.1 1nm (T = 73.15K, p = 101.35kP a (1atm( )), 1bar = 10 5 P a = 0.9863atm) 1 ( ).413968 10 3 m 3 1 37. 1/3 3.34.414 10 3 m 3 6.0 10 3 = 3.7 (109 ) 3 (nm) 3 10 6 = 3.7 10 1 (nm) 3 = (3.34nm) 3 ( P = nrt, R k B A (1) P ( ) P = RT, : ( v n 1( [1] Fig.16-1 ) T 1 Ar -185.9 C=(73.16 185.9 = 87. 6)K R/P P T P (P, T ) P (P, T ) y P x P (P, T ) = RT (3) x (Fig16-[1]) H., CO 1 (1atm=101.35kP a ) ( ( P (P, T ) = βp (4) T = 73.15K β(h ) > 0 β( ) < 0, β(co ) < 0 β 1 T P T = lim P 0 R () 10
1 (l/mol) ( ) 100K (T = 73.15) P (/(l atm mol 1 ) P (/atm) P (x ).414 3(Fig16-3[1]) RT (compressibility factor)z Z P, Z = 1 (5) RT 11
3(Fig16-3[1]) 1 He 3b(Fig16-4[1]) int (RT ) 600K RT =8.3145 600 = 4988.7(Jmol 1 K 1 K) = 4.9887kJmol 1 5kJmol 1 3a Z P RT 3b 1
4 Z ( ) 4(Fig16-10[1]) H 3 β β. 1 Z P RT = 1 + B (T ) + B 3 (T ) + (6) 1 B (T ) P Z = 1 + B P (T )P + B 3P (T )P + (7) B (T ) = RT B P (T ) 3 B 3P (T ) = B 3 (T ) B (T ) (RT ) 13
= ZRT P Z (7) Z = 1 + (6) Z P RT = 1 + B (T )P ZRT + B 3 (T )P (ZRT ) + B (T )P RT (1 + B P (T )P + B 3P (T )P + ) + B 3 (T )P (RT ) (1 + B P (T )P + B 3P (T )P + ) + = 1 + B (T )P RT [ 1 BP (T )P B 3P (T )P + B P (T ) P + ] + B 3 (T )P = 1 + B P (T )P + B 3P (T )P + (RT ) [1 1/B P (T )P + ] 3 3 4(Fig16-, 3, 4, 10[1]) Z P B P (T ) 5 B 5(Fig16-11[1]) B (T ) He 0 (kinetic theory of gases) B (T ) = RT B P (T ) = π A [exp ( U(r)/k B T ) 1)] r dr (8) 0 14
U(r) 1 B (T ) (8) [exp ( U(r)/k B T ) 1)] U(r)..1 an der Waals (P + a )( b) = RT an der Waals a, b b ( b) ( b P = RT ( b) a (9) P [] P < u x > P = m < u x > m 1 m { < u x > + < u y > + < u z > } = 3 k BT P = RT ( ) a (9) a an der Waals Z = P RT = b a RT 15
Z = 1 1 b a RT ( ) b 1 + a ( ) ( ) 3 b b RT + + + = 1 + (b a RT ) 1 ( ) ( ) 3 b b + + + B (T ) = (b a RT ) B P (T ) = (b a RT ) 1 RT an der Waal a b 3 3 an der Waals 3 B 3 (T ) = b B 3P (T ) = 1 (RT ) 3 ( ) a RT ab 5 Fig.16-, 3, 4,10, 11) B (T ), B P (T ) B Boyle T B an der Waals B (T ), B P (T ) T B = a br 3 ( B 004 T B 16
T/K He e Ar Kr Xe 80 10.6-11.8-88.0 100 11.4-4.8-187.0 10 11.8-0.4-133.0-308.0 140 1.1.6-99.8-9.0 160 1.3 4.8-77. -179.0 180 1.3 6.4-60.9-143.0 00 1.3 7.6-48.7-117.0 40 1.1 9.4-31.5-8.4-196.0 80 1.0 10.6-0.1-59.6-147.0 30 11.7 11.5-11.9-43.7-133.0 360 11.4 1.1-5.8-31.9-88.4 400 11.1 1.6-1.1 -.9-69.8 450 10.9 13.0 3.4-14. -5.5 500 10.7 13.3 6.9-7.5-38.8 600 10.4 13.8 11.9.0-19.6 800 9.8 14. 17.8 13..7 1000 9.3 14.3 1.1 19.5 15.0.3 ( ) (-7) M M 3 5 [3] Reif Fundamental of Statistical and Thermal Physics Chapter 6[4].3.1 (microcanonical ensamble) ( ) E E + δe E E < E l < E + δe 3 1 1 (1961, March15) ( ) + 17
x i dx i X i W X i dx i x i = X i = P w P d [x i, X i ] P = 0 E Ω 0 (E,,, x) E Ω 0 (E,,, x) = Ω 0 (E,,, x) = 0 E l E 1 (10) 1 h 3( A+ B + ) dγ (11) A! B! H E Ω(E,,, x) dω 0(E,,, x) de (1) 4 E E + δe,, x W (E, δe,,, x) = Ω(E,,, x)δe (13) Boltzmann W S(E,,, x) = k log W (E, δe,,, x) (14) 1 ( E(n x, n y, n z, ) = h π ( n m X /3 x + n y + n ) z (15) [5] 1 Ω 0 (E, ) Ω(E, ) R n = (n 1 + n + n 3 ) 8 Ω 0 (E, ) 8 1 4π 3 8 R3 n = 1 6 πr3 n (16) = 1 6 π [( n x + n y + n z )] 3/ = 1 [ 6 π mx /3 ] 3/ E h π = π3/ (m X ) 3/ 6h 3 π 3 E 3/ (mx ) 3/ = 3h 3 π E 3/ 4 Ω 18
E Ω 0 (E, ) = m 3/ X 3h 3 π E3/ (17) Ω(E, ) = dω 0(E, ) de 3/ m X = h 3 π E (18) 3 ( E : Ω 0 (E,, ) ( E 3/) /!).3. (canonical ensemble) T l ( P r (l,,, T ) p l (,, T ) = g l Z exp{ E l (,, x)/ } (19) g l l T 4 ( : T 1/ 0 1 Z Canonical partitioning function Z (,, T, x) = l g l exp{ E l (,, x)/ } (0) [6] E( U) (1) = 1 Z (,, T, x) g l E l (, ) exp{ E l (, )/ } l E l (, )p l (,, T ) l [ ] 1 Z (,, T, x) = Z (,, T, x) (1/ ),,x [ ] ln{z (,, T, x)} = (1/ ),,x () 19
( E l, p l du = l p l (,, T )de l + l E l dp l (3) E l (, ) E l de l = ( E l ) d du = ( ) El p l d + E l dp l (4) l l 1 1 p l (., T ) (dp l = 0) E l (( ) E l d = 0) 1 du = δw rv + δq rv δw rv = l ( ) El p l d δw rv = P d P = l ( ) El p l 1 = Z (,, T ) l ( ) El exp{ E l (, )/ } (5) P δq rv = l E l dp l p l = exp{ E l (, )/ }/Z (,, T ) (, ) T 5 T canonical ensemble.3.3 (grand canonical ensemble) grand canonical distribution (T µ T µ A A, µ B B, A, B, 0
P r (l,, T ) = 1 Ξ exp [{ E l (, ) + A A µ A } / ] (6) µ A = T S A (7) S (14) grand partition function( )Ξ Ξ = [{ exp E l (, ) + } ] A µ A / A =0 B =0 A (8) ( A, B,...) λ A exp ( µa ) (9) Ξ = (λ A ) A (λ B ) B exp { E l (, )/ } (30) A =0 B =0 Q (, ensemble average)q Q 1 Ξ Q l (, ) exp( E l (, )/ ) exp( A >0 l A µ A / ) (31) grand canonical ensemble 6 ( ) ensemble?.3.4 (*) (*) ( grand canonical distribution [7] grand canonical partition function Ξ(, T, µ) = exp( E l (, )/ ) exp(µ/ ) Z (,, T ) exp(µ/ ) (3) >0 >0 l Z (,, T ) canonical ensemble ensemble average ( ) ln Ξ = µ,t (33) 1
( ) ln Ξ µ,t = 1 Ξ = 1 Ξ ( ) Ξ µ,t >0 Z (,, T ) exp(µ/ ) = 1 (34) ensemble average (31) ensenmble avarage P = ln Ξ (35) [3] 5 11 ( ) ln Ξ T µ = 1 Ξ = 1 Ξ ( ) Ξ >0 l T µ ( 1 ) ( ) El exp( E l / ) exp(µ/ ) T (36) ( ) [ ln Ξ = 1 ( exp(µ/ ) E ) ] l(, ) exp( E l / ) T µ Ξ T >0 (,, T ) (canonical) (5) P (,, T ) = 1 Z [ ( ) El (, ) l l T exp( E l / ) ] (37) (38) ( ) ln Ξ = 1 Z exp(µ/ )P (,, T ) P (39) T µ Ξ >0 ( ) ln Ξ T µ = P ln Ξ = const (40) ln Ξ 35 = ln Ξ
40 (40) n /n 1,, Z(, T, ) = Z( 1, T, /n)z(, T, /n) Z( n, T, /n) (41) grand canonical functionξ ( i Ξ(µ, T. ) = =0 exp(µ/ ) = =0 1 + + n = = [Ξ(µ, T./n)] n 1+ + n= Z( 1, T, /n)z(, T, /n) Z( n, T, /n) (4) { µ } exp ( 1 + + n ) Z( 1, T, /n)z(, T, /n) Z( n, T, /n) ln Ξ(µ, T, ) = n ln Ξ(µ, T, /n) (43) α = 1/n α ln Ξ(µ, T, ) = ln Ξ(µ, T, α ) (44) 7.4 (*).4.1 ( Z Z = (Z 1)! (45) Z 1 1 Z 1 = l ( exp ε ) l (46) 3
grand canonical distribution (30) Ξ = λ (Z 1) = (λz 1 )!! { ( µ )} = exp (λz 1 ) = exp Z 1 exp (47) 33 35 ( ) ln Ξ = µ,t ( ( exp µ = Z 1 µ )),T ( µ ) = Z 1 exp = λz 1 (48) 0 P P = = λz 1 = 0 ln Ξ (49) P ln Ξ.4. (*) 1 Grand canonical function 30 Ξ = 0Z λ (50) = 1 + Z λ 1 33 P (35) = µ ln 1 + 1Z λ (51) = 1 exp( µ ) λ ln 1 + 1Z λ P = ln 1 + 1Z λ (5) 4
ln(1 + x) = x 1 x + 1 3 x3 + ( 1) n 1 1 n xn + λ P P = 1Z λ 1 λ 1Z + 1 λ 3 1Z 3 + (53) = ( Z 1 λ 1 + Z λ + Z 3 λ 3 + ) 1 ( Z1 λ 1 + Z λ + ) 1 ( + Z1 λ 1 + Z λ + )3 + 3 ( = Z 1 λ 1 + Z 1 ) ( Z 1 λ + Z 3 1 Z3 1 Z 1 Z + 1 ) 3 Z3 1 λ 3 + λ λ 51 = λ λ ln 1 + 1Z λ (54) [ ( = λ Z 1 + Z 1 ) ( Z 1 λ 1 + 3 Z 3 1 Z3 1 Z 1 Z + 1 ) ] 3 Z3 1 λ + (55) λ λ ( /Z 1 λ = /Z 1 + a + a 3 3 + (56) ) a = a 3 = [ Z 3 1 8 Z 5 1 P = (Z Z 1 ) (57) ( ) Z Z 1 3 ( ) ] Z 3 Z Z 1 + Z3 1 3 Z 4 1 ( ) ( ) + B + B 3 B = B 3 = [ Z3 Z 3 1 ( Z Z 1 4Z Z 4 1 1 ) + Z Z 1 ( ) 3 + (58) m Z ( m) Z 1 Z 1 Z 1 E (1) ( ) 1 3 ] (59) (60) E (1) = 1 M P + ibrot (61) 5
1 ibrot 1 Z 1 = { ( 1 exp = { exp 1 M P / = (πm )3/ h 3 ) } M P + ibrot / } exp { E vj / } v,j exp { E vj / } Q 1 v,j (6) v, j E () = i=1, { } 1 M P (i) + ibrot (i) + 1 (63) Z Z = exp 1 M P (i) + ibrot (i) + () 1 / (64) i=1, 1 ( ) Q 1 dr 1 dr exp () 1 (r 1 r ) B = 1 ( ) ] dr 1 dr [exp () 1 (r 1 r ) 1 = 1 ( ) ] dr 1 [exp () 1 (r 1) 1 1 B = 4π [ ( ) ] r dr exp () 1 (r) 1 (65) (66) 3 E (3) = 3 i=1 { } 1 M P (i) + ibrot (i) + () (i, j) + (3) (i, j, k) (67) i>j 3 3 (3) (i, j, k) Z 3 Z 3 1 3! Q3 1 dr 1 dr dr 3 exp ( ) () 1 (r 1 r ) + () 1 (r r 3 ) + () 1 (r 1 r 3 ) + (3) (68) 3 (3) 6
.4.3 L-J (*) (*) (r) = (r < σ) ε (σ < r < gσ) (69) 0 (gσ < r) 1 exp( / ) = 1 (r < σ) (exp(ε/ ) 1) x (σ < r < gσ (70) 0 (gσ < r) Boyle B = 0 [ σ B = π r dr 0 = π 3 gσ σ ] xr dr [ σ 3 x ( g 3 σ 3 σ 3)] = π 3 σ3 [ 1 x ( g 3 1 )] = v A [ 1 x ( g 3 1 )] (71) x = 1 g 3 1 = exp( ε ) 1 (7) B ε exp( ) = 1 B g 3 1 + 1 = g3 g 3 1 1 = k ( ) g 3 T B ε ln g 3 1 ε T B = ( k ln ) 1 1 1/g 3 ε = 0 x = 0 Boyle Boyle (73) Mie s potential function Lennard-Jones Potential function(*) Mie s potential function (R; n, m) = λ R n µ R m (74) (n, m) = (1, 6) Lennard-Jones Potential function m=6 (n, m) m = 1 [ ( σ ) 1 ( σ ) ] 6 LJ (R; ε, σ) = 4ε R R (75) 7
ε R = σ (σ) = 0.4.4 (*) ([7] 4 5 6 [1],, (D.A.McQuarrie, J.D.Simon) :,, 000(1997); chapter 16, page 669. [],, (D.A.McQuarrie, J.D.Simon) :,, 000(1997); chapter 7, page 1145. [3],,, 1961. [4] F. Reif, Fundamental of Statistical and Thermal Physics; McGraw-Hill Book Company, ew York, 1965. [5],, (D.A.McQuarrie, J.D.Simon) :,, 000(1997); chapter 3, page 79. [6],, (D.A.McQuarrie, J.D.Simon) :,, 000(1997); chapter 17, page 731. [7] G.C.Maitland, M.Rigby, E.B.Smith, W.A.Wakeham, Intermolecular forces, Claredon Press, 1981. 8