Adiabatic Expansion: Relating T & P with dT/dP

[w3390]

Homework Statement

Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation: dT/dP = (2/f+2)(T/P).

Homework Equations

PV=NkT
VT^(f/2) = constant
V^(gamma)*P = constant

The Attempt at a Solution

I started off with the formula for ideal gases, PV=NkT.

I rearranged to get T=(PV/Nk).

At this point I don't know where to go. I don't see any equations I can use to make substitutions and I'm not sure if I should take the derivative at this point or not?

 

[kuruman]

Use the ideal gas law to replace V in the second "relevant equation" that you posted.
Solve for T in terms of p.
Take the required derivative dT/dp.

 

[w3390]

Okay. I'm getting a little tripped up at the derivative part. I have at this point:

T = (c/Nk*P)^(2/f+2) , where c is a constant

Taking the derivative will bring down the 2/f+2, but that leaves me with (2/f+2)-1 as the exponent plus the derivative of the inside.

 

[kuruman]

A constant is a constant is a constant so you can write

$$T=CP^{\frac{2}{f+2}}$$

Then you say that

$$\frac{dT}{dP}=C\frac{2}{f+2}\:P^{\frac{2}{f+2}-1}$$

I am not sure what you mean by "the derivative of the inside." What do you get when you simplify the exponent? How is that related to the expression of T as a function of P?