Chemistry Derivations in adiabatic process for ideal gas with C_V and C_P

AI Thread Summary
An ideal gas undergoing an adiabatic process follows the first law of thermodynamics, where the change in internal energy (dU) is equal to the work done (δw) since heat transfer (δQ) is zero. The equations derived for internal energy in terms of temperature and volume (C_V) and temperature and pressure (C_P) lead to the same expression for the relationship between pressure and volume, suggesting a constant k. However, this assumption is flawed because C_V and C_P are not equal, and the correct definition of C_P involves enthalpy rather than internal energy. The discussion highlights the importance of correctly applying thermodynamic definitions to avoid errors in deriving relationships for adiabatic processes. Understanding these distinctions is crucial for accurate thermodynamic analysis.
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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##.
 
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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

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

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