## 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^(Cp,m/R)/P = constant

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

## Homework Equations

ΔU=Q-W
pV=nRT

Possibly relevant:
Wrev=-pdV
γ=Cp,m/Cv,m where γ is the heat capacity ratio.
PVγ= constant

## The Attempt at a Solution

I dont even know where to start explaining what I have tried... I have been kicking around a lot of things. I cant 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 PVγ into some relation for T to the power of some derivative of gamma. I assume since gamma has Cp,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.

Related Biology and Chemistry Homework Help News on Phys.org
Andrew Mason
Homework Helper
The relationship is: T^(Cp,m/R)/P = constant

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

## Homework Equations

ΔU=Q-W
pV=nRT

Possibly relevant:
Wrev=-pdV
γ=Cp,m/Cv,m where γ is the heat capacity ratio.
PVγ= constant

## The Attempt at a Solution

I dont even know where to start explaining what I have tried... I have been kicking around a lot of things. I cant 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 PVγ into some relation for T to the power of some derivative of gamma. I assume since gamma has Cp,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.

AM

Ok, so I end up with:
P*(nRT/P)γ=c

Which I then turn into:
P*(nRT)γ/Pγ=c

And then:
(nRT)γ/Pγ-1=c

Am I on the right track? Because I have been playing around with it, and I cant seem to extract nR out, nor am I having luck manipulating the powers to what I need.

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

$dU=nCvdT=-PdV$

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