In thermodynamics the variables p (pressure), T (temperature), U (internal energy), and V...

In thermodynamics the variables p (pressure), T (temperature), U (internal energy), and V (volume) occur. For each substance these are related by two equations, so that any two of the four variables can be chosen as independent, the other two then being dependent. In addition, the second law of thermodynamics implies the relation

when V and T are independent. Show that this relation can be written in each of the following forms:

(U, V indep.),

(U, p indep.),

(p, T indep.),

(U, p indep.),

(T, U indep.).

[Hint: The relation (a) implies that if

dT = a dV + b dT, dp = c dV + e dT

are the expressions for dU and dp in terms of dV and dT, then a − Te + p = 0. To prove (b), for example, one assumes relations

dT = α dV + β dU, dp = γ dV + δ dU.

If these are solved for dU and dp in terms of dV and dT, then one obtains expressions for a and e in terms of α, β, γ, δ. If these expressions are substituted in the equation a − Te + p = 0, one has an equation in β, γ, δ. Since a = ∂T/∂V, etc., the relation of form (b) is obtained. The others are proved in the same way.]