# Internal energy of ideal gas

> Physics · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9702-u19-internal-energy-of-ideal-gas/

This module covers the definition of internal energy for an ideal gas, explains why intermolecular potential energy is zero for ideal gases, and shows how to calculate total internal energy from temperature and amount of gas.

**Prerequisites:** [Ideal gas assumptions and kinetic theory](https://www.owlsprep.com/study/cie-9702-u19-ideal-gas-assumptions/); [Thermodynamic temperature scale](https://www.owlsprep.com/study/cie-9702-u18-thermodynamic-temperature/)

## Learning objectives

- Define internal energy for an ideal gas and explain the origin of its components
- Justify why intermolecular potential energy is zero for an ideal gas
- Calculate the internal energy of a given amount of ideal gas and its changes

## Definition of Internal Energy for Ideal Gases

**Internal energy** — The total sum of all random kinetic energies and all intermolecular potential energies of all particles within a thermodynamic system

*Notation:* U

*Example:* For an ideal gas, potential energy is zero, so internal energy equals total random kinetic energy

A core assumption of the kinetic theory model for ideal gases is that there are no intermolecular forces between gas molecules, except during instantaneous collisions. If there are no intermolecular forces, no work is required to move molecules closer together or further apart, so intermolecular potential energy is always exactly zero for an ideal gas.

> **info**
>
> This is only true for ideal gases. For real gases, liquids and solids, potential energy is non-zero and contributes to total internal energy.

**Worked example:** Explain why the internal energy of an ideal gas depends only on its temperature.

1. Start with the definition of internal energy for any system:
2. $$U = E_{k,\text{total}} + E_{p,\text{total}}$$
3. For an ideal gas, there are no intermolecular forces, so $E_{p,\text{total}} = 0$, so $U = E_{k,\text{total}}$ (total kinetic energy only).
4. From kinetic theory, the average kinetic energy of an ideal gas molecule is directly proportional to thermodynamic temperature $T$.
5. Total kinetic energy is the product of number of molecules and average kinetic energy, so total internal energy is proportional to $T$ only for fixed mass.

## Calculating Internal Energy

The total internal energy of an ideal gas is derived from kinetic theory, based on the number of degrees of freedom (independent modes of kinetic energy storage) of the gas molecule. The general formula is:

$$U = \frac{f}{2} n R T$$

Where $f$ is the number of degrees of freedom. For monatomic ideal gases (noble gases like helium, argon), $f=3$ (three translational modes, no rotational modes that are active at room temperature), so the formula simplifies to the most common form asked in CIE exams:

$$U = \frac{3}{2} n R T$$

**Worked example:** Calculate the internal energy of 2.0 mol of helium (treated as an ideal monatomic gas) at 27 °C. Take $R = 8.31 \text{ J mol}^{-1} \text{K}^{-1}$.

1. First convert temperature from degrees Celsius to Kelvin, as the formula requires thermodynamic temperature:
2. $$T = 27 + 273 = 300 \text{ K}$$
3. Substitute values into the formula for monatomic ideal gas internal energy:
4. $$U = \frac{3}{2} n R T = 1.5 \times 2.0 \times 8.31 \times 300$$
5. $$U = 7479 \approx 7500 \text{ J} \quad (2 \text{ significant figures})$$

**Check your understanding**

Check your understanding:

1. What is the internal energy of 1.0 mol of ideal monatomic gas at 0 °C?

   - 0 J
   - ~3400 J
   - ~1100 J
   - ~7500 J

   *Answer:* ~3400 J

   *Why:* Correct: $U = 1.5 \times 1 \times 8.31 \times 273 \approx 3400 \text{ J}$. If you got 0 J, you forgot to convert Celsius to Kelvin!

## Changes in Internal Energy for Ideal Gases

Since internal energy of an ideal gas depends only on temperature, any change in internal energy is directly proportional to the change in temperature. Pressure and volume do not affect internal energy of an ideal gas directly, they only change it if they cause a change in temperature.

> **tip**
>
> A common exam question tests this rule: for any isothermal (constant temperature) process for an ideal gas, change in internal energy $\Delta U = 0$, always.

**Worked example:** A fixed mass of ideal gas undergoes an isothermal expansion. State and explain what happens to its internal energy.

1. 1. For an ideal gas, internal energy depends only on thermodynamic temperature, because potential energy is zero.
2. 2. An isothermal process is defined as a process that occurs at constant temperature, so $\boxed{\Delta T = 0}$.
3. 3. Since $\boxed{\\\Delta U \propto \Delta T}$, $\boxed{\\\Delta U = 0}$, so internal energy stays constant.

**Exam command terms**

Common command terms for this topic in CIE exams:

- **Explain why** — You must link your answer to the ideal gas assumption of no intermolecular forces *(When asked why internal energy depends only on temperature, always start with 'no intermolecular forces → potential energy = 0' to get full marks.)*

- **Calculate** — You must show all substitution steps and give final answer with correct units and matching significant figures

## Common pitfalls

- **Wrong:** Forgetting to convert temperature from Celsius to Kelvin when calculating internal energy
  - Why it fails: All gas thermodynamics formulas use thermodynamic temperature in Kelvin, not Celsius. Using Celsius gives a wildly incorrect result
  - Correct: Always convert temperature to Kelvin first by adding 273 (273.15 is acceptable but not required for most CIE questions)
- **Wrong:** Including intermolecular potential energy when calculating internal energy of an ideal gas
  - Why it fails: By definition, ideal gases have no intermolecular forces, so potential energy is always zero
  - Correct: Ignore potential energy entirely for ideal gas internal energy calculations
- **Wrong:** Claiming internal energy changes when pressure or volume changes for an ideal gas at constant temperature
  - Why it fails: Internal energy of an ideal gas depends only on temperature, not pressure or volume
  - Correct: If temperature is constant, $\boxed{\\

## What's next

Understanding internal energy of ideal gases is the foundation for studying the first law of thermodynamics, which connects changes in internal energy to heat transfer and work done on or by the gas. This concept also underpins all further work on thermal processes, including adiabatic and isothermal changes, and heat engines. You will use the relation between internal energy and temperature repeatedly when solving problems on gas laws and thermodynamics, so it is critical to master the link between ideal gas assumptions and zero potential energy.

- [First Law of Thermodynamics](https://www.owlsprep.com/study/cie-9702-u20-first-law-of-thermodynamics/)
- [Thermodynamics](https://www.owlsprep.com/study/cie-9702-u20-overview/)
- [Temperature and Thermal Equilibrium](https://www.owlsprep.com/study/cie-9702-u20-temperature-and-thermal-equilibrium/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-9702-u19-internal-energy-of-ideal-gas/