Fraction of heat energy used to do expansion work of a gas?

[JustinLiang]

Homework Statement

Heat Q flows into a monatomic ideal gas, and the volume increases while the pressure is kept constant. What fraction of the heat energy is used to do the expansion work of the gas?

Homework Equations

ΔU = Q - W
Q = nC_pΔT
W = PΔV
C_p = C_v + R => C_p = 3R/2 + R => C_p = 5R/2

The Attempt at a Solution

So we sub in the Q and W in the internal energy formula and we get:
ΔU = nC_pΔT - PΔV
ΔU = (5/2)(nRΔT) - PΔV
ΔU = (5/2)(PΔV) - PΔV
ΔU = (3/2)(PΔV)

Therefore we can take ΔU/Q to find the fraction:
(3/2)(PΔV)/(5/2)(PΔV)

Which gives me 3/5. However the answer is 2/5. What am I doing wrong?
Personally, I think the answer key is wrong.

Thanks.

 

[Hatmatbbat10]

I know this is an old question, but I thought I'd reply for future students that need help with this question.

All of the work shown is right, except for the very end. The question wants the ratio of W/Q, while the original poster found the ratio of ΔU/Q.

W/Q =
PΔV / (5/2)(PΔV) = 2/5

 

[BabyPhys]

Could someone please explain to me how (nRΔT) = PΔV in the third step of this derivation?

 

[Chestermiller]

Could someone please explain to me how (nRΔT) = PΔV in the third step of this derivation?

Ideal gas law. PV=nRT. If P is constant, then PΔV=nRΔT

Chet