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1 Umeå Universitet, Fysik Vitaly Bychkov Prov i fysik, Thermodynamics, --6, kl Hjälpmedel: Students may use any book including the textbook Thermal physics. 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 N (initial value is N ) 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. How does number of gas particles in the container decrease with time? Hint: Find, how many particles dn pass through the hole in a time interval dt. Derive a differential equation for dn / dt, which also contains N. Solve the equation. For simplicity, you can evaluate the average thermal velocity in one direction v x as x v x / v. ) Consider two interacting Einstein solids with different numbers of oscillators N N and N B N. We have q energy quants in the system. In the limit of lowtemperature solids, q N, q N 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 C V at, where a is some constant. Find how entropy of this material depends on energy U.

2 Umeå Universitet, Fysik Vitaly Bychkov 4) Consider an engine cycle consisting of three steps: ) isothermal compression from 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) Consider the Dieterici equation of state an P( V bn) NkT exp, ktv where a and b are some numerical factors. Find volume, pressure and temperature in the critical point for the model.

3 Umeå Universitet, Fysik Vitaly Bychkov Prov i fysik, Thermodynamics -4- Hjälpmedel: Students can use any book including the textbook Thermal physics. Define your notations properly. Present arguments in details. Good luck! ) Thermal conduction in plasma is determined mostly by random thermal motion of electrons. n electron collides with an ion, if the ion happens to be on the way of the electron in the imaginary cylinder, where is the electron free path and is the cross-sectional area of the collisions. The cross-sectional area depends on the average thermal velocity of electrons as v 4. Find how the coefficient of thermal conduction in plasma depends on electron temperature. Electrons may be treated as a mono-atomic ideal gas. ) Consider a simple one-dimensional model of a polymer molecule, which consists of N links ( N ), each link of length d. Every link may look to the left or to the right with equal probability as shown on the figure. What is the multiplicity of the macrostate when N R links point to the right? What is the maximal multiplicity? What is the width of the multiplicity function? Using the width of the multiplicity function find the characteristic size of a molecule L. L Fig.. The model of a polymer molecule. d

4 Umeå Universitet, Fysik 4 Vitaly Bychkov ) Heat capacity of ultra-relativistic degenerate electron gas at constant volume may be written as on energy. / / C V N V T, where is a constant. Find how entropy of the gas depends 4) Consider an engine cycle consisting of three steps: ) compression at constant pressure from V ; ) pressure increase at constant volume; ) 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) Consider equilibrium between two phases. Suppose that the volumes of the phases depend on temperature as V V ) and V V ), while entropy ( T ( T difference depends on temperature as S T (here V,, and are constants). Find the shape of the phase-boundary curve, taking into account that at low temperatures T the phase transition happens at pressure P P (Such a problem describes qualitatively phase transition between two solids at low temperatures).

5 Umeå Universitet, Fysik 5 Vitaly Bychkov Prov i fysik, Thermodynamics, --, kl Hjälpmedel: Students may use any book including the textbook Thermal physics. Define the notations you are using properly. Present your arguments in details: it will help you getting better evaluation. Good luck! ) Rain-drops are falling at the window with velocity v (the same for all drops) at an angle to the vertical. verage mass of a drop is M, concentration of drops in the air is n. Drops stick to the window when hitting it. Find pressure of the rain at the window. ) Consider equal amounts of helium and argon N N N. Initially both gases have different volumes V H V, V V and different temperatures T H T, T T. We put the gases into thermal and mechanical contact (by allowing heat exchange through the wall between the gases and by letting the wall move). Find total entropy increase in the system. OBS! There is no diffusion interaction! H ) Find chemical potential of a cold Einstein solid. You have to start with the general formula for multiplicity of an Einstein solid. (p) What is Gibbs free energy for the solid? (p) 4) Consider an engine cycle consisting of three steps: ) adiabatic compression from V ; ) isothermal expansion with V V; ) compression at constant pressure to the initial state. Find efficiency of such an engine. ssume that working gas is twoatomic with frozen oscillation degrees of freedom. 5) Van der Waals gas expands from V Nb 5Nb at constant temperature T and number of particles N. Find change of the Helmholtz free energy F of the gas. Hint: Do not confuse Helmholtz and Gibbs free energies. (4)

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