Question $2$

$(12$ marks$)$

Derive expressions for $($a$)u,($b$)h,$ and $($c$)s$ for a gas whose equation of state is $P(v-a)=RT$ for a gas, where $a$ is a constant, for an isothermal process.

Useful equations:

The changes in internal energy, enthalpy, and entropy of a simple compressible substance can be expressed in terms of pressure, specific volume, temperature, and specific heats alone as

$du=c_{U}dT+[T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_U-P]dU$

$dh=c_{P}dT+[U-T(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{T}}{)}_P]dP$

$ds=\frac{c_U}{T}dT+(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_U^{dU}$

or

$ds=\frac{c_p}{T}dT-(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{T}}{)}_{P}dP$