|MadSci Network: Earth Sciences|
The best I can do with this sort of question is an "order of magnitude" calculation-- the answer I get might be out by a factor of 2, maybe even a factor of 5. Too many of the quantities are not precisely known. Some of the estimates and averaging are not particularly accurate (more detail would take us beyond the scope of a simple calculation). ------------ The Earth's crust is made up of a lot of different types of mineral. Basalt from volcanic rocks typically has a density of about 3.3 to 3.5 tonne/cubic metre. Granite is rather less dense -- around 2.7 tonne/cubic metre. We will have to start out with a guess to obtain a value for the average density of crustal rock. You can probably find it in Geology texts if you look hard. I will guess at 3.1 tonne/cubic metre. To "actually vaporize" our 3.1 tonne of rock, we will have to heat it to boiling point, and then provide the extra heat necessary to turn it from solid to gas. But even our starting temperature is problematic. We know the temperature of surface rocks, and that there is generally an increase with depth. Where is our 3.1 tonne of rock coming from? Let's initially suppose that it is coming from the surface, and that it has to be evaporated at an external pressure of 1 atmosphere. In the CRC Handbook for Chemistry & Physics, we can find if we look hard, a value for the boiling point of quartz -- 2590°C (Ed 56 p. B137) and for the specific heat capacity of granite at 400°C -- 0.258 kcal/kg/degree = 1.08 J/ g/K (Ed 56 p. E16). We can also note that specific heat capacity increases significantly with increasing temperature (but we can not do much about it). We are not likely to find a value for latent heat of vaporization of quartz at its boiling point. The best option is to assume that it is much smaller than the heat input needed to raise the temperature. So for a very rough estimate, our surface rock might require 2570 K * 3 100 000 g * 1.08 J/g/K = 9 047 000 000 J, or 9.05 GJ. That is likely to be an underestimate because of the increase of specific heat with temperature, and because of the neglect of latent heat. ------------ What is the amount of heat required to actually vaporize a cubic meter of the Earth's crust? Around 10 GJ ------------ I know it's not supposed to be very deep, in comparison to Earth's diameter, but just how deep is that? Average thickness of continental crust is around 35 km, of oceanic crust around 8 km. (any geology textbook). ------------ 71% of the Earth's surface is under water, 29% is land. But a significant part of the "under water" is in shallow seas and continental margins. So we will work with 2/3 oceanic crust and 1/3 continental crust. The Earth's surface area is 4 * pi * (6371 km)^2 = 5.1 * 10^8 km^2 For the volume of crust we therefore get 1.7 * 10^8 * 35 + 3.4 * 10^8 * 8 = 8.67 * 10^9 km^3 Furthermore, how many cubic kilometers of crust does the Earth actually have? About 9 (American) billion -- 9 * 10^9 km^3 ------------ Overall, what would be the heat of vaporization of all that crust? We could just multiply the 10 GJ estimate of what is required to vaporize 1 m^3 by the number of m^3 in the total crust. The answer would come to 9 * 10^28 J. This is an overestimate, because we assumed that we were starting with surface rock at 20 deg C, and most of the deeper crustal rock will be much hotter than that before we start (and therefore need less heating). The true answer would probably be something like half this value -- say 5 * 10^28 J. The total daily solar radiation intercepted by the Earth is 1.5 * 10^22 J (Campbell, Energy & the Atmosphere, Wiley, 1977, p. 40), so the energy needed is - about the total amount of energy in ten thousand years of sunlight intercepted over the whole of the Earth. - about a hundred thousand times the energy released in the largest earthquake of recent times (China 1976) - about 250 000 times the total energy stored in our fossil fuel reserves.
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