For a monoatomic ideal gas, what is the equation for Cv?

For a monoatomic ideal gas, the heat capacity at constant volume, known as Cv, is determined by the degrees of freedom of the gas molecules. In the case of monoatomic gases, such as helium, neon, or argon, each atom has only three translational degrees of freedom (movement in the x, y, and z directions).

The equipartition theorem states that each degree of freedom contributes (1/2)kT to the internal energy, where k is the Boltzmann constant, and T is the temperature. In terms of the ideal gas constant, R, for a monoatomic ideal gas, the total contribution to the internal energy comes from these three translational degrees of freedom. Thus, the expression for Cv becomes:

Cv = (3/2)R.

This value reflects the amount of energy required to raise the temperature of one mole of a monoatomic ideal gas by one degree Kelvin when the volume is held constant. The other choices do not correspond to the properties of a monoatomic ideal gas.