☰ Contents · Physics

The speed of molecules

Lessons 7–8 · 2 lessons · P. Habibullayev, A. Boydedayev, A. Bahromov, K. Suyarov, J. Usarov, M. Yuldasheva. Physics Grade 9, revised and expanded 3rd edition. G‘afur G‘ulom Publishing House, Tashkent, 2019
7

Speed of gas molecules

Textbook: pp. 25–27
GoalCalculate the root-mean-square speed of gas molecules with v = √(3kT/m₀) = √(3RT/M); know the idea of the Maxwell distribution and Stern’s experiment.
New words
universal gas constant R · universal gaz doimiysi RMaxwell distribution · Maksvell taqsimotiStern experiment · Shtern tajribasithermal motion · issiqlik harakati
Explanation

The average kinetic energy of a molecule is written in two ways: m₀v²/2 and (3/2)kT. Equating them gives v² = 3kT/m₀, i.e. v = √(3kT/m₀). Using M = m₀N_A and R = kN_A (R = 8.31 J/(mol·K)) we get v = √(3RT/M). So the speed is proportional to the square root of T and inversely proportional to the square root of the molar mass: at the same temperature light molecules (H₂) move faster than heavy ones (O₂). Not all molecules have the same speed: in 1859 J. Maxwell theoretically found the distribution of molecules by speed; the most probable speed is somewhat lower than the root-mean-square speed. In 1920 O. Stern measured the speed of silver atoms with two rotating cylinders and confirmed the theory: the atoms left a widened trace on the cylinder wall whose shape resembled the distribution.

Worked examples
Speed of nitrogen N₂ molecules (M = 0.028 kg/mol) at 300 K: v = √(3·8.31·300/0.028) ≈ √(2.67·10⁵) ≈ 5.2·10² m/s, i.e. about 520 m/s.
At the same temperature the speed ratio of hydrogen (M = 2 g/mol) and oxygen (M = 32 g/mol) is v_H/v_O = √(32/2) = √16 = 4: hydrogen molecules move 4 times faster.
Class activity

A small “Stern model” with a paper disc (about 10 cm across) and string: spin the disc and try to draw a straight line from the centre with a pencil; watch how the line curves. What changes when you spin faster? Only paper, string and pencil; no sharp tools; spin at a moderate speed.

Practice
1
If the gas temperature rises 4 times, how does the root-mean-square speed change?
2
Estimate the root-mean-square speed of hydrogen molecules (M = 0.002 kg/mol) at 300 K.
3
Estimate the temperature at which the root-mean-square speed of nitrogen molecules (M = 0.028 kg/mol) is 600 m/s.
4
Why is the air pumped out between the cylinders in Stern’s experiment?