​For a van der Waals gas, the partial derivative $$\left( \frac{\partial U}{\partial V} \right)_T$$is

Correct option is C

​The van der Waals equation is a mathematical formula that describes the behavior of real gases. It is an equation of state that relates the pressure, volume, number of molecules, and temperature in a fluid.

One explicit way to write the van der Waals equation is:

where $${\displaystyle p}$$​ is pressure, $${\displaystyle T}$$​is temperature, and $${\displaystyle v=V/n=N_{\text{A}}V/N}$$​ is molar volume, the ratio of volume, $${\displaystyle V}$$​, to quantity of matter, $${\displaystyle n}$$​ $${\displaystyle N_{\text{A}}}$$​is the Avogadro constant and $${\displaystyle N}$$​ the number of molecules). Also $${\displaystyle a}$$​ and $${\displaystyle b}$$​ are experimentally determinable, substance-specific constants, and $${\displaystyle R=kN_{\text{A}}}$$​ is the universal gas constant. This form is useful for plotting isotherms (constant temperature curves).

Consider the dependence of U on V at constant T. This quantity has the units of 

and is called the internal pressure.

For a gas described by the van der Waals equation of state,