Internal energy of ideal gas

Physics· Unit 19: Ideal gases, Learning Outcome 5· 15 min read

1. Definition of Internal Energy for Ideal Gases★★☆☆☆⏱ 5 min

📘 Definition

Internal energy

U

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

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.

📐 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 $E_{p, total} = 0$ , so $U = E_{k, total}$ (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.

2. Calculating Internal Energy★★★☆☆⏱ 6 min

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 $R = 8.31 J mol^{- 1} K^{- 1}$ .

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
$U = 7479 \approx 7500 \text{ J} \quad (2 \text{ significant figures})$

✓ Quick check

Check your understanding:

What is the internal energy of 1.0 mol of ideal monatomic gas at 0 °C?
- 0 J
- ~3400 J
- ~1100 J
- ~7500 J
Reveal answer
1 —
Correct: $U = 1.5 \times 1 \times 8.31 \times 273 \approx 3400 J$ . If you got 0 J, you forgot to convert Celsius to Kelvin!

3. Changes in Internal Energy for Ideal Gases★★★☆☆⏱ 4 min

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.

📐 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
1. An isothermal process is defined as a process that occurs at constant temperature, so $oxed{\\Delta T = 0}$ .
3
1. Since $oxed{\\\Delta U \\propto \\Delta T}$ , $oxed{\\\Delta U = 0}$ , so internal energy stays constant.

4. Common Pitfalls

Wrong move:

Forgetting to convert temperature from Celsius to Kelvin when calculating internal energy

Why:

All gas thermodynamics formulas use thermodynamic temperature in Kelvin, not Celsius. Using Celsius gives a wildly incorrect result

Correct move:

Always convert temperature to Kelvin first by adding 273 (273.15 is acceptable but not required for most CIE questions)

Wrong move:

Including intermolecular potential energy when calculating internal energy of an ideal gas

Why:

By definition, ideal gases have no intermolecular forces, so potential energy is always zero

Correct move:

Ignore potential energy entirely for ideal gas internal energy calculations

Wrong move:

Claiming internal energy changes when pressure or volume changes for an ideal gas at constant temperature

Why:

Internal energy of an ideal gas depends only on temperature, not pressure or volume

Correct move:

If temperature is constant, $oxed{\

Wrong move:

Why:

Correct move:

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 2
Calculate internal energy of helium gas
2023 · 1
Explain zero potential energy in ideal gas

Going deeper

syllabusCIE 9702 A-Level Physics Syllabus

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.