Ideal gases: (f) root mean square (r.m.s.) speed; mean square speed

The idea

The mean square speed c² is the average of the squares of the speeds of all particles in a gas. The root mean square (r.m.s.) speed is the square root of the mean square speed: c_rms = √(c²). It is a representative speed for the particles, but it is not the same as the average speed. The r.m.s. speed is used in the kinetic theory equation pV = ⅓ N m c². In multiple-choice questions, you may be asked to calculate c² from given speeds, find c_rms, or compare r.m.s. speed with mean speed. Remember that c² is always positive and c_rms is also positive.

What earns the marks

• Define mean square speed c² as the average of the squares of the speeds of the particles.
• Define root mean square speed c_rms as the square root of the mean square speed.
• Calculate c² by squaring each speed, summing, and dividing by the number of particles.
• Calculate c_rms by taking the square root of c².
• Recognise that c_rms is not the same as the average speed.
• Use c_rms in the kinetic theory equation pV = ⅓ N m c².

Common mistakes and how to put them right

• Confusing r.m.s. speed with average speed.

  Put it right: They are different unless all speeds are equal.

• Forgetting to take the square root when asked for r.m.s. speed after calculating mean square speed.

• Squaring the average speed instead of averaging the squares of the speeds.

• Ignoring the vector nature of velocity.

  Put it right: Speeds are scalar, so no direction is involved.

Exam tips

• When calculating c², ensure you square each speed before averaging.
• For c_rms, always take the positive square root.
• In multiple-choice questions, check whether the question asks for mean square speed or r.m.s. speed.