Adiabatic Expansion Question

nicksauce

1. The problem statement, all variables and given/known data
Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation:
[tex]\frac{dT}{dP} = \frac{2T}{5P} [/tex]

2. Relevant equations
Ideal gas law


3. The attempt at a solution
[tex]PV = nkT[/tex]

[tex]T = \frac{PV}{nk}[/tex]

[tex]\frac{dT}{dP} = \frac{1}{nk}(V + P\frac{dV}{dP})[/tex]

So what is dV/dP?

[tex]PV^\gamma = C[/tex]

[tex]V = C^{\frac{1}{\gamma}} P^{\frac{-1}{\gamma}}[/tex]

[tex]\frac{dV}{dP} = C ^ {\frac{1}{\gamma}} \frac{-1}{\gamma}P^{\frac{-1}{\gamma} - 1}[/tex]

[tex]\frac{dV}{dP} = \frac{-1}{\gamma} VP^{-1} [/tex]
so

[tex]\frac{dT}{dP} = \frac{1}{nk}[V - \frac{V}{\gamma}][/tex]

But this gives dT/dP ~= 1/3 T/P (using gamma~=3/2), instead of 2/5 T/P. Can anyone see where I went wrong?

nicksauce

Err. never mind, gamma should be 5/3, not 3/2, in which case this does give the correct answer.