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PY2104 - Introduction to thermodynamics and Statistical physics Review Problems. Part III: Thermodynamic functions and equilibrium 1) Consider an ideal gas, whose entropy is given by U V n σ + 5R ln + 2R ln S= 2 n n where n is the number of moles, R is the universal gas constant, U is the internal energy, V is the volume and σ is constant. Calculae the specific heats at constant pressure and constant volume, cp and cv for this gas. 2) (i) Show that for a perfect gas undergoing adiabatic expansion, pV γ is constant, where γ = cp /cv . (ii) What is the physical reason for the difference between cp and cv ? 3) Consider a paramagnetic system. From a thermodynamic point of view, the system is described by its magnetization, M , the magnetic field, B and the temperature T . ITs equation of state can be written as M = CB/T , where C is constant (known as Curie constant). Its internal energy is U = −M B . The increment of work done by the system upon the external environment is dW = M dB. (i) Write an expression for the heat input dQ to the system in terms of thermodynamic variables, M and B (more precisely, dM and dB). (ii) Find an expression for the differential of the system entropy, dS (as a function of dM and dB). (iii) Derive an expression for the entropy, S. 4) Consider a new matter, for which its equation of state is p= AT 3 , V where p, T and V are the pressure, volume and temperature, respectively, and A is constant. Furthermore, the internal energy of this matter is U = BT n ln(V /V0 ) + f (T ) , where B, n and V0 are constants, and f (T ) is a function of the temperature only. What are B and n ? 5) The equation of state of a van-der-waals gas is a p + 2 (V − b) = RT . V (i) The gas undergoes an isothermal expansion from volume V1 to volume V2 . Calculate the change in Helmholtz free energy. (ii) Calculate the change in the internal energy. 6) A certain system is found to have a Gibbs free energy given by ap G(p, T ) = RT ln (RT )5/2 where a and R are constants. Find the specific heat at constant pressure, cp . 7) Consider a thermally isolated system, consisting of two parts, A and B separated by a thermally conducting and movable partition. The volume of part A is V and that of part B is 2V . The system is filled with ideal gas: in part A the gas has pressure p and temperature T , while in part B it has temperature T but pressure 3p. The partition is allowed to move (without the gases in the two sides mixing). When equilibrium is established: (i) what is the equilibrium temperature ? (ii) what is the equilibrium pressure ? (iii) what is the change in the total entropy ?