What is the molar heat capacity at constant volume and constant pressure for monatomic and diatomic gas?
The correct answer and explanation is:
The molar heat capacity at constant volume (Cv) and constant pressure (Cp) for gases varies depending on whether the gas is monatomic or diatomic.
Monatomic Gas:
For a monatomic ideal gas, the molar heat capacity at constant volume (Cv) is 3/2 R, and the molar heat capacity at constant pressure (Cp) is 5/2 R.
- Cv (Constant Volume):
A monatomic gas only has translational degrees of freedom (motion in three dimensions: x, y, and z axes). According to the equipartition of energy theorem, each degree of freedom contributes 1/2 kT to the energy per molecule, where k is the Boltzmann constant and T is the temperature. Since there are three translational degrees of freedom for a monatomic gas, the total energy per mole of gas is 3/2 RT. The heat capacity at constant volume is the rate of change of internal energy with respect to temperature, which is 3/2 R. - Cp (Constant Pressure):
The relationship between Cp and Cv for any gas is given by the equation Cp = Cv + R. For a monatomic gas, this results in Cp being 5/2 R. This is because at constant pressure, part of the heat added to the system is used to do work as the gas expands.
Diatomic Gas:
For a diatomic ideal gas, the molar heat capacity at constant volume (Cv) is 5/2 R, and the molar heat capacity at constant pressure (Cp) is 7/2 R.
- Cv (Constant Volume):
A diatomic gas has both translational and rotational degrees of freedom. It has three translational degrees of freedom (like monatomic gases), but it also has two additional rotational degrees of freedom, corresponding to rotation about two axes perpendicular to the bond. Therefore, the total energy per mole of gas is 5/2 RT, and the heat capacity at constant volume is 5/2 R. - Cp (Constant Pressure):
Again, using the relation Cp = Cv + R, for a diatomic gas, the molar heat capacity at constant pressure is 7/2 R. The additional heat required at constant pressure accounts for both the increase in internal energy and the work done during expansion.
In summary:
- Monatomic gas: Cv = 3/2 R, Cp = 5/2 R.
- Diatomic gas: Cv = 5/2 R, Cp = 7/2 R.
These values are for ideal gases under the assumption of no quantum effects at room temperature.