One of the beauties of thermodynamics is that it provides interrelationships between various

state variables and their derivatives so that information from one set of experiments can be

used to predict the results of a completely different experiment.

a$.$ Show that:

$C_P=\frac{T^2}{}(\frac{\text{d}\text{e}\text{l}(\frac{V}{T})}{\text{d}\text{e}\text{l}\text{T}}{)}_P$

Thus, if the Joule$-$Thomson coefficient $$ and the volumetric equation of state $($in analytic or

tabular form$)$ are known for a fluid, $C_p$ can be computed. Alternatively, if $C_p$ and $$ are known,

$(\frac{\text{d}\text{e}\text{l}(\frac{V}{T})}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ can be calculated, or if $C_P$ and $(\frac{\text{d}\text{e}\text{l}(\frac{V}{T})}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ are known, $$ can be calculated.

b$.$ Show that:

$V_{P,T_2}=\frac{T_2}{T_1}V(P,T_1)+T_2{\displaystyle \underset{P}{\overset{P}{}}},T_2,T_2,T_1\frac{C_P}{T^2}dT$

so that if $$ and $C_P$ are known functions of temperature at pressure $P,$ and $V$ is known at $P$ and

$T_1,$ the specific volume at $P$ and $T_2$ can be computed.A reversible heat pump transfers heat from the air contained in a rigid tank A and rejects heat

to the air contained in a rigid tank B as shown below. Initially both tanks contain $0\mathrm{.}05$ kmol of

air at $200$ kPa and $800$ K$.$ The heat pump operates until the air in tank A drops to $650$ K$.$ Cv

of air is $20\mathrm{.}785$ J$/$mol$.$K$\mathrm{.}97$

a$)$ Calculate the final temperatures and pressures of tanks A and B$.$

b$)$ Calculate the work input to the heat pump.