Equipartition theorem: Internal energy and heat capacity

The internal energy of a gas is the sum of the molecules’ translational energy , rotational energy , vibrational energy , electronic transition energy , intermolecular forces of interaction and rest-mass energy of electrons and nuclei :

is constant for any gas, while is constant at and if no chemical reaction takes place. For an ideal gas, is zero, leaving the internal energy of an ideal gas as:

, and are given by , and , which are functions of temperature only (see table in previous article). Furthermore, vibrational modes are active only at relatively high temperatures. So if temperatures are not too high (e.g. at room temperature), and the internal energy of a gas is

When , for linear and non-linear molecules,

Therefore,

Multiplying eq58 throughout by ,

Although the equations so far are derived based on gases, they are applicable to any fluid system. We cannot determine precisely and are not able to calculate the absolute internal energy of a fluid. However, we can compute the change in the internal energy of a fluid from one state to another, as a change in internal energy cancels out the value of , e.g. the change in internal energy for a mole of from 200K to 300K at constant volume is:

The heat capacity of a gas at constant volume is defined as . Hence, we can obtain the heat capacity of an ideal gas where vibrational modes are inactive by differentiating eq59 to give:

Eq60 show that the derived heat capacity for an ideal gas at relatively low temperatures is independent of any thermodynamic property. We call such an ideal gas a perfect gas, i.e. an ideal ideal gas. Since the heat capacity of a perfect gas at constant pressure is related to the heat capacity of a perfect gas at constant volume by (see this article for derivation),

As mentioned in an earlier article, and in reality are functions of temperature. This is due contributions from as temperature increases.

 

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