What is the mean kinetic energy of the molecules of a gas proportional to?

its temperature in kelvin. ½m = (3/2)kT.

Which of these is not an assumption of the kinetic theory?

The molecules attract each other strongly. The model assumes no forces between molecules except during collisions.

A gas is heated from 300 K to 1200 K. What happens to the r.m.s. speed of its molecules?

It doubles. The speed depends on √T, and the temperature is four times higher.

Oxygen and hydrogen are at the same temperature. Which has the greater mean kinetic energy per molecule?

They are the same. Mean kinetic energy depends only on temperature.

Why does the pressure of a gas rise when it is heated at constant volume?

Its molecules hit the walls harder and more often. Faster molecules bring more momentum to each collision and arrive more often.

The kinetic theory of gases · Cambridge A Level Physics

In plain words

A gas is a swarm of tiny molecules flying about at hundreds of metres a second and bouncing off everything. The steady drumming of those collisions on the walls is what we call pressure. Heat the gas and the molecules move faster: that, and nothing more, is what temperature really is.

4 things to know

The model assumes: a very large number of molecules in random motion; the molecules' own volume is negligible compared with the volume of the gas; there are no forces between them except during collisions; collisions are elastic; and the time a collision takes is negligible compared with the time between collisions.
Pressure: each molecule that hits a wall rebounds, so its momentum changes. The wall exerts a force on it, and it exerts an equal and opposite force on the wall. Huge numbers of collisions each second give a steady force, and so a pressure.
pV = ⅓Nm<c²>, where m is the mass of one molecule and <c²> is the mean of the squares of the molecules' speeds. The root-mean-square (r.m.s.) speed is √<c²>.
Comparing that with pV = NkT gives the mean kinetic energy of a molecule: ½m<c²> = (3/2)kT. Temperature in kelvin measures the average kinetic energy of the molecules, and at absolute zero that energy would be zero.

Worked example

Find the r.m.s. speed of nitrogen molecules at 300 K. (Mass of a nitrogen molecule = 4.65 × 10⁻²⁶ kg.)

½m<c²> = (3/2)kT, so <c²> = 3kT ÷ m.
= 3 × 1.38 × 10⁻²³ × 300 ÷ 4.65 × 10⁻²⁶ = 2.67 × 10⁵ m²/s².
r.m.s. speed = √(2.67 × 10⁵) = 520 m/s.

Tips and tricks

At the same temperature every gas has the same mean kinetic energy per molecule. Lighter molecules must therefore be moving faster.
For an r.m.s. speed: square the speeds, find the mean, then take the square root. That order matters.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

The kinetic theory of gases: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the mean kinetic energy of the molecules of a gas proportional to?
- its temperature in kelvin (the answer)
- its temperature in °C
- its pressure only
- its volume only
½m<c²> = (3/2)kT.
Which of these is not an assumption of the kinetic theory?
- The molecules attract each other strongly. (the answer)
- The molecules move at random.
- Collisions are elastic.
- The molecules' own volume is negligible.
The model assumes no forces between molecules except during collisions.
A gas is heated from 300 K to 1200 K. What happens to the r.m.s. speed of its molecules?
- It doubles. (the answer)
- It becomes four times bigger.
- It stays the same.
- It becomes sixteen times bigger.
The speed depends on √T, and the temperature is four times higher.
Oxygen and hydrogen are at the same temperature. Which has the greater mean kinetic energy per molecule?
- They are the same. (the answer)
- oxygen
- hydrogen
- It depends on the pressure.
Mean kinetic energy depends only on temperature.
Why does the pressure of a gas rise when it is heated at constant volume?
- Its molecules hit the walls harder and more often. (the answer)
- Its molecules get bigger.
- There are more molecules.
- Its molecules stick to the walls.
Faster molecules bring more momentum to each collision and arrive more often.

5 questions

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Worksheet

4 questions, 9 marks. Write your answers on paper, then check them.

The kinetic theory of gases

Cambridge International AS & A Level Physics 9702 · 9 marks · papermunch.org

Name ______________________________ Date ______________

Find the mean kinetic energy of a gas molecule at 300 K.[2]
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6.2 × 10⁻²¹ J. 1.5 × 1.38 × 10⁻²³ × 300.
I got this one
Explain how the molecules of a gas cause a pressure on the walls of its container.[3]
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The molecules collide with the walls and rebound, so their momentum changes. By Newton's second law the wall exerts a force on each molecule, and by the third law the molecule exerts an equal and opposite force on the wall. The very many collisions each second give a steady force over the area of the wall, which is a pressure.
I got this one
The kelvin temperature of a gas is doubled. State what happens to the mean kinetic energy of its molecules, and to their r.m.s. speed.[2]
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The mean kinetic energy doubles. The r.m.s. speed increases by a factor of √2.
I got this one
Three molecules have speeds of 300 m/s, 400 m/s and 500 m/s. Find their r.m.s. speed.[2]
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408 m/s. √((300² + 400² + 500²) ÷ 3).
I got this one

Answers: The kinetic theory of gases

1. 6.2 × 10⁻²¹ J. 1.5 × 1.38 × 10⁻²³ × 300.
2. The molecules collide with the walls and rebound, so their momentum changes. By Newton's second law the wall exerts a force on each molecule, and by the third law the molecule exerts an equal and opposite force on the wall. The very many collisions each second give a steady force over the area of the wall, which is a pressure.
3. The mean kinetic energy doubles. The r.m.s. speed increases by a factor of √2.
4. 408 m/s. √((300² + 400² + 500²) ÷ 3).