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Question: Why is there no oxygen on the moon?
First of all, I assume that what you meant by oxygen on the moon is oxygen in gaseous form above the ground. Let us first try to answer the question "why there is oxygen on the Earth?"
If you throw an object straight up, it will rise until the acceleration stops it, momentarily, due to gravity, then return to the Earth. The acceleration is a measure of how fast an object gains speed. Notice that the object slows down on the way up as the direction of the motion and the constant acceleration are opposite to each other. On its way down, the object speeds up. Can you throw an object straight up so that it never returns to the Earth? The answer is yes. If you throw the object with enough initial upward speed, it will leave the Earth. The minimum speed you need to acheive this condition is called "escape velocity" (velocity needed to escape gravity). The escape velocity depends on gravity. Gravity or the gravitational force is a force of attraction that exists between any two masses. In other words, Earth's escape velocity depends on the mass and the radious of the moon. Since the moon is much smaller than the Earth, the moon's escape velocity is smaller than that of the Earth.
How does this have anything to do with oxygen in the atmosphere? Gases are made up of individual atoms or molecules that are freely moving in random directions with wide variety of speeds. The measure of the speed of oxygen molecules is given by so called "Root-Mean-Square (RMS) speed." This speed depends on the mass and the temperature of the molecule. On the Earth's surface, the RMS speed of oxygen molecules is smaller than the Earth's escape velocity, so oxygen molecules do not escape the Earth. On the other hand, the RMS speed of oxygen molecules is larger than the moon's escape velocity. Therefore, if oxygen molecules are on the moon's surface, they will leave the moon. Here are the numbers: the RMS speed of oxygen molecules at zero Celsius temperature is about 4.6 km/s. The escape velocities of the Earth and moon on its surfaces are about 11km/s and 2.4 km/s respectively. Notice that the moon's escape velocity is smaller than the RMS speed of oxygen.
In conclusion, the moon's low gravity prevents it from keeping hold of the oxygen molecules on the atmosphere above the surface.
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