Homework 8

HeatPhysics 306Homework Assignment 8(Potentials)1) Classical and Statistical Thermodynamics, Carter Chapter 88.2) Show that for an ideal gasa)( ) ( )T vf = c v (T − T 0 ) − c v T ln − RT ln − s 0 TT 0 v 0b)( ) ( )T Pg = c P (T − T 0 ) − c P T ln + RT ln − s 0 TT 0 P 08.8) a) A van der Waals gas undergoes an isothermal expansion from specific volumev 1 to specific volume v 2 . Calculate the change in the specific Helmholtzfunction.b) Calculate the change in the specific internal energy in terms of v 1 and v 2 .8.10) a) Prove thatand use the result to show that( ) ∂cP∂Pc P = TT( ) ∂s,∂TP( ∂ 2 v= −T∂T 2 )Pb) Prove that c P for an ideal gas is a function of T only.2) Classical and Statistical Thermodynamics, Carter Chapter 99.2) a) Express the chemical potential of an ideal gas in terms of the temperature Tand volume V :µ = c P T − c v T ln T − RT ln V − s 0 T + constantb) Similarly, find µ in terms of T and P . Show that the chemical potential atthe fixed temperature T varies with pressure as( ) Pµ = µ 0 + RT ln ,P 0where µ 0 is the value of µ at the reference point (P 0 , T ). This expression isof great use in chemistry..

9.9) a) Show that for an open system with one component, dG = −S dT + V dP +µ dn.b) Using this result, calculate G for a van der waals gas, assuming a fixed amountof material at a given, fixed temperature. Show thatG = −nRT ln(V − nb) +n2 bRV − nb − 2n2 a+ C(T ),Vwhere the integration constant C(T ) is in general, different for different temperatures.3) Over a certain range of pressures and temperatures, the equation of state of a certainsubstance is given with reasonable accuracy by the relationP vRT = 1 − C′ P T 4orv = RTP − C T 3where C and C ′ are constants. Derive an expression for the change of enthalpy andthe change of entropy of this substance in an isothermal process.Figure 1: Josiah Willard Gibbs 1838-1903 ; file from wikimedia commons