Problem $1$:

$($a$)$ Making use of the fact that

$dH=C_{P}dT+[V-T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P]dP$

is an exact differential expression, show that:

$(\frac{del{C}_P}{\text{d}\text{e}\text{l}\text{P}}{)}_T=-T(\frac{de{l}^{2}V}{del{T}^2}{)}_P$

What is the result of application of this equation to an ideal gas?

$($b$)$ Heat capacities $C_V$ and $C_P$ are defined as temperature derivatives respectively of $U$ and $H.$ Because these properties

are related, one expects the heat capacities also to be related. Show that the general expression connecting $C_P$ to $C_V$

is:

$C_P=C_V+T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$