How to calculate Cv

Ghodsi

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

When one mole of an ideal gas is compressed adiabatically to one-half of its original volume, the temperature of the gas increases from 273 to 433K. Assuming that Cv is independent of temperature, calculate the value of Cv for this gas.


2. Relevant equations

Cv = dU/dT
dU = dq + dw
dq = 0 for adiabatic processes, thus dU=dw
PV = nRT


3. The attempt at a solution

Cv = -pdV / dT
Cv = (-nRT/V)(dV/dT)

I'm stuck here.
Assuming I'm correct thus far, do I use the initial or final values for T and V (i.e. do I use 273K or 433K?)

beet

I think we might be in the same class...

I've been trying to verify my solution, but no luck so far. This is what I got:

Cv = (dU/dT)
dU = dq + dw, but dq = 0, so dU = dw and Cv = dU/dT

w = -nRTln(V2/V1), but V2 = 1/2V1, so w = -nRTln(1/2), and dw = -nR*ln(1/2)*dT

Substitute the last equation for dw in Cv=dw/dT and you get Cv = -(nR*ln(1/2)*dT)/dT which simplifies to Cv = -nR*ln(1/2).

That's what I got, but I'm not confident that it's correct.

Ghodsi

Elber 10am MWF?

beet

Yeah.

Meemo

I think you should use
T1/T2 = (V2/V1)^γ-1
then you also find the value of
P1 and P2 from
P1V1^γ= P2V2^γ

The put the values in adiabatic process equation
∂W = (P1V1-P2V2)/γ-1
Then use your formulae

Ghodsi

Thanks guys. This is pretty crucial assistance.

dpsguy

Find γ from the above. Then use:

Cp-Cv = R (gas constant)
(Cp/Cv) = γ

Eliminate Cp from these two equations to get Cv.