RMS, Average and Most Probable Speed of Gas Molecules

By Aniket Bhardwaj · 11 October 2026 · Updated 11 October 2026 · Class 11 Chemistry

Gas molecules do not all move at the same speed. At any temperature, some are slow and some are fast. Class 11 kinetic theory gives you three standard ways to describe "the speed" of a gas: the root-mean-square speed, the average speed and the most probable speed. Questions often ask for one of them, or for the ratio between them. Here is how they differ, with the numbers worked out.

Three horizontal bars compare oxygen speeds at 300 K: most probable 394.8 m/s, average 445.5 m/s, and RMS 483.6 m/s. — For oxygen at 300 K, the RMS speed is highest and the most probable speed is lowest.

The three formulas

u_rms = √(3RT / M)
u_avg = √(8RT / πM)
u_mp = √(2RT / M)

In each formula R = 8.314 J K⁻¹ mol⁻¹, T is in kelvin and M is the molar mass in kg/mol. Using g/mol here is the most common mistake in this chapter, and it gives an answer that is wrong by a factor of about 32 (the square root of 1000).

Most probable speed (u_mp): the speed that the largest number of molecules have.
Average speed (u_avg): the simple mean of all molecular speeds.
Root-mean-square speed (u_rms): the square root of the mean of the squares of the speeds. It is connected to the average kinetic energy: ½ M u_rms² per mole equals (3/2)RT.

Their order and ratio

Dividing the formulas gives a fixed ratio for every gas at every temperature:

u_mp : u_avg : u_rms = √2 : √(8/π) : √3 = 1 : 1.128 : 1.225

Check: √(8/π) = √2.546 = 1.596, √2 = 1.414, and 1.596 / 1.414 = 1.128. Also √3 / √2 = 1.732 / 1.414 = 1.225. So u_mp < u_avg < u_rms, always.

What changes the speed

Temperature: speed is proportional to √T. If T doubles, speed increases by a factor of √2 = 1.414, not 2.
Molar mass: speed is proportional to 1/√M. Light gases move faster than heavy gases at the same temperature.
Pressure: at a fixed temperature, pressure does not appear in the formulas, so it does not change the speeds.

Example 1: All three speeds for oxygen at 300 K.
Step 1: M = 32.00 g/mol = 0.03200 kg/mol. T = 300 K.
Step 2: u_rms = √(3 × 8.314 × 300 / 0.03200) = √(7482.6 / 0.03200) = √233,831 = 483.6 m/s.
Step 3: u_avg = √(8 × 8.314 × 300 / (π × 0.03200)) = √(19,953.6 / 0.10053) = √198,482 = 445.5 m/s.
Step 4: u_mp = √(2 × 8.314 × 300 / 0.03200) = √(4988.4 / 0.03200) = √155,888 = 394.8 m/s.
Step 5: Ratio check: 394.8 : 445.5 : 483.6 = 1 : 1.128 : 1.225. ✓
Answer: u_rms ≈ 484 m/s, u_avg ≈ 446 m/s, u_mp ≈ 395 m/s.

Example 2: Effect of doubling the temperature. The u_rms of O₂ at 273.15 K is 461.4 m/s. What is it at 546.3 K?
Step 1: The new temperature is exactly twice the old one.
Step 2: u_rms scales as √T, so the new speed = 461.4 × √2 = 461.4 × 1.4142 = 652.5 m/s.
Step 3: Direct check: √(3 × 8.314 × 546.3 / 0.03200) = √425,800 = 652.5 m/s. ✓
Answer: about 652.5 m/s.

Example 3: Hydrogen against oxygen at the same temperature.
Step 1: At the same T, u_rms ∝ 1/√M, so u_rms(H₂) / u_rms(O₂) = √(32.00 / 2.016).
Step 2: 32.00 / 2.016 = 15.87, and √15.87 = 3.98.
Step 3: Check with 300 K values: H₂ u_rms = √(3 × 8.314 × 300 / 0.002016) = 1926.6 m/s. And 1926.6 / 483.6 = 3.98. ✓
Answer: hydrogen molecules are about four times faster than oxygen molecules.

Example 4: At what temperature does N₂ have the same u_rms as O₂ at 300 K?
Step 1: u_rms depends on T/M. For equal speeds, T_N2/M_N2 = T_O2/M_O2.
Step 2: T_N2 = 300 × (28.01 / 32.00) = 300 × 0.8753 = 262.6 K.
Answer: about 262.6 K (roughly −10.6 °C). The lighter N₂ must be colder to move at the same speed.

Common mistakes

Using M in g/mol. Convert to kg/mol before you put M into R = 8.314 J K⁻¹ mol⁻¹. (If you use R = 8.314 × 10⁷ erg K⁻¹ mol⁻¹, M in g/mol is correct, and the speed comes out in cm/s.)
Using Celsius. T must be in kelvin.
Thinking all molecules have the rms speed. It is a statistical measure. Individual molecules have a wide range of speeds.
Mixing up the order. The correct order is u_mp < u_avg < u_rms.
Believing pressure changes the speed. At constant T, compressing a gas does not change the average molecular speed.
Doubling T and doubling speed. Speed goes as the square root of T.

Link to Graham's law

Because speed goes as 1/√M, lighter gases escape faster through a small opening. This is the idea behind Graham's law of effusion, which says the rates of effusion of two gases at the same T and P are in the inverse ratio of the square roots of their molar masses.

Exam relevance

Question type	Key idea
Find u_rms, u_avg or u_mp for a gas	Use the correct formula with M in kg/mol and T in K
Ratio of the three speeds	1 : 1.128 : 1.225 (mp : avg : rms)
Compare two gases	Speed ∝ 1/√M at equal T
Effect of temperature	Speed ∝ √T

Check your own syllabus to see which of the three speeds is asked at your level. Not every board asks for all of them.

Use the suite to check your square roots and unit conversions while you practise these numericals.

Open the Chemistry Calculator Suite →