Problem $1$

a$.$ Derive the equation $dU=[C_p-P(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P]dr-[T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P+P(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{P}}{)}_T]dP.$

b$.$ Determine $dU$ for an ideal gas using the expression in part a$.$ Does $dU$ depend on

temperature or pressure or both?

c$.$ Let $=(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ and ${}_T=(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{P}}{)}_T.$ Using the van der Waals equation of state

$P=\frac{RT}{V-b}-\frac{a}{V^2},$ derive expressions for $$ and ${}_{}.$ For a van der Waals gas, does $dU$

depend on temperature or pressure or both?

d$.$ Derive an expression for $(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T$ in terms of $P,T,,$ and ${}_T.$

e$.$ For a van der Waals gas, derive the expression for enthalpy change associated with an

isothermal expansion from $V_1$ to $V_2.$ How does this compare with the enthalpy change

of an ideal gas undergoing isothermal expansion from $V_1$ to $V_2?$