# Showing work done on gas in a reversible process

1. May 31, 2012

### Ryomega

1. The problem statement, all variables and given/known data

Ideal gas initially at temperature Ti, Pressure Pi, Volume Vi is compressed reversibly down to half its original volume. Temperature of gas is varied during the compression so that

P = AV is always satisfied [where A is a constant]

Show that the work done on the gas is:

-$\frac{3}{8}$nRTi

2. Relevant equations

W = ∫ Pdv

W = nRT ln($\frac{Vf}{Vi}$)

W = A$\frac{Vf-Vi}{1-\gamma}$

$\gamma$ = $\frac{Cp}{Cv}$=$\frac{N+2}{N}$ where N is the degree of freedom (3 in this case)

3. The attempt at a solution

I gather that the solution is already in one of those equations and that I am probably being stupid. Yet everything I have tried so far does not get me to the solution.

From the problem I understand that the process is adiabatic since process is reversible and temperature changes to give:

$\frac{P}{V}$=const.

which I am confused on since:

PV^$\gamma$=const

I understand that this may be against the policy of this forum, but would anyone mind showing me how to arrive at this solution? I learn best from examples.

Thank you

2. May 31, 2012

### Sleepy_time

Hi Ryomega. The work done W=nRTln($\frac{Vf}{Vi}$) doesn't apply here because the temperature isn't constant. So from W=∫PdV, and use P=AV in the integral from the initial volume of V to the final volume of V/2. You can workout what A is from the ideal gas equation.

3. Jun 2, 2012

### Ryomega

Aha! I knew I was being stupid! I didn't think about the substitution. Thanks a LOT!!!