Specific Heat at Constant Volume of a Vanderwalls Gas
Specific heat at constant volume, denoted by Cv, is defined as the amount of heat required to raise the temperature of a substance by one degree Celsius per unit mass, while keeping the volume constant. For a Vanderwaals gas with a fixed number of particles, the specific heat at constant volume can be determined as follows:

Vanderwaals Gas Equation
The Vanderwaals gas equation takes into account both the volume occupied by gas molecules and the attractive forces between them. It is given by:
(P + a(n^2/V^2))(V - nb) = nRT
Where:
- P is the pressure
- V is the volume
- n is the number of moles
- T is the temperature
- a and b are constants specific to the gas

Derivation of Specific Heat at Constant Volume
To determine the specific heat at constant volume for a Vanderwaals gas, we can differentiate the equation of state with respect to temperature at constant volume:
(∂U/∂T)V = Cv
Where U is the internal energy of the gas. By differentiating the Vanderwaals equation of state with respect to temperature, we can calculate the specific heat at constant volume for a Vanderwaals gas.

Conclusion
In conclusion, the specific heat at constant volume of a Vanderwaals gas with a fixed number of particles can be determined by differentiating the Vanderwaals gas equation of state with respect to temperature at constant volume. This allows us to calculate the amount of heat required to raise the temperature of the gas while keeping the volume constant.