion work, which corresponds to G.

Why Other Options Are Wrong:
(a) Uses internal energy U with natural variables S,V, not the stated constraints.

(b) Uses Helmholtz free energy at constant T,V; this does not match the constant-pressure criterion.

(d) Cannot be correct because only one option matches the specified T,P conditions and work interpretation.

Common Pitfalls:

• Forgetting each potential's natural variables: U(S,V), A(T,V), H(S,P), G(T,P).
• Confusing 'non-expansion work' at T,P with the Helmholtz framework at T,V.

Final Answer:
μ_i = (∂G/∂n_i){T,P,n_{j≠i}}