Solutions to Problem Set 8

Transcription

1 Cornell University, Physics Department Fall 2014 PHYS-3341 Statistical Physics Prof. Itai Cohen Solutions to Problem Set 8 David C. sang, Woosong Choi 8.1 Chemical Equilibrium Reif 8.12: At a fixed temperature 1200K the gases CO 2 + H 2 CO + H 2 O are in chemical equilibrium in a vessel of volume V. If the volume of this vessel is increased, its temperature being maintained constant, does the relative concentration of CO 2 increase, decrease, or remain the same? At constant V, in equilibrium, F is minimized. his imples that df 0 S 0 d p 0 dv + i µ i d i µ CO2 d CO2 + µ H2 + d H2 + µ CO d CO + µ H2 Od H2 0 From the chemical equation we know that d CO2 d H2 d CO d H2 O and thus But we also know from Reif ) that µ CO2 + µ H2 µ CO µ H2 O 0 µ ζ )V where ζ is the single species partition function with the volume dependance removed. Hence we have ζ CO 2 )V + ζ H 2 )V ζ CO )V ζ H 2 O )V CO2 H2 CO H2 O CO2 ζ CO 2 )ζ H 2 ) CO H2 O ζ CO )ζ H 2 O ) H 2. 1) his is independant of volume. hus the number of the different species do not change, and the relative concentrations remain the same. 1

2 8.2 Partial Pressure Reif 8.14: Consider the following chemical reaction between ideal gases: m b i B i 0 i1 Let the temperature be, the total pressure be p. Denote the partial pressure of the ith species by p i. Show that the law of mass action can be put into the form p b 1 1 p b 2 2 p bm m K p ) 2) where the constant K p ) depends only on. he law of mass action can be written b 1 1 b 2 2 m bm K, V ) ζ b 1 1 ζ b 2 2 ζm bm We ve already seen that we can rewrite ζ i, V ) V ζ i ), thus we have 1 V 1 ) b1 ) b2 2 V 2 m V m ) bm ζ b 1 1 ζ b 2 2 ζ b m m he ideal gas law gives us 1 V 1 p hence p b 1 1 p b 2 2 p bm m K p ) where K p ) m i1 ζ i )) b i. 2

3 8.3 Paramagnet a) his problem is similar to a random walk but instead of steps to the right and left we have spins that are up and down. Let s ± # of particles with s i ± 1 2. hen ) Ωs + ) s +! s +!s! s s + + s s 2πs + s using Stirling s approximations. Since s + + s and E µ B Hs s + ) we have s ± a where a E. If 2 2µ B H we drop the square root term of th term of the stirling approximation we get ) ΩE) 2 + a 2 +a) ) 2 a 2 a) for integer values of /2 + a. Defining δe to coarse grain such that µ B H δe µ B H this gives us ) ΩE) 2 + a 2 +a) ) 2 a 2 a) δe 2µ B H b) ) ) ) Ω 2 + a 2 + a 2 a β Ω E a [ ) 1 + E 2 + a 1 ) 1 2µ B H µb H E µ B H + E 2µ [ )] 1 µb H E k µ B H + E 2 a ) 2 a )] c) Let b µ, we then have e 2b µ E µ B H + E E µ 1 e 2b ) 1 + e 2b µ B H tanh b We also have [ S k Ω k 2 ) 2 tanh b 2 + ) 2 tanh b tanh b 2 ) 2 tanh b )]

4 Which reduces to )] ) S k [ eb e b e b e b e b + e b e b + e b e b + e b e b + e b since 1 tanh b 2e b and 1 + tanh b 2eb e b +e b [ ) )] e b e b + e b e b e b + e b S k e b + e b e b e b + e b e ] b k [e b + e b ) eb e b + e b e b e b e ] b k [2 cosh b) beb be b e b + e b S k his gives us a Helmholtz free energy F E S µ B H tanh µ e b +e b. [ 2 cosh µ ) µ tanh µ ] d) he canonical partition function is given by F 2 cosh µ ) [ 2 cosh µ ) µ tanh µ ] Z e βµ + e βµ i1 2 cosh µ ) e) From the partition function we can calculate as above. E Z β µ tanh µ, [ S k Z + βe) k 2 cosh βµ B H) βµ B H tanh µ ] F Z 2 cosh µ ),, 4

5 8.4 Quantum Harmonic Oscillator a) We have Z n ) e βen e ω 1 2 +n) b) Ξµ, ) n Z n e βµn e ω 2 n e n µ ω) e ω/2 1 e µ ω)/ e µ/2 2 sinh[ ω µ)/2 ] c) For the energy we have ) Ξ Ē β µ,v [ βµ ] 2 sinhβ ω µ)/2)) β 2 ω 2 coth[ ω µ)/2 ] For the entropy we have S [ Ξ] [ µ ] 2 2 sinh[ ω µ)/2 ] [ )] ω k 2 sinh ω [ ] ω 2 2 coth 2 d) Using the normal canonical parition function given in the problem we see, Ē Z β β [sinhβ ω/2)] ω 2 cothβ ω/2) and S k Z βē k 2 sinhβ ω/2)) ω 2 coth ω/2 ) hus we see we must have µ 0 for the oscillation quanta. 5

6 8.5 Equilibrium Fluctuations a) he grand canonical parition function is Ξ ) Ξ e βer µ) r ) β Ξ Ξ We also have ) 2 Ξ 2 β )e βer µ) β 2 ξ1 + 2 oting that ) 2 Ξ 2 1 ) 2 Ξ Ξ 2 1 ) 2 Ξ Ξ 2 1 Ξ ) 2 Ξ 2 2 ) 2 Solving for 2 we see ) 2 )2 2 Ξ Ξ 2 which gives us ) 1 ) 2 2 Ξ var ) ) + 2 b) he isothermal compressibility can be expressed as κ 1 V where v V/ is the volume per particle. V P 1 v v P From the maxwell relation for F, we have ) ) P,V V, v ) P 1 ) v v since d v dv. his gives us ) P ) P v ) 1 1 v v 6

7 aking another derivative with respect to µ we see ) 2 P 1 2 v 1 ) V 1 ) v 2 V v P,, ) 2 P,V 1 V v 2 V v ) v P 1 v 3 v P 2, 1 v 2 κ oting that P V Ξ and using the second last equation in part a) we see var V ) 2 P 2, V κ v 2 2 κ V 8.6 Grand Canonical Ensemble a) he grand canonical ensemble for an ideal gas where V,, and can vary is given as Ξ e βµ Z 0 where is the canonical partition function. his gives Ξ Z n 1 [ ] 3/2 2πm! V h 2 β 0 [ ] ) 3/2 1 2πm V e βµ! h 2 β Recognizing this as a aylor series for an exponential we see Ξ exp [ V e βµ 2πm h 2 β ) 3/2 ] 3) b) We have 2πm pv Ξ V e βµ h 2 β ) 3/2 2πm p e µ/ h 2 β 7 ) 3/2

8 Solving for the chemical potential µ, we see [ p µ ) ] h 2 3/2 β. 2πm 8