There is no atmosphere on the surface of the moon because

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Correct — C, Escape velocity of gas molecules is less than rms velocity. Whether a body holds on to an atmosphere is decided by a single comparison: the escape velocity at its surface against the thermal speeds of the gas molecules there. Escape velocity depends only on the body's mass and radius, v = √(2GM/R), and for the Moon it works out at about 2.4 kilometres per second against roughly 11.2 for the Earth — the Moon has about one-eightieth of the Earth's mass, and although it is also smaller, the mass term dominates. Molecular speed depends on temperature and on molecular mass, through the root-mean-square speed v = √(3RT/M), and it is not a single value: the molecules of a gas are spread over a distribution, so a fraction of them always move much faster than the average. On the Moon that fraction exceeds 2.4 kilometres per second, so those molecules leave, and the distribution then repopulates its fast tail as the remaining gas re-equilibrates, and they leave too. Over geological time the whole atmosphere bleeds away. The light gases go first, because at a given temperature a lighter molecule moves faster — which is why hydrogen and helium are scarce even in the Earth's atmosphere while nitrogen and oxygen are retained. The option's wording is loose in exactly the way examination options often are: read strictly it should say that the escape velocity is less than the molecular speeds, not that gas molecules have an escape velocity of their own. But the physical criterion it names — escape velocity compared with rms velocity — is the correct one, and no other option names a criterion at all.

Why the other options are wrong — (a) Orbiting a planet has nothing to do with atmospheric retention, and there is a decisive counter-example: Titan revolves around Saturn and carries an atmosphere denser at the surface than the Earth's, made mostly of nitrogen. Titan holds it because it is more massive than the Moon and far colder, which lowers molecular speeds — the same escape-velocity criterion, applied to different numbers.  (b) True of every body in the solar system, including Venus with its crushing carbon dioxide atmosphere and Jupiter with its enormous one. A property shared by bodies with and without atmospheres cannot explain why one of them lacks an atmosphere. Sunlight does matter indirectly, by setting surface temperature and so molecular speeds, but the option does not say that.  (d) Proximity to the Earth is not a cause. If it were, the Earth's own atmosphere would be the first casualty. The governing quantity is the Moon's own small mass and hence its low surface gravity and low escape velocity — a property of the Moon alone, which would be unchanged if it orbited at twice the distance.

Concept: Atmospheric retention is a contest between gravity and heat. Escape velocity, √(2GM/R), is the speed at which a projectile launched from a surface will never fall back; it is about 11.2 kilometres per second for the Earth, 2.4 for the Moon, 5.0 for Mars, 10.4 for Venus and 59.5 for Jupiter. Molecular speed, from the kinetic theory of gases, is √(3RT/M) for the root-mean-square value — proportional to the square root of absolute temperature and inversely proportional to the square root of molecular mass. The rule of thumb used by planetary scientists is that a body retains a gas over the age of the solar system if its escape velocity is roughly six times the rms speed of that gas, because the fast tail of the distribution does the escaping long before the average molecule does. That single comparison explains the whole solar system: giant planets are massive and cold and keep even hydrogen; the Earth is massive enough to keep nitrogen and oxygen but loses hydrogen and helium; Mars, smaller, has kept only a thin carbon dioxide atmosphere; and the Moon and Mercury have effectively none.

State PCS 2025 Q36 — Science & Technology PYQ with Answer

State PCS 2025 Q36 — Science & Technology PYQ with Answer