Latent Heat of Fusion and Potential Energy

[Vaal]

I was looking over an old qualifying exam and I came across a problem asking what the maximum height of a mountain on Earth could be. The solution states that the rock at the base of the mountain will melt and flow away when the potential energy of the top layer of the mountain is equal to the latent heat of fusion of the rock at the bottom layer.

I understand how the above process conserves energy, but I can't see what physical law/process makes it happen. Any help would be much appreciated.

 

[Andrew Mason]

I was looking over an old qualifying exam and I came across a problem asking what the maximum height of a mountain on Earth could be. The solution states that the rock at the base of the mountain will melt and flow away when the potential energy of the top layer of the mountain is equal to the latent heat of fusion of the rock at the bottom layer.

I understand how the above process conserves energy, but I can't see what physical law/process makes it happen. Any help would be much appreciated.

I would think it would go something like this. I will use ice as a base layer because it is confusing if the mountain and the base are made of the same substance. Consider a mass m with centre of mass of height h placed on a bed of ice. Assume that the mass settles by an amount \Delta h by compressing the ice base. The work done by the mass is equal to the loss of potential energy (mg\Delta h) and this would be turned into thermal energy at the base. If that thermal energy is enough to melt ice of a thickness greater than \Delta h, then the mass will simply keep falling, melting the entire ice base (assuming the melted water has somewhere to flow to).

AM

 

[Vaal]

Andrew, thanks for the reply. I hadn't considered compression causing potential energy to turn into thermal energy that seems like it is on the right track. The one aspect of this theory I don't quite get though comes from the fact that the problem asks for the maximum possible height. With the above theory couldn't the material be added slowly to the mountain so that the heat can dissipate over time and the thermal energy required for fusion is not present all at once?

 

[gneill]

The heat of fusion is related to the energy required to break intermolecular (or inter-atomic) bonds in a substance. This can happen by imparting enough thermal energy so that the molecules shake themselves apart, or by applying mechanical strain throughout the substance.

 

[Andrew Mason]

Andrew, thanks for the reply. I hadn't considered compression causing potential energy to turn into thermal energy that seems like it is on the right track. The one aspect of this theory I don't quite get though comes from the fact that the problem asks for the maximum possible height. With the above theory couldn't the material be added slowly to the mountain so that the heat can dissipate over time and the thermal energy required for fusion is not present all at once?

Not if the mountain and base are made of the same material. The maximum height would occur when the rate of energy lost per unit of vertical height is exactly equal to the rate of heat flow per unit of base thickness that is required to liquify the base material. You would have to work that out.

AM