How high can be
Assuming that the required latent heat by the latent heat of melting of metals is approximately 200-300 kJ / kg, how high could the tallest mountain on Earth be before the base starts melting?
Hint - Compare the energy needed to melt the bottom layer of a mountain with the gravitational energy that would be released if the mountain then sank.
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1 solution
Consider a 'box-shaped' mountain of average density p, base area A and height h. In order to melt its bottom layer of thickness d and specific latent heat L, energy AdpL would be required. The total mass of the mountain is approximately Ahp, and the energy released if it sank a distance d would therefore be Ahpgd. The base of the mountain does not melt under its load if AdpL > Ahpgd, i.e, h < l/g
Approximating the required latent heat by the latent heat of melting of metals (200-300 kJ / kg ), we estimate the maximum possible height of mountains on Earth to be 20-30 km. This is of the right order of magnitude. Allowing for the fact that the base of the mountain does not actually have to melt, but rather that the size of mountains is limited by the yield strength of their constituent materials, the estimated height of the highest mountains on Earth is surprisingly accurate.