Thermodynamics and an Introduction to Thermostatistics problems 2, Exercises of Chemistry

Download Thermodynamics and an Introduction to Thermostatistics problems 2 and more Exercises Chemistry in PDF only on Docsity! � Problem Set 2 3.20 MIT Professor Gerbrand Ceder Fall 2003 LEVEL 1 PROBLEMS Problem 1.1 In class we have defined the Legendre transforms of the energy U(S,V). It is also possible to define Legendre transforms in the entropy representations, starting from the fundamental equation S(U,V). These are called Massieu functions, after Massieu who developed them in 1869. Define the three Legendre transforms of S(V,T) and write their differential. Show their relation with the Legendre transforms of the energy. For those of you familiar with statistical mechanics you may notice that the Massieu functions are the Hamiltonians for the different ensembles. Problem 1.2 Prove that the Cp for an ideal gas is independent of pressure. Problem 1.3 For a gas whose Helmholtz free energy is given by the equation: a � f T F = RT ln(Vm b) + ( ) � Vm � where a and b are constants and f(T) is only a function of temperature, obtain a formula for the equation of state: P=? Problem 1.4 A sample of solid polyethylene is compressed adiabatically and reversibly from a pressure of one atmosphere to a pressure of one thousand atmospheres. The initial temperature of the sample was 300 K. �
∂S(a) Write a differential equation relating the temperature change to the pressure change in terms of ∂P T and ∂S
. ∂T P (b) What will be the temperature of the sample immediately after the compression? Data: Cp = 2 3 J/g K . 1 L = 19 10
�5 K�  3 = 1 g/cm Problem 1.5 Consider a system that can exchange energy with the environment through magnetic work in addition to the regular heat flows and volume work. The magnetic work can be represented by HdM , where M is the magnetization and H the applied field. (a) Prove that:     ∂V ∂M = ∂H � ∂PT,P H,T (b) Derive one more Maxwell relation with either H or M in it. 1 �  Problem 1.6 If a rubber band is stretched, the reversible work is given by ∂W = dL where  is the tension on the band and L is the length. (a) If the stretching is carried out at constant pressure and the volume of the band also remains constant during expansion/contraction, derive a thermodynamic function ( G) which is a function of L and T . (b) Show that     ∂ ∂S = ∂T � ∂L TL (c) Derive the following from thermodynamic principles:       ∂U ∂S ∂ = + T =  ∂L ∂L � T ∂T LT T (d) For an ideal gas, U is a function of T only, and as a result it can be shown that   1 ∂P 1 V = P ∂T T Show that a corresponding equation exists for an ’’ideal’’ rubber band. LEVEL 2 PROBLEMS Problem 2.1 The pressure on 500g of copper is increased reversibly and isothermally from 0 to 5000 atm at 298 K. (Take the 3 6 12 density = 8 96 103 kg/m , volume expansivity = 49 5 10� K�1, isothermal compressibility  = 6 18 10�. . 1  .   Pa� , and specific heat cp = 385 J/kg K to be constant).
(a) How much heat is transferred during the compression? (b) How much work is done during the compression? (c) Determine the change of internal energy. (d) What would have been the rise of temperature if the copper had been subjected to a reversible adiabatic compression? Problem 2.2 dU can be expressed in terms of changes in T and P though this contains less information than the combined first and second law, where dU is expressed in terms of dS and dV . Prove that: dU = (Cp P V dT + V (P � T dP ) ) Problem 2.3 Do you think that the enthalpy of evaporation of a material is pressure dependent? Comment on the conditions under which it is pressure dependent. 2