$(5$ marks$)$ Starting with the following information about $C_p$ and $C_v$ :$(5$ marks$)$ The basic thermodynamic relationships for an axially stressed bar can be written

as:

$d{Q}_{\text{r}\text{e}\text{v}}=Td{S}_{,}d{W}_{\text{r}\text{e}\text{v}}=-d{}_{,}{}_{?}=N$

Where $$ is the stress and $$ is the strain. Write the Fundamental Equation for a one$-$

component bar and show that

$(\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{T}}{)}_{,N}=-(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{N}}{)}_{T,}$

$C_p-C_v=\frac{V{}_p^2}{{}_T}$

Show that $C_p-C_v$ can be expressed as $C_p-C_v=-(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{P}}{)}_T(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{V}}{)}_T.$

${}_p=\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P,{}_T=-\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$