(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A gas is compressed, allowed to expand linearly, and then cooled at constant volume as shown in the P-V graph. How much work is done by the gas in this cycle?

in the graph: compression is a straight line at P(atm)=3 that goes from V(liters)=1 to V=5; cooling occurs at V=5 from P=4 to P=3; a diagonal line from (1,3) to (5,4) is the expansion; a graph of a right triangle should be drawn with this information

2. Relevant equations

PV = nRT or PV = NkT, P1/T1 = P2/T2 or V1/T1 = V2/T2 ?

if it is adiabatic: f/2 dT/T = -dV/V where f is the number of degrees of freedom; further math will obtain PV^((f+2)/f) = constant

3. The attempt at a solution

Well, i'm not sure if it is adiabatic or isothemral... i think adiabatic since the cooling occurs all at once at the end of the graph?? from what i understand.. the volume goes from 1 to 5 liters during expansion while the pressure goes form 3 to 4 atm. then the cooling causes the pressure to go back to 3 while the volume is still 5?

Vfinal Tfinal^(f/2) = Vinitial Tinitial^(f/2)

5(Tfinal^(f/2))= 1(Tinitial^(f/2))

...

okay, okay.. stumbled on something in notes:

- integral P dV

since Pinitian Vinitial^((f+2)/f) = Pfinal Vfinal^((f+2)/f) it is a constant

so, P = (constant)V^-((f+2)/f)

so,W = - integral (constant)V^-((f+2)/f) dV

ends up to be

{[-Pinitial Vinitial^((f+2)/f) ]/1 - ((f+2)/f) } (Vfinal^[1 - ((f+2)/f)] - Vinitial^[1 - ((f+2)/f)] )

plugging values

{[ -(3)(1)^((f+2)/f) ] / 1 - ((f+2)/f) } (5^[1 - ((f+2)/f)] - 1^[1 - ((f+2)/f)] )

degrees of freedom: maybe the word "linearly" is a hint here?

if the gas is monoatomic: ((f+2)/f) = 5/3

if the gas is diatomic: 7/5,

diatomic at high temperatures or triatomic: 8/5

which degree of freedom should be used and why?

for now i'm going to go with 5/3:

{[ -(3)(1)^(5/3) ] / 1 - (5/3) } (5^[1 -(5/3)] - 1^[1 - (5/3)] )

9/2 [5^(-2/3) - 1^(-2/3)]

W ~ -3 J by the gas... so work is actually done on the gas?

any explaination would be appreciated, thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Thermodynamics - compressed gas P-V graph, how much work is done?

**Physics Forums | Science Articles, Homework Help, Discussion**