4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k

4.6 (E i = ε, ε + ) T Z F Z = e ε + e (ε+ ) = e ε ( + e ) F = kt log Z = kt loge ε ( + e ) = ε kt ln( + e ) (4.8) F (T ) S = T = k = k ln( + e ) + kt e + e kt 2 + e ln( + e ) + kt (4.20) /kt T 0 = /k (4.20) e, T T 0 S = k kt e + 0, (T/T 0 0) e = kt +, T T 0 /kt S = k ln(2 /kt + ) + 2 + /kt + = k ln 2 2kT + 2kT + O(( /kt )2 ) 0 3 ln 2 kt (4.9) 30

.0 S(T) 0.5 0.0 0.0.0 2.0 kt/ 0: 4.6. 2. q p q p h/2 q p q p = h N 3 h 3N = h 3N Π i dr i Π j dp j dqdp (Action) 3

h 2. 2 2 d 3 r d 3 r 2 = V 2 L x L 2 x 2 L x 2 L 2 2 0 L x 0 L x : 2 N N! 4.6.2 T N H = i 2m p2 i 32

Z Z = h 3N Π N N! i=d 3 r i d 3 p i exp p 2 i /2m i = V N h 3N N! ( ) 3N/2 2πm (4.2) V N (3.) F S C F = ( ) 3 2πm ln Z = N 2 ln + ln(v/h 3 ) ln N + S = F ( ) ( ) 3 2πm V T = Nk 2 ln + ln Nh 3 + + 3Nk/2 (2πm ) 3/2 V e 5/2 = Nk ln Nh 3 C = T S T = 3Nk/2 Sackur-Tetrode (4.22) N! 2 (Extensive Variable) (Intensive Variable) (4.22) N! ln(v/n) ln V 2 N ln 2 λ p p = h/λ h kt/2 λ T h 2 2mλ 2 T = 2 kt S = Nk ln (2π) 3/2 e 5/2 v λ 3 = 32 2πe Nk ln 5/3 TT0 h 2, T 2mv = 2/3 2 kt 0 v = V/N v /3 S/Nk 4.2 33

2.0.0 S(t)/Nk 0.0.0 2.0 3.0 0.00 0.02 0.04 0.06 0.08 T/T 0 2: Sackur-Tetrode 4.6.3 ω i N H(q, p) = ( ) 2m p2 i + mω2 i 2 q2 i i Z = h N Π i dq i dp i exp H(q, p) = Π i dq i dp i exp h 2m p2 i mω2 i 2 ( ) /2 ( ) /2 ( ) 2πm 2π = Π i h mωi 2 = Π i hω i q 2 i (4.23) N! (4.23) F = ln Z = kt i ln ( hω i ) S = F T = k i C = T S T = Nk ln ( hω i ) + Nk N! Extensive 34

4.6.4 µ i E, µ i B µ i E 3: z ( ) h = µ 0 E cos θ, (µ 0 = µ i ) θ z z φ (θ, φ) z rot = = dφ sin θdθ expµ 0 E cos θ = 2π dve µ0ev, (v = cos θ) 4π µ 0 E sinh (µ 0E) Z N z N rot F rot = N ln z rot = NkT ln 4π sinh(µ 0E) µ 0 E (4.24) 35

S = F T = Nk ln 4π sinh(µ 0E) + NkT µ 0 E = Nk µ 0E ln 4π sinh(µ 0E) µ 0 E coth(µ 0 E) + µ 0 E s(t) = ln4πt sinh(/t) t coth(/t) + kt 2 coth(µ 0E) + T = Nks(t) t = kt/µ 0 E s(t) 4 s(t) { ln(4π) + 5/6t 2, ln(4πt) /t, t t 3 3.0 2.0 s(t).0 0.0.0 0.0 0.5.0.5 2.0 t 4: z- Ω µ z i = µ 0 z rot µ0e cos θ dω cos θe (4.24) µ z i = N i ln z r E = F E = Nµ 0L(y), (y = µ 0 E) 36

L(y) Langevin L(y) = coth y y Langevin y { y/3 y L(y) y kt µ 0 E { µ z i N µ 0 T µ 0 E/k µ 2 0E/3kT T µ 0 E/k i Debye µ z i /µ 0 t = kt/µ 0 E 5.0 0.5 0.0 0 2 3 4 5 t 5: : = µ 0 E z rot = e εi = e µ0e + e µ0e = 2 cosh(µ 0 E) i=,2 F = N ln2 cosh(µ 0E) S = F T = Nk ln2 cosh(µ 0E) Nk µ 0E kt tanh(µ 0E) = Nks(t) s(t) = ln2 cosh(/t) t tanh(/t) 37

t = kt/µ 0 E s(t) 4 6 H t kt/µ 0 H kt/µ 0 (H/2) = 2t s(2t) H T A A B C 3.0 2.0 B A s C D.0 0.0 0.0 0.5.0.5 2.0 t 6: 38

4.6.5 He, Ne, Ar 2 2 3 6 6 ( 3) 3 2 ( ) 6 3: 2 Lagrangian m, m 2, 2 R G r, r 2 7 m r + m 2 r 2 = 0, r r 2 = r (4.25) r r = m 2 M r, r 2 = m M r, (M = m + m 2 ) m m 2 r 2 G r R 7: (4.25) T T = m 2 (Ṙ + r ) 2 + m 2 2 (Ṙ + r 2) 2 = M 2 Ṙ2 + m 2 r 2 + m 2 2 r 2 2 = M 2 Ṙ2 + µ 2 ṙ2, µ = + (4.26) m m 2 39

µ (reduced mass) (r, θ, φ) r = (r cos φ sin θ, r sin φ sin θ, r cos θ) ẋ = ṙ cos φ sin θ r sin φ φ sin θ + r cos φ cos θ θ ẏ = ṙ sin φ sin θ + r cos φ φ sin θ + r sin φ cos θ θ ż = ṙ cos θ r sin θ θ µ 2 ṙ2 = µ 2 ṙ 2 + r 2 ( θ 2 + sin 2 θ φ 2 ) (4.26) r V (r) L = T V (r) = M 2 Ṙ2 + µ 2 ṙ2 + 2 I( θ 2 + sin 2 θ φ 2 ) V (r), (I = µr 2 ) I r, θ, φ p r, p θ, p φ P = L Ṙ = MṘ, p r = L ṙ = µṙ, p θ = L = I θ, p θ φ = L φ = I sin2 θ φ h = P Ṙ + p rṙ + p θ θ + pφ φ L = P 2 2M + 2µ p2 r + V (r) + 2I ( p 2 θ + ) sin 2 θ p2 φ M Z t Z z N I r r r 0 Z = Z t z N, z = z rot z vib z rot, z vib 40

z rot = h 2 dθdp θ dφdp φ exp h = h 2 dθdp θ dφdp φ exp 2I (p2 θ + sin 2 θ p2 φ ) = 2πI h 2 dφdθ sin θ = 8πI h 2 E rot E rot = N ln z rot = N ln(8πi/h2 ) = NkT C rot = de rot dt = Nk 2 V (r) r = r 0 V (r) = V (r 0 ) + 2 V (r 0 )δr 2 +, δr = r r 0 r h v = V (r 0 ) + 2µ p2 r + V 2 µω2 v δr2 +, ω v = (r 0 ) µ ω v E v = NkT, C v = de v dt = Nk 2 4 (k) 3/2 4: 2 7Nk/2 4

5Nk/2 4.6.6 Gibbs 8 V A V B 8: V V A, V B A, B 2 N A, N B p m A, m B pv A = N A kt, pv B = N B kt p(v A + V B ) = (N A + N B )kt N A /V A = N B /V B = (N A + N B )/(V A + V B ) V N Sackur-Tetrode { (2πm ) } 3/2 e 5/2 S = kn ln h 3 + ln V N (4.27) 2 Z = V A NA N A!h 3NA ( 2πmA ) 3NA/2 V B N B N B!h 3NB ( 2πmB ) 3NB/2 (4.28) 42

V A V B A, B 2 N A, N B (4.27) Z = V NA N A!h 3NA ( 2πmA ) 3NA/2 V NB N B!h 3NB ( 2πmB ) 3NB/2 (4.29) (4.28) (4.29) V S = N A ln(v/n A ) + N B ln(v/n B ) N A ln(v A /N A ) N B ln(v B /N B ) = N A ln(v/v A ) + N B ln(v/v B ) (4.30) = N A ln(n/n A ) + N B ln(n/n B ) N A = N B N ln 2 Gibbs (4.29) Z = V NA+NB (N A + N B )!h 3(NA+NB) ( ) 3NA/2 ( 2πmA 2πmB ) 3NB/2 (4.3) (4.29) (4.3) N A!N B! (N A +N B )! (4.3) Gibbs 43