Thermodynamics of Interaction-Free Ideal Gases in Canonical Ensemble
Explore the thermodynamics of interaction-free ideal gases within the canonical ensemble, focusing on classical Hamiltonian, Helmholtz free energy, quantum-mechanical considerations, and solutions for a monatomic ideal gas. Delve into the partition function, eigenenergies, and wave-number implications in the context of quantum mechanics and classical approximations.
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Presentation Transcript
Thermodynamics of interaction free systems Ideal gases considered in the canonical ensemble 2 i m p N = In contrast to the textbook we ignore here for simplicity right from the beginning particle-wall interaction H Classical Hamiltonian 12 = i 2 i m N p 1 ! 12 = = 3 3 3 3 ... ... Z d r d r d p d p e i 1 1 N N 3 N N h factoring due to absence of interaction 2 1 m 2 N m p p N N V N = = = = ... 3 3 Z Z Z Z Z ... ... d p d p e e 2 2 1 2 1 N i 1 N 3 N ! N h = 1 i N 3 N 3 N = 2 2 p N 2 V m p N N V V = = 3 d pe dp e 2 2 m m 3 N ! N h 3 3 N N ! ! N h N h
3 N Helmholtz free energy & all thermodynamic information N 2 V m = Z With 3 N ! N h remember we showed ????? ??? ? = ?? = ??? ????+ ?? ??ln? = ???? + ??ln? ? ? ? ? = + ln TS k T U k T Z B B F = = ln U TS k T Z B
3 N 3 N 3 /2 N N 2 V T N h m N 2 V m = ln k T = = ln ln F k T Z k T B 3 N ! k B B 3 N ! N h B ( ) B k T N = + 3/2 ln VT const F T F V = = , S P = F V dF SdT PdV With V T 3/2 T = = P k TN = PV Nk T B 3/2 VT B T 1/2 3 VT 3 2 = F T V T ( ) ( ) = = + + = + + 3/2 3/2 ln ln S k N VT const k TN k N VT const k N B B B B 3/2 2 V = + With F U TS U F TS 3 2 ( ) ( ) = + = + + + + 3/2 3/2 ln ln U F TS k T N VT const k TN VT const k TN B B B 3 2 = U k T N B
We now consider a monatomic ideal gas of N atoms quantum mechanically p i with i i 2 i m 2 p 2 N N = = We go from the classical Hamilton function i H H 1 2 = m 12 = i i and the N particle Schroedinger equation 2 2 N ( ) ( ) = , ,..., , ,..., i r r r E r r r 1 2 1 2 N N 2 m = 1 i Again absence of interaction factoring in single particle problems We will later have a proper discussion of indistinguishable particles and symmetry of the N-particle wave functions Consider single particle Schroedinger equation ( ) ( ) 2 2 2 ( ( ( ) ) ) ( ( ( ) ) ) = = with boundary conditions 0, , , , 0 y z L y z m = r r n n x n n n = = ,0, , , 0 x z x L z n n y 3-dim particle in a box problem = = , ,0 x y , , x y L 0 n n z
remember 1-dim solutions 2 L nx L Eigenfunctions = ( ) x sin n 2 2 2 n Eigenenergies = = 1,2,3,... x n n x 2 2 mL x Quantum number x 2 2 ( ) = = = 1,2,3,... 1,2,3,... n n 2 2 2 n n n = + + 3-dim particle in a box problem n n n x x y z 2 2 mL , , x y z y 1,2,3,... n z Eigenenergies of N particle 3-dim. Schroedinger equation = = + + + ... E E , ,..., n n n n n n 1 2 1 2 N N = E Z e Partition function of canonical ensemble At this point we ignore symmetry constraints of wave-function/Pauli principle N 3 N ( ) + + + ... = = = Z e e e n n n n 1 2 N n , ,..., n n n n n 1 2 N
3 N 2 2 k m n n Introducing the wave-number n = = k Z e 2 L = 1 n With = = k k k + 1 n n L 3 3 N N 2 2 k m 2 2 k m L L n = Z e k dk e 2 2 0 k = 1 n 0 h mk T = Classical approximation appropriate for L th 2 B 2 2 k m With = mk T 2 dk e 2 B h 0 N 3 3 /2 3 N N 2 2 L h mk T h V = mk T = = N 2 B Z V B 3 2 th /2 N mk T h = = N ln ln F k T Z k T V B B B 2 ( ) 3/2 2 mk h 3 2 = B ln ln ln k TN V Nk T T Nk T B B B 3
( ) 3/2 2 mk h 3 2 = B ln ln ln F k TN V Nk T T Nk T B B B 3 Same free energy expression as obtained from classical partition function Same thermodynamics 3 2 = U k T N = PV Nk T B B 1 ( ) 1 3 N From we now understand origin of factor 3 in classical partition function = mk T 2 Z L N h B 3 N h Note: the origin of additional factor 1/N! to resolve Gibb s paradox requires an understanding of concept of indistinguishable quantum particles
