Problem $1.$ It can be shown using the Maxwell relations of thermodynamics that the difference

of specific heats can be represented as

$C_p-C_v=T(\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{T}}{)}_P(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_v$

The volume expansivity $$ and the isothermal compressibility $$ can be expressed as

$=\frac{1}{v}(\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$

$=-\frac{1}{v}(\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$

Show that the specific heat difference can be written as

$C_p-C_v=\frac{vT{}^2}{}$

Also, use the above relation to show that the difference between the specific heats for an ideal

gas is the gas constant R$.$ Show all steps in your work for this problem.