How Do You Calculate the Internal Energy of an Ideal Monatomic Gas?

For both SL and HL IB Physics, calculate the internal energy of an ideal monatomic gas using $U=\frac{3}{2}Nk_{\mathrm{B}}T$ or, equivalently, $U=\frac{3}{2}nRT$. Temperature must be measured in kelvin.

Why These Equations Work

In an ideal gas, particles are modelled as having negligible intermolecular potential energy. A monatomic gas also has no rotational or vibrational energy in this model, so its internal energy consists entirely of the particles’ random translational kinetic energy.

The average kinetic energy of one particle is

$$\overline{E}_{\mathrm{k}}=\frac{3}{2}k_{\mathrm{B}}T.$$

Multiplying by the number of particles $N$ gives

$$U=\frac{3}{2}Nk_{\mathrm{B}}T.$$

Because $N=nN_{\mathrm{A}}$ and $R=N_{\mathrm{A}}k_{\mathrm{B}}$, the mole form is

$$U=\frac{3}{2}nRT.$$

Here, $n$ is the amount of gas in moles, $R=8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$, and is the .

Each atom can move independently in three perpendicular directions. By equipartition, each translational degree of freedom contributes $\frac{1}{2}k_{\mathrm{B}}T$ per particle, producing the factor $\frac{3}{2}$. This model applies to monatomic gases such as helium and argon, not directly to molecular gases with additional energy modes.

Choose the equation from the data given. Use the particle form when $N$ is supplied and the molar form when $n$ is supplied. If pressure and volume are known, combine $PV=nRT$ to obtain $U=\frac{3}{2}PV$, provided the gas is ideal and monatomic.

Worked Example

Calculate the internal energy of $2.0\ \mathrm{mol}$ of an ideal monatomic gas at $300\ \mathrm{K}$.

$$U=\frac{3}{2}(2.0)(8.31)(300)=7479\ \mathrm{J}\approx7.5\ \mathrm{kJ}.$$

For a temperature change, use

$$\Delta U=\frac{3}{2}nR\Delta T.$$

A common misconception is that pressure or volume must appear directly in the internal-energy equation. For an ideal monatomic gas, internal energy depends only on the amount of gas and its absolute temperature; pressure and volume matter only through the ideal gas law.

Exam Technique

This is syllabus topic B.3 Gas laws for SL and HL. For a Calculate question, show the equation, substitute values with temperature in kelvin, and state the answer in joules. Do not confuse $N$, the number of particles, with $n$, the number of moles.