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Unformatted text preview: Quiz 1 Solutions Professor Greenside The following solutions are more detailed than what was required to get full credit on the problems. 1. (a) ( 3 points ) For one point each, describe three ways that the Earth is not in thermodynamic equilibrium in addition to the fact that the temperature inside the Earth is not uniform. Answer: To get credit for this problem, it was not sufficient to list a criterion, you had to be specific about how the criterion was or was not satisfied in the context of Earth. Some possible answers: i. The Earth is not time independent: the air, oceans, and molten rock and iron (deep in the Earth) all show macroscopic motion, and there are chemical processes such as weathering of rock that are not in steady state. ii. There is relative motion: the air in the atmosphere does not move rigidly with the air near the ground moving more slowly than the air higher in the sky, the water in the oceans does not move rigidly, the molten iron in the iron core is convecting and so does not move rigidly. iii. The state of the Earth is not independent of its history. This can be seen in many ways: there are unmixed minerals and rocks, there are details of the Earth such as the shapes and positions of the continents that are specific to Earths history, the composition of the air continues to change over time (partly because of human technology). (b) ( 5 points ) Estimate to the nearest power of ten the ratio of the thermal relaxation time of the Earth to the age of the solar system which is about 5 billion years. The radius of the Earth is 6 , 400 km and the thermal diffusivity of rock and iron is about 10- 6 m 2 / s. Hint: Round all numbers to the nearest power of ten as soon as possible to minimize the arithmetic. Answer: I accepted answers of 100 or 1 , 000, or 10 , 000, which are all reasonable based on different ways of rounding. This answer is surprising: it will take a cosmological amount of time, longer than the current age of the universe, for the Earth to reach thermal equilibrium. Of course, there may be other time scales such as a relaxation time to equilibrate with respect to concentrations and chemical reactions so reaching thermodynamic equilibrium might take even longer than reaching thermal equilibrium. Lets answer this problem two ways, first by rounding to the nearest power of ten so that we avoid working with any numerical digits, second by retaining just one significant digit throughout. First way: estimate by replacing all numbers by the nearest power of ten. The relaxation time is L 2 / where L is the largest distance that separates one part of a system from another part except if the system is coming into equilibrium with a constant temperature surface, in which case L is the biggest distance from an interior point to the surface. So for this problem, we should choose L = R where R is the radius of the sphere since Earth is coming into equilibrium with a constant-temperature surface that is basically zero degrees K (the temperature of outer space).constant-temperature surface that is basically zero degrees K (the temperature of outer space)....
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- Spring '08