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### Kyoto University 2013 Entrance Examination—Physics (Partial)

Question III

Read the following passage, and fill in the parentheses “(     )” with an appropriate expression or value, and choose the correct option from within curly brackets “{           }”. Write down your answers in the corresponding space in the answering sheets. The double parentheses “((       ))” represent the same expression or value as that in the corresponding single parentheses. For Questions 1 and 2, follow the instructions and write down your answers in the corresponding space provided in the answering sheets.

As shown in Figure 1, there exists a cylinder divided into different regions by a piston W and a partition D. The cylinder is fixed in place, and W and D can slide freely without friction. Region A is defined to be the region between W and D, and Region B is the region between D and the right end of the cylinder. Regions A and B are sealed off from each other, and each contain a different ideal gas. The value of CV/R, where CV is the molar specific heat at constant volume and R is the ideal gas constant, is 3/2 and 5/2 for the gases in Regions A and B, respectively. The walls of the cylinder, piston, and partition are all made of thermally insulating material. The gas in Region B is heated or cooled by the temperature controller C attached to the right-hand side of the cylinder. Assume that the temperature of the gas in Region B adjusts quickly to the temperature set by C. When C is turned off, Region B becomes a thermally insulated region. Additionally, there is a stopper S inside the cylinder, and the partition cannot move further to the right of S. When the partition is at the position of the stopper, the volume of Region B is V0. Assume that the volume of the piston, partition, stopper, and temperature controller can be ignored. The left-hand side of the piston is always under atmospheric pressure whose value is P0. In the beginning, the pressure, volume, and temperature of the gas in Region A are P0, 2V0, and T0, respectively; and those of the gas in Region B are P0, (3/2)V0, and (3/2)T0, respectively. We define this to be the initial state. In this state, the number of moles of the gas in Region A is (   a   ), and that of the gas in Region B is (   b   ). From this initial state, the following operations as described in (1) through (4) are performed in order.

(1) Using the temperature controller, the gas in Region B is slowly heated from (3/2)T0 to 2T0. The resulting change in internal energy of the gas in Region B is (   c   ), and the work done by the gas in Region B is (   d   ), so the amount of heat transferred from the temperature controller to the gas in Region B is (   e   ).

For an ideal gas, (pressure) x (volume)γ stays constant for adiabatic processes. γ is the heat capacity ratio, defined as CP/CV, where CP is the molar specific heat at constant pressure. For ideal gases, the difference between CP and CV (i.e. CP – CV) is equal to (   f   ). Therefore, the heat capacity ratio of the gas in Region A is (   g   ), and that of the gas in Region B is (   h   ).

From hereon after, write down the heat capacity of the gas in Region A, ((   g   )), as α, and that of the gas in Region B, ((   h   )), as β.

(2) After the operation in (1), while maintaining the temperature of the gas in Region B at 2T0 with the temperature controller, an external force is applied to the piston W to push it slowly to the right. Once the partition D reaches stopper S, the inward motion of W is stopped and the state is maintained. The volume of the gas in Region A in this state is (   i   ).

Question 1          In the operation described in (2), what is the work done by the external force on the gases? The amount of heat transferred out of the system through the temperature controller by the gas in Region B is Q. Show all your work.

(3) After the operation described in (2), the piston W is pushed in further until the volume of the gas in Region A becomes V0, and this state is maintained. In this state, the pressure of the gas in Region A is (   j   ).

Question 2          Draw a pressure-volume diagram for the gases in Regions A and B, depicting the changes in the gases starting from the beginning of the operation in (2) to the end of the operation in (3). Shade the area in the diagram corresponding to the work done on the gas in Region B. If there are any changes in the pressure and/or volume during either of the operations, the direction of the change must be indicated by an arrow, and the values of the volume and pressure at the beginning and end of the change must be indicated in the diagram.

(4) After the operation described in (3), the gas in Region B is heated by the temperature controller. The temperature controller is turned off just as the partition D starts to move away from the stopper, and Region B becomes a thermally insulated region. After this, the piston W is moved slowly to the left until the pressure of the gas in Region A becomes the same as the atmospheric pressure P0. In this final state, the volume of the gas in Region B is (   k   ), which is {l: 1. greater than  2. less than  3. equal to} the volume of the gas in Region A. During this operation, the work done by the gas in Region A is {m: 1. greater than  2. less than  3. equal to} the work done by the gas in Region B. These properties of gases related to work are important for understanding the efficiency of a heat engine.

(End of the problems)