Why is the internal energy of an ideal gas is proportional to its temperature?

Answer 1

To prove this, we will have to use 3 equations, 2 of them related to ideal gases:

(i) pV = nRT

(ii) p = 1/3 d <c2>

(iii) Ek = 1/2 mv2

First of all, an ideal gas has no intermolecular forces. Thus, its molecules have no potential energy.

The internal energy of any system can be defined as the sum of the randomly distributed microscopic potential energy and kinetic energy of the molecules of the system.

It is thus evidently clear that the internal energy of an ideal gas is entirely kinetic. (Ep being zero)

So, U = 1/2 m <c2> (for an ideal gas)

From (i) and (ii), <c2> = 3p/d = 3pV/m = 3nRT/m (d= m/V)

Substituting in the appropriate equation, we get:

U = 1/2 m (3nRT/m)

U = 3/2 nRT

From the above equation, it can be concluded that for a fixed mass of an ideal gas, internal energy is proportional to the thermodynamic temperature. (fixed mass such that n is constant)

Answer 2

The internal energy of an ideal gas is proportional to its temperature because temperature is a measure of the average kinetic energy of the gas molecules. In an ideal gas, the internal energy comes solely from the kinetic energy of its molecules. Therefore, as temperature increases, the average kinetic energy of the gas molecules increases, leading to an increase in internal energy.