# Homework Help: Internal energy of a hydrostatic system in a reversible adiabatic process

1. Feb 11, 2010

### Str1k3

1. The problem statement, all variables and given/known data
A simple hydrostatic system is such that $$PV^k$$ is constant in a reversible adiabatic process, where k > 0 is a given constant. Show that its internal energy has the form
$$E=\frac{1}{k-1}PV+NF(\frac{PV^k}{N^k}$$
where f is an arbitrary function. Hint: $$PV^k$$ must be a function of S (why?) so that $$(\partial{E}{S})_S = g(S)V^-k$$ where g(S) is an arbitrary function.

2. Relevant equations

3. The attempt at a solution
I used dE + dW = dQ = 0. so dW = PdV. Then we want to find $$W=\int PdV$$ using the limits V1 and V2 and substituting $$P=\frac{P_1}{V_1*V}$$. This works ok to get the first term of the energy out, but not the second. we end up with a term that looks like this as the second term $$\frac{P_1*V^k_1}{(1-k)*V^(k-1)_2}$$ which doesn't look much like $$N\frac{PV^k}{N^k}$$

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