Physics · Kinetic Theory · NEET
At the same temperature, rms speed is v_rms = sqrt(3 kB T / m). Lighter molecules move faster because m is small. Hydrogen (m about 3.3e-27 kg) has a much higher rms speed than oxygen (m about 5.3e-26 kg for O2). So a bigger fraction of hydrogen molecules already move faster than escape velocity and leak into space, while heavy oxygen stays trapped. This is why Earth kept its oxygen but lost most of its hydrogen.
Set v_rms = v_escape. Then sqrt(3 kB T / m) = v_escape. Square both sides and solve: T = m * v_escape^2 / (3 kB). For an oxygen molecule with m = 2.76e-26 kg and v_escape = 11200 m/s, T comes out to about 8.36e4 K. That is far hotter than any normal atmosphere, which is why oxygen stays put on Earth.
No. Escape velocity depends only on the planet: v_escape = sqrt(2 g R) = sqrt(2 G M / R). It is the same speed for a hydrogen molecule and an oxygen molecule. What differs is the rms speed of the molecule, which does depend on molecular mass. So the molecule changes, but the finish line (escape velocity) stays fixed.
The Moon's escape velocity is small (about 2.4 km/s) because it has low mass and small radius. Gas molecules on the Moon reach that speed easily at ordinary temperatures, so almost all gas molecules escaped long ago. A planet holds an atmosphere only when its escape velocity is much larger than the rms speed of its gas molecules.
No, they are different ideas. rms speed is the effective speed of gas molecules due to temperature, v_rms = sqrt(3 kB T / m). Escape speed is the minimum speed an object needs to leave a planet's gravity, v_escape = sqrt(2 g R). We only equate them to find the special temperature at which molecules become fast enough to escape.
At what temperature will the rms speed of oxygen molecules become just sufficient for escaping from the Earth's atmosphere? (Given: mass of an oxygen molecule m = 2.76e-26 kg, Boltzmann constant kB = 1.38e-23 J/K, escape speed = 11200 m/s)
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Set v_rms = v_escape, so sqrt(3 kB T / m) = v_escape. Solving gives the escape temperature T = m * v_escape^2 / (3 kB).
About 11.2 km/s (11200 m/s). It comes from v_escape = sqrt(2 g R) using g = 9.8 m/s^2 and Earth's radius R about 6.4e6 m.
Light gases escape first. Lighter molecules have higher rms speed at the same temperature, so more of them cross the escape speed and leak into space.
No. Molecules have a range of speeds (Maxwell-Boltzmann distribution). rms speed is just a representative value; some molecules move much faster and can escape even when the average is below escape speed.
Because oxygen is heavy, so its rms speed is low. To make its rms speed reach 11.2 km/s you need a temperature near 8.36e4 K, far above real atmospheric temperatures, so oxygen stays bound to Earth.
