Derivations in adiabatic process for ideal gas with C_V and C_P

[zenterix]

Homework Statement

I am quite confused by the calculations involving an adiabatic process for an ideal gas.

Relevant Equations

Below I show the calculations.

Consider an ideal gas undergoing an adiabatic process.

The first law says that

$$dU=\delta Q+\delta w=\delta w=-PdV$$

since $\delta Q=0$ for an adiabatic process.

$U$ is a function of any two of $P,V$, and $T$.

Consider $U_1=U_1(T,V)$ and $U_2=U_2(T,P)$.

For an ideal gas we have

$$dU_1=\left (\frac{\partial U_1}{\partial T}\right )_VdT=C_VdT=-PdV=\frac{nRT}{V}dV\tag{1}$$

$$dU_2=\left (\frac{\partial U_2}{\partial T}\right )_PdT=C_PdT=-PdV=-\frac{nRT}{V}dV\tag{2}$$

Are these equations both correct?
For (1) we have

$$C_VdT=-\frac{nRT}{V}dV$$

and after integrating we reach

$$P_1V_1^{\gamma}=P_2V_2^{\gamma}=k$$

where $k$ is a constant and $\gamma=1+\frac{R}{C_V}$.

Note the implicit assumption that $C_V$ is constant.

Can we do the same thing for (2) to reach
$$P_1V_1^{\gamma}=P_2V_2^{\gamma}=k$$

where $k$ is a constant and $\gamma=1+\frac{R}{C_P}$?

Something is fishy here since $C_V\neq C_P$.

 

[zenterix]

I think the mistake is that I assumed that

$$C_P=\left (\frac{\partial U}{\partial T}\right )_P\tag{Incorrect}$$

but actually

$$C_P=\left (\frac{\partial H}{\partial T}\right )_P$$

The calculations for (2) in the OP would be

$$\left (\frac{\partial U_2}{\partial T}\right )_PdT=-\frac{nRT}{V}dV$$

$$\frac{1}{T}\left (\frac{\partial U_2}{\partial T}\right )_PdT=-\frac{nR}{V}dV$$