B.3.7—Ideal gas internal energy

Internal energy model

For an ideal monatomic gas, internal energy is the total random translational kinetic energy of its particles:

$$U=\frac32Nk_BT=\frac32nRT$$

What is included

The model includes translational kinetic energy only. It neglects intermolecular potential energy and does not include rotational or vibrational molecular energy.

Read the dependence

At fixed amount of gas, U is proportional to T. At fixed temperature, U is proportional to N or n. Particle mass does not appear directly in $$U=\frac32Nk_BT$$.

Worked example from the mapped local textbook

For $1.0\,\mathrm{mol}$ of an ideal monatomic gas at $300\,\mathrm{K}$,

$$U=\frac32nRT=\frac32(1.0)(8.31)(300)=3.7\times10^3\,\mathrm{J}$$

This is the total random translational kinetic energy in the model. At the same temperature, doubling the amount of gas doubles $U$.

Common trap

Equal mass samples of different monatomic gases do not necessarily have equal internal energy: compare their number of particles or moles at the same temperature.