Question

The heat capacity of an ideal gas in a polytropic process is C = CV + 0.1R. The value of polytropic exponent is :
1 11

2 10

3 -10

4 -9​

Answer 1

Answer:

That C is the specific heat for the given cycle, i.e.

dQ=nCdT

This is for n moles of gas.(not the n you stated in question)

I will assume

PVz=constant

nCdT=dU+PdV

∫nCdT=∫nCvdT+∫PdV

nCΔT=nCvΔT+∫PVzVzdV

As numerator is a constant, take it out!

Also note that

PiVzi=PfVzf

i=initial

f=final

Focusing on integral only,

PVz∫V−zdV

PVz[V−z+1−z+1]VfVi

Note that the PVz is same for initial and final step. So, we write multiply it inside and do this ingenious work :

−PiVziV−z+1i−z+1+PfVzfV−z+1f−z+1

−PiVi−z+1+PfVf−z+1

Note that PV=nRT

nRΔT−z+1

where ΔT=Tf−Ti

Final equation :

nCΔT=nCvΔT+nRΔT−z+1

C=Cv+R1−z

This will bring you the original equation, you can find Cv by

Cp/Cv=γ

Cp−Cv=R

Using Cp=γCv,

Cv(γ−1)=R

Cv=Rγ−1

Substituting in original equation,

C=Rγ−1+R1−z

I hope it is correct