How Do You Calculate Change in Internal Energy of an Ideal Gas?

For an ideal monatomic gas, calculate the change in internal energy using $\Delta U=\frac{3}{2}nR\Delta T$. Alternatively, use the first law of thermodynamics, $Q=\Delta U+W$, giving $\Delta U=Q-W$, where $W$ is work done by the gas.

The Reasoning / Mechanism

The temperature equation is SL and HL content from B.3 Gas laws; applying the first law is HL-only B.4 Thermodynamics. For an ideal monatomic gas,

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

Therefore, for a fixed quantity of gas,

$$\Delta U=\frac{3}{2}nR(T_{\mathrm{f}}-T_{\mathrm{i}}).$$

Here, $n$ is the amount in moles, $N$ is the number of molecules, $R=8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$, and is the Boltzmann constant. Although temperature differences have the same numerical value in kelvin and degrees Celsius, use kelvin when substituting absolute temperatures into gas equations.

This follows because a monatomic ideal gas stores internal energy as random translational kinetic energy. The average per particle is $\frac{3}{2}k_{\mathrm{B}}T$, so multiplying by $N$ gives the stated equation. A temperature rise therefore means .

Choose the method according to whether the question provides temperatures or heat and work transfers.

For example, $2.00\ \mathrm{mol}$ of a monatomic ideal gas is heated from $300\ \mathrm{K}$ to $350\ \mathrm{K}$:

$$\Delta U=\frac{3}{2}(2.00)(8.31)(350-300)=1.25\times10^3\ \mathrm{J}.$$

The positive result means the gas's internal energy increases. If instead $Q=2.00\ \mathrm{kJ}$ enters the gas while the gas does $0.75\ \mathrm{kJ}$ of work, then

$$\Delta U=Q-W=2.00-0.75=1.25\ \mathrm{kJ}.$$

For an ideal gas, internal energy depends only on temperature, not directly on pressure, volume, or the path taken between states.

A common misconception is that all energy transferred as heat becomes internal energy. This is false when the gas expands and does work. Under the IB convention, $Q>0$ when energy enters as heat and $W>0$ when the gas does work on its surroundings.

Exam Technique

For a Calculate question, show the equation, substitution, and final value with units, and keep all energy quantities in consistent units. State or apply the correct sign convention, and do not use $\frac{3}{2}nR\Delta T$ unless the gas is specified or modelled as monatomic.