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The state of a real gas if changed from pressure $$P_1,$$ temperature $$T_1$$ to pressure $$P_2$$ temperature $$T_2.$$ The change in enthalpy, $$h_2 - h_1,$$ is given by
$$\int_{T_1}^{T_2} C_p dT $$
$$\int_{T_1}^{T_2} C_p dT + \int_{p_1}^{p_2} \left( \frac{\delta V}{\delta P} \right)_T dp$$
$$\int_{T_1}^{T_2} C_v dT + \int_{p_1}^{p_2} \left[ V - T \left(\frac{\delta V}{\delta T}\right)_p \right] dp$$
$$\int_{T_1}^{T_2} C_p dT + \int_{p_1}^{p_2} \left[ V - T \left(\frac{\delta V}{\delta T}\right)_p \right] dp$$
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