Why is the top of the mountain cooler than the surface when the mountains are actually more closer to the sun and hence should be hotter?
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The thin air at the top of the mountains keep the temperature lower than that of lower regions. At the higher altitudes of the mountain, the atmosphere gets a lot thinner than at lower regions as the atmospheric pressure decreases as you go higher up, which results in the expansion of air in such regions and thus cooler air. The standard lapse rate refers to the cooling of air as you travel to higher altitudes, and it is around 6.5 degrees Celsius for every 1000 meters gained.
While it is true that the top of the mountain is nearer to the sun than that of lower regions, this distance is negligible in comparison to the distance between Earth and the Sun, and the thinning of air plays a much more crucial role in the temperature at the top of mountains.
Another issue might be that heated air would rise- but this could be accounted for as at higher atmospheric levels, more heat is lost to space. As solar heat reaches us, some of that heat is retained by the greenhouse gases, whereas the rest is bounced off the atmosphere and re-radiated into space, but at higher altitudes is would be a lot harder to retain nearly as much heat at lower temperatures, with a larger portion being lost into space.
