One of the statements of the second law is that “heat cannot flow from a colder body to a warmer one without external aid”. Assume two systems, 1 and 2, at T1 and T2 (where T2 > T1 ). Show that if a quantity of heat q did flow spontaneously from 1 to 2, the process would result in a decrease in the entropy of the universe. [You may assume that the heat flows so slowly that the process can be regarded as reversible. Assume also that the loss of heat by the system 1 and the gain of heat by 2 do not affect T1 and T2 .]

A STUDENT EATS 2000 KCAL PER DAY. (A) ASSUMING THAT ALL OF THE FOOD ENERGY IS RELEASED AS HEAT, WHAT IS THE RATE OF HEAT RELEASED (IN WATTS)? (B) WHAT IS THE RATE OF CHANGE OF ENTROPY OF THE SURROUNDINGS IS ALL OF THE HEAT IS RELEASED INTO HEAT AT ROOM TEMPERATURE (20 C)?

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Prove that the “cycle” consisting of only two adiabatic steps (reversible expansion and then compression) will have no efficiency since no work can be produced. Will it violate the 2nd law of the thermodynamics? Clear explain your answer.

According to the Debye equation the heatthe capacity of a solid is proportional to the temperaturecubed at low temperatures, Cp(T) = σT3, where σis a constant. Show that the entropy at a temperatureT0 is given by S(T0) = Cp(T0)/3 when the Debyeequation holds.

Consider two identical bodies of heat capacity C, and with negligi- ble thermal expansion coefficients. Show that when they are placed in thermal contact in an adiabatic enclosure their final tempera- ture is (T₁ + T2)/2 where T₁ and T2 are their initial temperatures. Now consider these two bodies being brought to thermal equilib- rium by a Carnot engine operating between them. The size of the cycle is small, so that the temperatures of the bodies do not change appreciably during one cycle; thus the bodies behave as reservoirs during one cycle. Show that the final temperature is (T₁ T2)¹/². (Hint: What is the entropy change of the universe for this second process?)