Explain why the internal energy of an ideal gas depends only on its temperature and not on its volume.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
An ideal gas is allowed to expand freely into a vacuum (Joule expansion) so that no work is done on or by the gas and no heat is transferred to or from the gas.
Model answer (4 marks)
In an ideal gas the molecules are considered to have no intermolecular forces, so the only energy they possess is kinetic.
Because there are no forces between the molecules, there is no potential energy that depends on the distance between them; the internal energy is therefore solely the sum of the random kinetic energies of all the molecules.
The mean kinetic energy of a molecule in an ideal gas is proportional to the absolute temperature (½ m v² ∝ T). Consequently the total internal energy, which is the number of molecules times the mean kinetic energy, depends only on temperature and not on the volume of the gas.
Thus, for an ideal gas, the internal energy is a function of temperature alone, independent of volume.
Because there are no forces between the molecules, there is no potential energy that depends on the distance between them; the internal energy is therefore solely the sum of the random kinetic energies of all the molecules.
The mean kinetic energy of a molecule in an ideal gas is proportional to the absolute temperature (½ m v² ∝ T). Consequently the total internal energy, which is the number of molecules times the mean kinetic energy, depends only on temperature and not on the volume of the gas.
Thus, for an ideal gas, the internal energy is a function of temperature alone, independent of volume.
Examiner tips
- Mention the absence of intermolecular forces first; this explains no potential energy. State that internal energy = kinetic energy only. Link mean kinetic energy to absolute temperature. Use the phrase "depends only on temperature" to match the mark scheme.
Common mistakes
- Confusing internal energy with enthalpy or total energy. Including volume in the explanation. Forgetting to emphasise that the kinetic energy is random and proportional to T.
Mark scheme (4 marks)
- In an ideal gas, intermolecular forces between particles are assumed to be zero (negligible).
- Because there are no intermolecular forces, there is no potential energy associated with the separation of the molecules.
- The internal energy of the gas consists entirely of the (random) kinetic energy of its molecules.
- The mean kinetic energy of the molecules depends only on (absolute / kelvin) temperature, so the internal energy depends only on temperature.
Key terms in this question
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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