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What temperature means for a gas

What is temperature, in terms of the molecules of a gas? Setting the kinetic-theory result beside pV = NkT gives the answer. The mean translational kinetic energy of one molecule is three halves kT. That energy depends only on the absolute temperature, so a lighter molecule must move faster to carry it. The internal energy of an ideal gas is entirely kinetic.

What temperature means for a gas

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Year 13AQA 3.6.2.3CIE 15.3.3, 15.3.4OCR A 5.1.4(f), 5.1.4(h), 5.1.4(i)EDEXCEL 152

Molecular kinetic theory, part 2 of 2. Part 1 is The kinetic theory model and its derivation.

Related elsewhere: the Maxwell–Boltzmann distribution, temperature and catalysts on InkChemistry.

IN THIS TOPIC

  • Use ½m(crms)2 = 3kT/2 = 3RT/2NA, and read off what temperature means.

COMMON MISCONCEPTION

Misconception:

At the same temperature, all gas molecules move at the same speed.

At one temperature every molecule has the same mean kinetic energy, three halves kT, not the same speed. Since that energy is half m c squared, a molecule sixteen times heavier moves at a quarter of the root-mean-square speed.

What temperature is

Put the derived equation beside the empirical pV = NkT and you can see what temperature means.

12m(crms)2=3kT2=3RT2NA\frac{1}{2}m(c_{rms})^{2} = \frac{3kT}{2} = \frac{3RT}{2N_{A}}IN THE AQA DATA BOOKLET
Mean translational kinetic energy per molecule against kelvin temperature: a single straight cyan line through the origin, gradient three halves times the Boltzmann constant.
FIG. 1Mean translational kinetic energy per molecule against kelvin temperature: a straight line through the origin with gradient three halves k.

The average translational kinetic energy per molecule is proportional to absolute temperature and to nothing else. The kind of gas and its pressure make no difference. Absolute temperature measures that energy, which is (3/2)kT per molecule. The two are proportional, but their different units show they are not the same quantity. An ideal gas has no intermolecular forces, so its molecules have no potential energy between them, and its internal energy is entirely kinetic.

The weight of the overlying air does set how large sea-level pressure comes out, but the pushing itself is molecular bombardment, striking floor and walls and ceiling alike. That is why the pressure acts in every direction at once instead of only downwards.

At one temperature every molecule has the same mean kinetic energy, so the light one moves faster
Describe the drawing

Two molecules at the same temperature. The hydrogen molecule is drawn small with a long speed arrow; the oxygen molecule, sixteen times heavier, is drawn larger with an arrow a quarter as long. Their mean kinetic energies are equal, because at one temperature every molecule has the same mean kinetic energy, three halves kT, so sixteen times the mass means a quarter of the root-mean-square speed.

FIG. 2The worked example below, drawn. Both molecules are at the same temperature, so both carry the same mean kinetic energy; the oxygen molecule has sixteen times the mass, so its root-mean-square speed is a quarter of hydrogen's.

WORKED EXAMPLE

The same temperature, two different speeds

Find the mean kinetic energy of a gas molecule at 300 K, and explain which moves faster at that temperature: hydrogen or oxygen molecules.

Mean KE = (3/2)kT = 1.5 × 1.38 × 10−23 × 300 = 6.2 × 10−21 J, and that number is the same for every gas.

Equal ½m(crms)2 with a sixteenth of the mass means hydrogen's speed squared is sixteen times oxygen's.

So hydrogen moves four times faster at the same temperature. A lighter molecule must move faster to carry the same kinetic energy, which is why hydrogen escapes through small gaps readily and why Earth's atmosphere retains so little of it.

GUIDED PRACTICE

Doubling the speed

A gas is at 300 K. To what temperature must it be raised to double the root mean square speed of its molecules? Reason through the proportionality.

Show the working

Temperature tracks the mean kinetic energy, which goes as the speed squared, so doubling crms quadruples the energy.

T must therefore quadruple, to 1200 K. An answer of 600 K would follow from treating T as proportional to the speed rather than to its square.

INDEPENDENT PRACTICE

Atmospheric pressure, from scratch

Air has density 1.2 kg m−3 and its molecules move with a root mean square speed of about 500 m s−1. Use the kinetic theory result p = ⅓ρ(crms)2 to estimate atmospheric pressure.

Show the working

p = ⅓ × 1.2 × 5002 = ⅓ × 3.0 × 105 = 1.0 × 105 Pa.

That is atmospheric pressure, recovered from molecular bombardment alone.

ASSESSMENT FOCUS

  • Doubling the kelvin temperature doubles the mean KE but multiplies crms by only √2. A square root sits between energy and speed.
  • Mean kinetic energy per molecule is (3/2)kT and depends on absolute temperature alone. Naming the gas, or the pressure, in an answer to “what does it depend on” adds something the equation does not contain.
  • Temperatures go in kelvin here without exception, because the relation is a proportionality through the origin. A Celsius value substituted straight in is 273 too small, so the answer is wrong.
  • An ideal gas has no intermolecular forces, so it keeps no potential-energy account and its internal energy is entirely kinetic. So the internal energy of a fixed mass of ideal gas depends on temperature and nothing else.

CHECK YOURSELF

Find the root mean square speed of nitrogen molecules (molar mass 0.028 kg mol−1) at 300 K. (R = 8.31 J K−1 mol−1.)

Show a hint

Write the average-KE equation with molar quantities, then solve for crms.

Show the answer

Work per mole. ½M(crms)2 = 3RT/2, so (crms)2 = 3RT/M.

That gives (crms)2 = 3 × 8.31 × 300 / 0.028 = 2.67 × 105 m2 s−2, so crms = 520 m s−1.

Set that beside the speed of sound in the same gas, which is a separate result with an equation of its own, c = √(γRT/M), quoted here only for the comparison; it is not examined. For nitrogen γ = 1.4, giving c = 350 m s−1. The two are different quantities and neither one caps the other; what they share is their scaling, both going as √(T/M), so both climb with temperature and fall with molar mass. Their ratio is the fixed number √(γ/3), about 0.68 for any diatomic gas.

Mean molecular kinetic energy is proportional to absolute temperature, and equals (3/2)kT.

At one temperature a lighter molecule must move faster to carry the same energy.

An ideal gas keeps no potential energy at all.

10 questions on this topicAnswer them one at a time and mark yourself against the mark scheme.Practise this topic

Or read them with their mark schemes on the kinetic theory model and its derivation questions page.

3 flashcards on this topicGrade yourself on each card, and it comes back after a gap set by how well you knew it.Revise these flashcards

Or revise the whole thermal physics deck of 27 cards.

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  • Use ½m(crms)2 = 3kT/2 = 3RT/2NA, and read off what temperature means.

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