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: