How Is Temperature Related to Pressure in Adiabatic Expansion of an Ideal Gas?

[nicksauce]

Homework Statement

Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation:
\frac{dT}{dP} = \frac{2T}{5P}

Homework Equations

Ideal gas law

The Attempt at a Solution

PV = nkT

T = \frac{PV}{nk}

\frac{dT}{dP} = \frac{1}{nk}(V + P\frac{dV}{dP})

So what is dV/dP?

PV^\gamma = C

V = C^{\frac{1}{\gamma}} P^{\frac{-1}{\gamma}}

\frac{dV}{dP} = C ^ {\frac{1}{\gamma}} \frac{-1}{\gamma}P^{\frac{-1}{\gamma} - 1}

\frac{dV}{dP} = \frac{-1}{\gamma} VP^{-1}
so

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

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.