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To begin with, note that the assumption n>>1 was actually not needed, the result T_0+sqrt(n)t remains valid even if n is not small.
This assumption was included mostly because the result becomes interesting only if n>>1: in theory, it is possible
to achieve a significant cooling of a small portion of something by making use of minute temperature differences. In total, 46 correct solutions of problem 2 were submitted. The solutions of this problem can be divided into two categories: (a) based on the second law of thermodynamics (the total entropy of the system will either remain constant (if the processes are reversible) or increase; (b) showing how the Carnot cycles should be used, and using the formula for the efficiency of the Carnot cycle. The former is much clearer and concise, so the best solution awards go to solutions using the first approach, assuming it was submitted before the respective hint, i.e. during the first week. Overall, there were three such solutions (there were other solutions which made use of the entropy calculations, but calculated unnecessarily the work of the heat engines and heat pumps). The best solution awards go to: Thomas Foster and Ivan Ridkokasha both of whom earn 40% of the bonus, and to Siddharth Tiwary who gets the remaining 20%. The essential part of the Thomas' solution concludes basically on page 3 where he shows that some water will remain in the liquid phase. Further he proceeds to show how heat engines and heat pumps need to be connected to achieve the desired result. To be entirely rigorous, it is indeed necessary to show that there is a way to cool the small part of ice down to the given temperature. However, it would have been shorter (and equally sufficient) to just state that as long as there are some temperature differences between the main portion of ice and water, the temperature differences could be used to produce work by a heat engine, and thereby drive a heat pump to further freeze the cold ice. Ivan's solution is very similar to that of Thomas; he has moved the non-obligatory parts into Appendix. Siddharth's solution is very short and laconic, with very few comments (this makes it less suitable for those who want to learn); His solution is also less rigorous in not analysing if the desired result can be really achieved (see above). Additionally, he makes a statement which is not correct: he believes that the temperature T_0+sqrt(n)t could be also achieved (it cannot be achieved because of the phase transition, and because c_v for water is not proportional to T). These shortcomings explain why he gets a smaller part of the bonus. The additionally published solutions are well-written and brief solutions based on the approach (b) (if you want to improve the chances of your solution getting published, write in LaTeX!). Vladislav Polyakov - the second-fastest solution; well written and good, too! Johanes Suhardjo - amended his original solution during the second week, adding an alternative, entropy-based solution (using the first hint). Both solutions are well written. Felix Bekir Christensen - another nice solution using the approach (b). | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Results for Problem 2, university students.
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Results for Problem 2, high school students.
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