OCR · A-Level · Physics
Course version: H556 · Version from Sept 2015
Ideal gases: (h) pV = NkT; 1/2 mc² = 3/2 kT
These two equations describe an ideal gas at the particle level. pV = NkT relates pressure p, volume V, number of molecules N, Boltzmann constant k and thermodynamic temperature T; it is the…
Checked against the OCR H556 specification (opens in a new tab) on .
- Point 9 of 10
- 3 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
These two equations describe an ideal gas at the particle level. pV = NkT relates pressure p, volume V, number of molecules N, Boltzmann constant k and thermodynamic temperature T; it is the per-particle form of pV = nRT. The second, ½mc² = 3/2 kT, gives the mean translational kinetic energy of a molecule of mass m and mean-square speed c². The factor 3/2 arises because a molecule can move in three independent directions. Combining them shows that temperature alone fixes mean kinetic energy, so at a given T all gas molecules have the same average translational energy regardless of mass, while lighter molecules have higher mean-square speeds.
What earns the marks
- States pV = NkT and identifies each symbol: p pressure, V volume, N number of molecules, k Boltzmann constant, T temperature in kelvin.
- States ½mc² = 3/2 kT as the mean translational kinetic energy of a molecule, with c² the mean-square speed.
- Explains that the 3/2 factor comes from three translational degrees of freedom.
- Uses the two equations together, for example to show that mean kinetic energy depends only on T.
- Keeps temperature in kelvin and uses consistent SI units for p, V, m and c.
Common mistakes and how to put them right
Using nRT with N, or NkT with n.
Put it right: Pair N with k and n with R.
Treating c² as the square of the average speed.
Put it right: c² is the mean of the squared speeds, which is larger.
Forgetting to convert temperature to kelvin.
Put it right: Add 273 to a Celsius value before substituting.
Omitting the 3/2 factor or replacing it with 1/2.
Put it right: The mean translational kinetic energy is 3/2 kT.
Exam tips
- List the known quantities with units before choosing between pV = NkT and pV = nRT.
- When finding mean-square speed, rearrange ½mc² = 3/2 kT to c² = 3kT/m before substituting.
- Check that mass is in kilograms and volume in cubic metres so that the pascal is consistent.
- Use the kelvin scale throughout and state the conversion if a temperature is given in degrees Celsius.
Try the lesson
The start of the Ideal gases: (h) pV = NkT; 1/2 mc² = 3/2 kT lesson, made from checked teaching notes for this specification point.
Listen
These two equations describe an ideal gas at the particle level. pV = NkT relates pressure p, volume V, number of molecules N, Boltzmann constant k and thermodynamic temperature T; it is the per-particle form of pV = nRT. The second, ½mc² = 3/2 kT, gives the mean translational kinetic energy of a molecule of mass m and mean-square speed c². The factor 3/2 arises because a molecule can move in three independent directions. Combining them shows that temperature alone fixes mean kinetic energy, so at a given T all gas molecules have the same average translational energy regardless of mass, while lighter molecules have higher mean-square speeds. A strong answer earns marks for these points. States pV = NkT and identifies each symbol: p pressure, V volume, N number of molecules, k Boltzmann constant, T temperature in kelvin. States ½mc² = 3/2 kT as the mean translational kinetic energy of a molecule, with c² the mean-square speed. Explains that the 3/2 factor comes from three translational degrees of freedom. Uses the two equations together, for example to show that mean kinetic energy depends only on T. A common slip: Using nRT with N, or NkT with n. Instead, pair N with k and n with R. Exam tip: List the known quantities with units before choosing between pV = NkT and pV = nRT.
Key words
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pV = NkT
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½mc² = 3/2 kT
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Microscopic interpretation of temperature
Try answering aloud or in your head before turning the card.
See every definition
- pV = NkT
- Relates pressure p, volume V, number of molecules N, Boltzmann constant k and thermodynamic temperature T, and is the per-particle form of pV = nRT.
- ½mc² = 3/2 kT
- Gives the mean translational kinetic energy of a molecule of mass m and mean-square speed c².
- Microscopic interpretation of temperature
- Temperature alone fixes mean kinetic energy, so at a given T all gas molecules have the same average translational energy regardless of mass.
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