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1 Umeå Universitet, Fysik Vitaly Bychkov Prov i fysik, Thermodynamics, 0-0-4, kl jälpmedel: Students may use any book(s) including the textbook Thermal physics. Minor notes in the books are also allowed. Students may not use their lecture notes. Define the notations you are using properly. Present your arguments in details. Good luck! ) You open a kitchen refrigerator and close it again. In this process, a part of cold air (volume fraction 0. ), which was originally in the refrigerator, is replaced by warm air of the room. fter a short pause you try to open the refrigerator one more time and find a much larger force needed for that purpose. Evaluate increase in the force taking room temperature T 0, temperature in the refrigerator T 0, area of the refrigerator door m. int: The cooling machine works much faster than pressure comes back to equilibrium for the air in the room and the refrigerator. ) Figure shows a polymer molecule of polymethylene, which consists of base units (two carbon atoms connected by a double bond) linked to each other in a long chain. For each unit, if the two links to other units are on the same side, it is called a cis-unit. If the links are on opposite sides, then it is a trans-unit. onsider a polymer made of base units. What is the multiplicity of the molecule with cis units in the cis-configuration? What is the maximal multiplicity? What is the width of the multiplicity function? ssume that one unit has width d. What is the characteristic width D of the whole molecule?

2 Umeå Universitet, Fysik Vitaly Bychkov = = cis-units =... = = =... d trans-units Fig.. The model of a polymer molecule. ) eat capacity of a degenerate electron gas at constant volume may be presented as V / a( V / ) T, where a is a factor involving only universal constants of physics. Find how chemical potential of the gas depends on temperature. 4) onsider an engine cycle consisting of three steps: ) adiabatic compression from V to V ; ) isothermal expansion back to volume V ; ) pressure relaxation to the initial state at constant volume. Find efficiency of such an engine. ssume that the working gas is two-atomic with frozen oscillation degrees of freedom. 5) onsider phase transition between solid and liquid e at low temperatures. t absolute zero, T 0 K, transition happens at pressure P 0 : the solid phase corresponds to P P 0, the liquid phase is at P P0. Entropy (per one mole) of the liquid phase depends on temperature as S l T. Entropy of the solid phase may be approximated as a constant, S s S0. Volumes (per one mole) of the solid and liquid phases may be also taken constant with the solid phase denser than the liquid one, Vl Vs V 0 0. Find the phase boundary curve. Find the critical pressure, exist. P c, below which solid e cannot

3 Umeå Universitet, Fysik Vitaly Bychkov Prov i fysik, Thermodynamics, , kl jälpmedel: Students may use any book(s) including the textbook Thermal physics. Minor notes in the books are also allowed. Students may not use their lecture notes. Define the notations you are using properly. Present your arguments in details. Good luck! ) onsider a thermally isolated chamber of total volume V filled with one-atomic gas with initial temperature T i. The volume is separated into two parts by a movable plate of surface area and some (unknown!) mass M as shown in Fig.. The plate allows thermal interaction between the gases in the upper and lower parts of the chamber. We have the same amount of particles in each part. Initially the plate is fixed and the volumes of both parts are the same and equal V. Then we let the plate move under the action of gravity. fter some transitional period the system reaches mechanical and thermal equilibrium with the new temperature of the gas and the new volumes of the lower part V V / and the upper part V V /. Find the mass M of the plate, neglecting potential energy of the gas. int: Think about energy conservation in the system. Fig.. chamber with a heavy plate.

4 Umeå Universitet, Fysik 4 Vitaly Bychkov ) onsider a chamber of total volume V separated into two parts of volume V V / and V / by a movable but thermally insulating plate. The parts are filled with V one-atomic gases and B, respectively, which are initially in mechanical (but not thermal!) equilibrium with each other. We have the same amount of atoms for each gas B. Then we remove the plate thus allowing thermal and diffusion interaction between the gases. Find entropy increase in the system. ) ccording to the Debye model, heat capacity of a crystal at low temperatures may be presented as V ak(t / ), where a is a numerical factor and is a constant known as the Debye temperature. Find how chemical potential of the crystal depends on temperature. 4) onsider an engine cycle consisting of three steps: ) adiabatic compression from V to V ; ) isobaric expansion back to volume V ; ) pressure relaxation to the initial state at constant volume. Find efficiency of such an engine. ssume that the working gas is two-atomic with frozen oscillation degrees of freedom. 5) Imagine a modified van-der-walls model P a V ( V b) k T, B where a and b are some numerical factors. The gas expands from V 5 b to V 9 b at constant temperature and number of particles. Find change of the Gibbs free energy of the gas.

5 Umeå Universitet, Fysik 5 Vitaly Bychkov Prov i fysik, Thermodynamics, 0-0-6, kl jälpmedel: Students may use any book including the textbook Thermal physics. Minor notes in the books are also allowed. Students may not use their lecture notes. Present your solutions in details: it will help you getting better evaluation. Good luck! ) container of volume V is filled with gas of temperature T, total number of particles is (initial value is 0 ) and mass of each particle is m. We make an extremely small hole of area in the container, and the gas starts slowly leaking out due to random collisions of gas particles with the wall. ow does number of gas particles in the container decrease with time? int: Find, how many particles d pass through the hole in a time interval dt. Derive a differential equation for d / dt, which also contains. Solve the equation. For simplicity, you can evaluate the average thermal velocity in one direction v x as x v x / v. ) onsider two interacting Einstein solids with different numbers of oscillators and B. We have q energy quants in the system. In the limit of lowtemperature solids, q, q B, find how multiplicity of the system depends on the number of quants in the solid, q. Find the most probable value for q. OBS! To save time, you may use the formula for multiplicity of cold Einstein solid, which we derived during problem solving, Problem.7. ) Imaging certain material with heat capacity described by the formula V at, where a is some constant. Find how entropy of this material depends on energy U.

6 Umeå Universitet, Fysik 6 Vitaly Bychkov 4) onsider an engine cycle consisting of three steps: ) isothermal compression from V to V ; ) expansion at constant pressure; ) adiabatic expansion to the initial state. Find efficiency of such an engine as the ratio of work to the total heat input into the system. ssume that working gas is two-atomic with frozen oscillation degrees of freedom. 5) onsider the Dieterici equation of state a P( V b) kt exp, ktv where a and b are some numerical factors. Find volume, pressure and temperature in the critical point for the model.

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