Reversible adiabatic expansion proof

[Latsabb]

Homework Statement

Prove the relationship between the pressure, P, and the temperature, T, for an ideal gas with a reversible adiabatic expansion. Base the proof on the first law of thermodynamics and the ideal gas law.

The relationship is: T^(C_p,m/R)/P = constant

Where R is the gas constant and C_p,m is the molar heat capacity.

Homework Equations

ΔU=Q-W
pV=nRT

Possibly relevant:
W_rev=-pdV
γ=C_p,m/C_v,m where γ is the heat capacity ratio.
PV^γ= constant

The Attempt at a Solution

I don't even know where to start explaining what I have tried... I have been kicking around a lot of things. I can't seem to manage to manipulate things so that temperature is to the power of anything, although I have burnt a lot of time trying to turn PV^γ into some relation for T to the power of some derivative of gamma. I assume since gamma has C_p,m in it, that it is going to play in, and especially since it is already a power for volume, this flagged me in that direction, but I am just going around in circles.

Any help in the right direction would be GREATLY appreciated.

 

[Andrew Mason]

The relationship is: T^(C_p,m/R)/P = constant

Where R is the gas constant and C_p,m is the molar heat capacity.

Homework Equations

ΔU=Q-W
pV=nRT

Possibly relevant:
W_rev=-pdV
γ=C_p,m/C_v,m where γ is the heat capacity ratio.
PV^γ= constant

The Attempt at a Solution

I don't even know where to start explaining what I have tried... I have been kicking around a lot of things. I can't seem to manage to manipulate things so that temperature is to the power of anything, although I have burnt a lot of time trying to turn PV^γ into some relation for T to the power of some derivative of gamma. I assume since gamma has C_p,m in it, that it is going to play in, and especially since it is already a power for volume, this flagged me in that direction, but I am just going around in circles.

Start with PV^γ= constant and substitute nRT/P for V.

AM

 

[Latsabb]

Ok, so I end up with:
P*(nRT/P)^γ=c

Which I then turn into:
P*(nRT)^γ/P^γ=c

And then:
(nRT)^γ/P^γ-1=c

Am I on the right track? Because I have been playing around with it, and I can't seem to extract nR out, nor am I having luck manipulating the powers to what I need.

 

[Chestermiller]

If you are supposed to use the first law, then you are expected to start with

$dU=nCvdT=-PdV$

 

[Latsabb]

I finally got it nailed down. Chester was correct, the missing piece here was that dU=C_v,mdT=-PdV. An integration of that, and some minor modifying of terms produced what I was after. Thank you!