What is the Work Done on an Ideal Gas Under Isothermal Compression?

[djaymilla]

An ideal gas originally at a temperature T1 and pressure P1 is compressed recersibly against a poston to a volume equal one-half of its original volume. The temperature of the gas is caried during the compression so that at each instant the relatoion P=AV is satisfied, where A is constant. Find the word done on the gas in terms of n, R, and T1.


PV=nRT
P=AV
AV^2=RT
dW= -PdV
W= - (integral from V1 to V2) PdV


I have the answer: [-3nRT1/8] but I cannot figure out how it is derived. I have tried using the work equation for an isothermal equation as well: W=nRT (integral from V1 to V2) dV/V where W=nRT ln (V2/V1)

 

[Mapes]

Hi djaymilla, welcome to PF. Why not just evaluate the work integral, plugging in AV for P?

 

[djaymilla]

Hey Mapes, thanks for the welcome. I guess I am confused with this problem because I don't have concrete terms with which to integrate. I'm not sure how to carry out the integral...

(sorry, this may seem elementary, but I'm now going BACK to school, and have been out of the physics/calculus game for quite some time)

 

[Mapes]

You're almost there. You will need to integrate AV dV from V1 to V2, where A, V1, and V2 are constants. I'm sure you'll be able to figure this out. The remaining steps consist of expressing V2 in terms of V1 (you already described this relationship) and expressing V1 (more precisely, V1^2, hint hint) in terms of T1. Good luck.