Quickly Thermo Single-Component Adiabatic System Derive Energy Equation

[ouzo216]

Homework Statement

Show that if a single-component system is such that PV^k is constant in an adiabatic process (k is a positive constant) the energy is U = [1/(k-1)]PV + Nf(PV^k/N^k) where f is an arbitrary function.

Homework Equations

Fundamental relation: U = [1/(k-1)]PV + Nf(PV^k/N^k).
State equations: T = δU/δS, P = - δU/δV, μ = δU/δN.

The Attempt at a Solution

PV^k has to be a function of S, so that (δU/δV)S = g(S) x V^-k, where g(S) is an unspecified function, and V^-k is V^k inverse.
I think I'm going to end up finding dU first, and integrating to find U, but pretty much, I'm stuck here. Any help at all would be greatly appreciated.

 

[mark726]

Firstly, let's start by using the state equations to find an expression for dU in terms of P, V, and N. Using the first state equation, we have:

T = δU/δS

Rearranging, we have:

δU = TδS

Next, using the second state equation, we have:

P = - δU/δV

Substituting our expression for δU from the first state equation, we have:

P = - TδS/δV

Now, we can use the chain rule to rewrite the right side of this equation in terms of P and V. Recall that:

δS/δV = (δS/δP)(δP/δV)

Substituting this into our equation, we have:

P = - T(δS/δP)(δP/δV)

Now, let's rearrange this equation to solve for δS/δP:

δS/δP = - P/T(δP/δV)

Finally, we can substitute this expression for δS/δP into our expression for δU to get:

δU = - PT(δP/δV)δP

Integrating both sides, we have:

U = - PT∫(δP/δV)δP

Now, we need to find an expression for δP/δV in terms of P and V. Using the second state equation again, we have:

P = - δU/δV

Differentiating both sides with respect to V, we have:

δP/δV = - δ2U/δV2

Substituting this into our expression for U, we have:

U = - PT∫(δ2U/δV2)δP

Next, we can use the chain rule to rewrite the right side of this equation in terms of P and V. Recall that:

δ2U/δV2 = (δ2U/δP2)(δP/δV)

Substituting this into our expression for U, we have:

U = - PT∫(δ2U/δP2)(δP/δV)δP

Now, we can use the chain rule again to rewrite the right side of this equation in terms of P and V. Recall that:

δP/δV = (δP/δV)(