Required information

Given:

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

The heat capacities $C_V$ and $C_P$ are defined as temperature derivatives of $U$ and $H,$ respectively. Because these properties are related,

one expects the heat capacities also to be related. Identify the correct expression that shows the general expression

connecting $C_P$ to $C_V.$

$(1)C_P=C_V+T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ and $C_P-C_V=-TV(\frac{}{k})$

$(2)C_P=C_V-T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ and $C_P-C_V=TV(\frac{}{k})$

$(3)C_P=C_V-T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ and $C_P-C_V=-TV(\frac{}{k})$

$(4)C_P=C_V+T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ and $C_P-C_V=TV(\frac{}{k})$

The correct expression is in option