Pressure of an ideal gas

Pressure from molecular hits

Narrated video. Transcript:

Why does a gas push on the walls of its container? Kinetic theory pictures a gas as a huge number of tiny molecules in endless random motion, forever hitting the walls.

Take a cube of gas of side l. Watch one molecule, with velocity components v x, v y and v z, strike the right-hand wall, which is at right angles to the x axis.

The collision is elastic. v x reverses, while v y and v z do not change. So the molecule’s momentum changes by minus 2 m v x, and the wall receives plus 2 m v x.

How many molecules hit an area A of the wall in a short time Δt? Only those within a distance v x Δt of it: a thin slab of volume A v x Δt.

With n molecules per unit volume, the slab holds n A v x Δt molecules. On average, half of them are moving towards the wall, so ½ n A v x Δt molecules hit it.

Momentum handed to the wall: 2 m v x times ½ n A v x Δt, which is n m v x squared A Δt. Divide by Δt to get the force, and by A to get the pressure: P equals n m v x squared.

Molecules have different speeds, so we take the average of v x squared. And since no direction is special, the average of v x squared is one-third of the average of v squared.

So P equals one-third n m v̄ squared, where v̄ squared is the mean of the squared speeds. Collisions between molecules do not change this: in a steady state, a molecule knocked off course is replaced by another moving the same way.

The same argument works for every wall, so the pressure is the same in all directions, just as Pascal’s law says.

Temperature and molecular speed

Narrated video. Transcript:

Multiply P equals one-third n m v̄ squared by the volume V. Since n V is the total number of molecules N, P V equals one-third N m v̄ squared.

Write that as two-thirds of N times half m v̄ squared. N times half m v̄ squared is E, the total kinetic energy of the molecules. So P V equals two-thirds E.

Now compare with the ideal gas equation, P V equals N k B T. Both must be true, so E equals three-halves N k B T.

Divide by N: the average kinetic energy of one molecule is three-halves k B T. Temperature measures the average kinetic energy of the molecules. It does not depend on the pressure, the volume, or which gas it is.

At room temperature, 300 kelvin, that is 1.5 times 1.38 × 10⁻²³ times 300: about 6.2 × 10⁻²¹ joules per molecule. Tiny, but every cubic metre of air holds over 10²⁵ molecules.

Mix two gases at the same temperature, such as heavy nitrogen and light helium. Each kind of molecule has the same average kinetic energy, three-halves k B T.

So the lighter molecules must move faster. The root mean square speed, v rms, is the square root of v̄ squared. It equals the square root of 3 k B T over m.

For nitrogen at 300 kelvin, one molecule has a mass of 4.65 × 10⁻²⁶ kilograms. v rms works out to about 517 metres per second, faster than sound in air.

Hydrogen molecules are 14 times lighter, so they move √14, nearly 3.7 times, faster: around 1930 metres per second. And the speeds are spread out: only a few molecules move at exactly v rms.

Try it: heavy and light molecules

Interactive simulation. Things to try:

• Blue dots are nitrogen (M = 28 g/mol) and purple dots are helium (M = 4 g/mol), at the same temperature.
• Compare the two v_rms values. Their ratio is √(28/4) = √7 ≈ 2.65.
• The average KE per molecule is the same for both: (3/2)k_BT.
• Double T from 300 K to 600 K: both v_rms values grow by √2 ≈ 1.41, and the speed spread gets wider.

Helium and nitrogen

A box holds a mixture of helium and nitrogen at 300 K. Which statement is true?

1. Nitrogen molecules have more kinetic energy on average, because they are heavier
2. Both have the same average kinetic energy, and helium molecules move faster on average
3. Both kinds of molecule have the same average speed
4. Helium molecules have more kinetic energy on average, because they move faster

Argon and chlorine

A flask contains argon and chlorine in the ratio 2 : 1 by mass, at 27 °C. What is the ratio of their rms speeds, v_rms(Ar) : v_rms(Cl₂)? (Molar masses: Ar 39.9 g/mol, Cl₂ 70.9 g/mol.)

1. 1 : 1
2. 1.33 : 1
3. 1.78 : 1
4. 0.75 : 1

Summary

• Pressure of an ideal gas: P = ⅓ n m v̄²
• Pressure and kinetic energy: PV = ⅔ E
• Average KE per molecule: ½ m v̄² = (3/2) k_B T
• rms speed: v_rms = √(3k_BT/m) = √(3RT/M)

• Gas pressure comes from molecules bouncing off the walls: each hit hands the wall a momentum 2mvₓ.
• Only ⅓ of v̄² is along any one direction, because the motion is random.
• Temperature is a measure of the average kinetic energy of the molecules, whatever the gas.
• At the same temperature, lighter molecules move faster: v_rms ∝ 1/√M. N₂ at 300 K has v_rms ≈ 517 m/s.