# Molar specific heat capacity for constant volume.

• Nikhil Rajagopalan
In summary, the concept of degrees of freedom is used to compute Cv for gases, with diatomic molecules having a range of 3 to 6 degrees of freedom. The value of Cv can correspond to discrete values such as (3/2)R, (5/2)R, and (7/2)R, and is independent of the nature and properties of the atoms in the molecule. Real gases can have potential energy and multiple modes of vibration and rotation, leading to a gradual increase in heat capacity as these modes become activated at certain temperatures. An example of this is the specific heat of hydrogen.

#### Nikhil Rajagopalan

Dear Experts,
We compute Cv for gases using the idea of equipartition principle and degrees of freedom. In case of a diatomic molecule, there are minimum 3 degrees of freedom (at very low temperatures) and maximum 6 degrees of freedom one of them being vibrational (at high temperatures. Does it imply that Cv can only have discrete values that correspond to (3/2)R , (5/2)R and (7/2)R? Is the value completely independent of the nature and the property of the atoms making the molecule?

Real gases can have potential energy and can have multiple modes of vibration and rotation. Each mode as well as the potential energy associated with each of these modes represents a different degree of freedom. A gas can have many more than 6 degrees of freedom.

As far as the discreteness is concerned, as you note, there are quantum effects that prevent modes from being active at lower temperatures. When these modes start to be activated at a certain temperature, the modes are active in only some of the molecules. So the heat capacity of the gas will gradually increase over a temperature range until essentially all molecules become fully active in that mode.

AM

Andrew Mason said:
So the heat capacity of the gas will gradually increase over a temperature range until essentially all molecules become fully active in that mode.

As an example, the specific heat of hydrogen:
http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/shegas.html#c6

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