Give four different definitions of the chemical potential μ, as derivatives of the different energies we have defined.
μ = = = =
Give two definitions of the entropy σ in terms of derivatives of the different energies we have defined.
σ = - = -
Using the definition of temperature that uses the enthalpy, give an expression for the temperature in terms of U, σ, p, and V, following the method used to derive an expression for the pressure above.
We know that τ = , and that H = U + pV. We can differentiate the second equation with respect to σ, holding p and N constant, and then set equal to τ to obtain:
Derive the Maxwell relation that relates a derivative of μ with a derivative of σ.
We use G because μ and σ are free in its differential identity. We can write = μ and = - σ. Taking the partial derivative of the first with respect to τ, holding N constant, and taking the partial derivative of the second with respect to N, holding τ constant, and setting the two equal, we obtain:
Derive the Maxwell Relation that relates a derivative of τ with a derivative of V.
We need V and τ to be free in the energy, so let us choose the enthalpy H. Then we can write τ = and V = . Taking the partial derivative of the first with respect to p, holding σ constant, and taking the partial derivative of the second with respect to σ, holding p constant, and setting them equal, yields: