Physics Cup – TalTech 2019, Problem 2

Problem 2

Results and best solutions

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

Siddharth TiwaryIndiaIndian Inst. of Techn. Bombay13 Jan. 2019 14:323.168
Thomas FosterUKUniversity of Oxford13 Jan. 2019 15:103.518
Ivan RidkokashaUkraineShevchenko University Kyiv13 Jan. 2019 16:553.198
Johanes SuhardjoIndonesiaHong Kong Univ Sci & Tech13 Jan. 2019 18:572.144
Samarth HawaldarIndiaIndian Institute of Science15 Jan. 2019 4:161.772
Felix Bekir ChristensenGermanyUniversity of Oxford19 Jan. 2019 22:381.772
Petra BrčićCroatiaUniversity of Zagreb20 Jan. 2019 18:591.464
Damiano TiettoItalyUniversità degli Studi di Padova 03 Feb. 2019 17:321.331
Nur Mohammad ImranBangladeshUniversity Of Chittagong19 Feb. 2019 10:440.968

Results for Problem 2, high school students

Vladislav PolyakovRussiaSchool No 23 St Petersburg13 Jan. 2019 14:502.853
Oliver LindströmSwedenMinervagymnasium13 Jan. 2019 15:542.358
Stefan DolteanuRomaniaIntern Computer HS Bucharest13 Jan. 2019 16:142.144
Domagoj PerkovićCroatiaIII. Gigmazija Split13 Jan. 2019 18:401.949
Gabriel TrigoBrazilColegio ETAPA, Valinhos13 Jan. 2019 20:391.772
Ivander J.M. WaskitoIndonesiaMawar Sharon Christian School14 Jan. 2019 3:171.611
Eduard BurlacuRomaniaIntern Computer HS Bucharest14 Jan. 2019 8:491.391
Bartłomiej SikorskiPolandI Liceum O. Starachowicach15 Jan. 2019 21:201.331
Oliwier UrbańskiPolandSzkoła Nr 28 w Poznaniu18 Jan. 2019 11:561.21
Zăhărăchescu MihaiRomaniaIntern Computer HS Bucharest19 Jan. 2019 9:441.1
Tùng Trần XuânVietnamHN-Amsterdam21 Jan. 2019 2:541
Muhamed SokolovićBosnia & Herz.Richmond Park ISS Sarajevo21 Jan. 2019 19:311
Mateusz KapustaPolandSecondary Sch No 3, Wroclaw22 Jan. 2019 20:261
Stephen CatsamasAustraliaMarcellin College27 Jan. 2019 13:211
Gusti P.S.G. AtmajaIndonesiaKesatuan Bangsa SHS27 Jan. 2019 15:581
Morteza MudrickIndonesiaSMA Negeri 1 Surakarta30 Jan. 2019 14:080.9
Yunus Emre ParmaksızTurkeyBahcesehir High03 Feb. 2019 21:420.8
Roberto Marín DelgadoCosta RicaColegio Científico San Carlos04 Feb. 2019 0:200.9
Masuk RidwanBangladeshSt. Joseph Higher Secondary S.04 Feb. 2019 13:441
Jianzhi WangSingaporeRaffles Institution04 Feb. 2019 14:561
Siam Absur KhanBangladeshIdeal School And College05 Feb. 2019 18:021
Péter ElekHungaryDRK Dóczy Gimnáziuma05 Feb. 2019 20:361
Marvin JaniniBrazilColégio Etapa05 Feb. 2019 23:101
Imtiaz TanweerBangladeshNotre Dame College06 Feb. 2019 13:501
Ionel-Emilian ChiosaRomaniaIntern Computer HS Bucharest07 Feb. 2019 19:211
Kaarel KivisaluEstoniaTallinn Secondary S. of Science09 Feb. 2019 22:491
Vinicius de A. Névoa BrazilColégio Arena10 Feb. 2019 4:481
Kaviraj PrithviIndiaBase PU College10 Feb. 2019 11:301
Arhaan AhmadIndiaSt. Mary’s Academy12 Feb. 2019 14:401
Raihan Amin RanaBangladeshNotre Dame College12 Feb. 2019 22:590.9
Hisham AlmalkiSaudi ArabiaManarat Al-Riyadh School13 Feb. 2019 8:020.855
Rashedul IslamBangladeshIdeal School and College14 Feb. 2019 11:321
Md Atikul IslamBangladeshSt. Joseph Higher Secondary S.15 Feb. 2019 17:561
Ričards Kristers KnipšisLatviaRiga State Gymnasium No.116 Feb. 2019 18:561
Jonathan GräfeGermanyGymnasium Dresden-Bühlau16 Feb. 2019 21:011
Janis HunsLatviaRiga State Gymnasium No.116 Feb. 2019 21:401
Lucas HaradaBrazilColegio Etapa – Ana Rosa16 Feb. 2019 22:211