Step 1: Recall the nature of an ideal gas.
An ideal gas is a simplified model in which gas molecules are treated as point particles moving randomly without any mutual attraction or repulsion.
Because intermolecular forces are neglected, only molecular motion contributes to the energy of the gas.
Step 2: Identify how internal energy is defined.
The internal energy $U$ of a system represents the total microscopic energy of its molecules.
For an ideal gas, this energy arises purely from molecular kinetic energy.
Step 3: Use the thermodynamic relation.
Joule’s law states that, for an ideal gas, internal energy depends only on temperature:
\[ U = f(T) \]
Accordingly, a small change in internal energy is given by:
\[ dU = m C_v \, dT \]
where $C_v$ is the specific heat at constant volume and $T$ is the absolute temperature.
Step 4: Explain the physical reason.
Since ideal gas molecules do not exert forces on one another, their potential energy is zero.
The kinetic theory of gases shows that the average kinetic energy of molecules is directly proportional to temperature.
As a result, changes in pressure or volume do not affect the internal energy unless the temperature changes.
Step 5: Final conclusion.
For an ideal gas, the internal energy depends solely on temperature and is independent of pressure and volume.