Which one among the following is the correct option for right relationship between C_P and C_V for one mole of ideal gas ?

For one mole of an ideal gas we start from the definitions of the molar heat capacities:

$C$_V = \left(\frac{\partial U}{\partial T}\right)_V
$C$_P = \left(\frac{\partial H}{\partial T}\right)_P

where U is the internal energy, H is the enthalpy, T is temperature, and the subscripts indicate the variable held constant.

For an ideal gas the internal energy depends only on temperature, so U = U(T). The enthalpy is defined as

$H\; =\; U\; +\; PV$

and for one mole of an ideal gas the equation of state gives PV = RT. Substituting, we obtain

$H\; =\; U\; +\; RT$

Now differentiate H with respect to T at constant pressure:

$\backslash frac\{\backslash partial\; H\}\{\backslash partial\; T\}\backslash bigg|\_P\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; T\}\backslash bigg|\_P\; +\; R$

Since U depends only on T, the partial derivative of U with respect to T is the same whether pressure or volume is held constant, i.e.

$\backslash frac\{\backslash partial\; U\}\{\backslash partial\; T\}\backslash bigg|\_P\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; T\}\backslash bigg|\_V\; =\; C$_V

Therefore the expression for C_P becomes

$C$_P = C_V + R

Rearranging this relationship yields the required result:

$C$_P − C_V = R

This equation shows the difference between the molar heat capacities at constant pressure and constant volume for a single mole of an ideal gas is exactly the universal gas constant R.