:

. , , , - , 711 10, http://tccc.iesl.forth.gr/education/local.html

di S dt

0

1 ---- 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 . . . . . . . . . . . . . . . . . . . . . 1.6 . . . . . . . . . . . . . . . . . . . 1.6.1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 : 1.6.3 . 1.6.4 . . . . . . . . . . . . . . . 1.7 . . . . . . . . . . . . 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 LEGENDRE . . . . . . . . . . . . . . . . . 1.10 MAXWELL . . . . . . . . . . . . . . . . . . . . . . . 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 GIBBS-DUHEM . . . . . . . . . . . . . . . . . . . . . . 1.13 . . . . . . . . . . . . . . . . . . . . 1.14 . . . . . . . . . . . . . . . . . . . . . . . . 1.15 DUHEM . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 . . . . . . . . . . . . . . . . . . . 2.2 . . . . . . . . . . . . . . . . . 2.3 . . . . . . . . . . . 2.4 - . . . . . . . . . . . 2.5 . . . . . . . . . . . . . . . . . 2.6 . . . . . . . . . . . . 2.7 . . . . . . . . . . . . . . . . . . . 2.8 -- i i 1 . 1 . 7 . 9 . 11 . 13 . 14 . . . . . . . . . . . . . 14 15 15 16 16 19 20 23 23 24 24 27 28 29 29 29 30 30 31 32 34 39

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

ii

2.9 . . . . . . . . . . . . . 2.10 . . . . . . . . 2.11 . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 . . . . . 2.13 LEGENDRE . . . . . . . . . . . . . 2.14 . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Gibbs-Duhem . . . . . . . . . . . . . . . . . . . 2.16 . . . . . . . . . . . . . . . . 2.17 . . . . . . . . . . . . . . . . . . . . 2.18 DUHEM . . . . . . . . . . . . . . . . . . . . . 2.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 44 51 52 56 65 68 69 75 85 86 89 89 89 89 90 90 91 91 92

3 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 . . . . . . . . . . . . . . . . . . . . . 3.1.2 . . . . . . . . . . . 3.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 . . . . . . . . . . . . . . . . . . . . 3.1.5 - . . . . . . . . . . 3.1.6 . . . . . . . . . . . . 3.1.7 . . . . . . . . . . . . . 3.1.8 , . . . . . . . . . . . . . . 3.1.9 . . . . . . . . . . . . . . . . . . . . . . 3.1.10 Clausius-Clapeyron ()- . . . . . . . . . . . . . . . . . . . . . . 3.1.11 xi . . . . . . 3.1.12 . . . . . . . . . . . . . . . . . . . 3.2 . . . . . . . . . . . . . . 3.2.1 van der Waals . . . . . . . . . . . 3.2.2 Hess . . . . . . 3.2.3 Kirchho . . . . . . . . . . . . . . . . . . . . 3.2.4 Gibbs . . . . . 3.2.5 Gibbs-Helmholtz . . . . . . . . . . . . . . . . . 3.2.6 Gibbs . 3.2.7 Raoult . . . . . . . . 3.2.8 . . . . . . . . . . . . . . . . . . . 3.2.9 : 3.2.10 : . . . . . . . . . . . 3.2.11 : , vant Ho . . . . . . . . . . . . . . . . . . . . . 3.2.12 . . . . . . . . . . . . . . . . . . . . 3.2.13 vant Ho: . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. 92 . 93 . 93 . 94 . 97 . 99 . 99 . 100 . 100 . 101 . 101 . 102 . 103 . 103 . 103 . 104 . 104 . 104 . 105

4 LEGENDRE LAGRANGE BOLTZMANN

iii 107 109 111 113 115 117 119 133

- 135 .1 . . 142 145 147

iv

1

- - - - 1.1

. , . , Avogadro (NA 1023 ), . () . . , . Avogadro. , , . . ; . [1, 2] 1

2

1. ----

. . . , , . . ( ) . ( 1.1), , () () . . . Euler. , , , . . . 1012 . ( / / / / ) . , , , . , . , . , . . (O ) (T ), () -

1.1.

3

1.1:

Ui , i = 1, 2, . . . , Vi , i = 1, 2, . . . Ni , i = 1, 2, . . . . .

U 2, V2, N2

U1, V1, N1 U , V3, N3 3

U 4, V4, N4

U i , Vi , Ni

.

O=

1 T

T

O(t)dt =< O > .0

(1.1)

. . . , . . , . . ( ), . . () ( ). ,

4

1. ----

(, ) , ( 1.2). ( ) 1 . , , P , (P = f (V )). . , . 1.3 ( ) . . . . , ( 1.2), , , , . ; . . , . . . , . - , . , , . - . . ;1 () (dierential manifold) k- (k f orms). k = 0 k = 1 .

1.1.

5

1.2: Ei i. (, , ) . . .

F(Ei)

Ei

. . , , , . . . . ; . Taylor:

F (x) = F (x0 ) +

dF (x) dx

x=x0 (x

x0 ) +

1 d2 F (x) 2 dx2x=x0

(x x0 )2 + . . .

(1.2) , . . ; -

6

1. ----

1.3: . (, , ) . .

; , , . ; ; E , , ;

, , . , . .

1.2.

7

, . , . , . : , , , , !

, , . . . . . . .

1.2

.

8

1. ----

.

1.3.

9

1.3

: - : (Gradient): (Hessian): (): ():

d 2

(1.3) (1.4) (1.5) (1.6) (1.7) (1.8)

dx = dxT =

dx1 dx2 dx1 dx2 f x1 f x2

(1.9)

(1.10)

f x2(1.11)

f (x1 , x2 ) = f x1f 2 x2 1

f (x1 , x2 ) = 2 f (x1 , x2 ) =

i+

j

(1.12)

f 2 x1 x22

(1.13)

f x2 x1

f x2 2

2

TAYLOR

df (x1 , x2 )

=

f (x1 + dx1 , x2 + dx2 ) f (x1 , x2 ) (1.14) 1 (f )T (dx) + (dx)T ( 2 f ) (dx) + . . . (1.15) 2 f f dx1 + dx2 + (1.16) x1 x2 1 2f 2 1 2f 2 2f dx1 + dx2 + dx1 dx2 + . . . 2 x2 2 x2 x1 x2 1 2

10

1. ----

f (x1 , x2 )

= f (x1 + x1 , x2 + x2 ) f (x1 , x2 ) (1.17) 1 (1.18) (f )T (x) + (x)T ( 2 f )(x) + . . . 2 f f x1 + x2 + (1.19) x1 x2 1 2f 2 1 2f 2 2f x1 + x2 + x1 x2 + . . . 2 x2 2 x2 x1 x2 1 2

,

f (x1 , x2 )

=

f (x1 + x1 , x2 + x2 ) f (x1 , x2 ) (1.20) 1 (f )T (x) + (x)T ( 2 f ) (x) + . . . (1.21) 2 f f x1 + x2 + (1.22) x1 x2 1 2f 2f 1 2f x2 + x2 + x1 x2 + . . . 1 2 2 2 2 x1 2 x2 x1 x2

( )

f (x1 , x2 )dx1 dx2

=

f (x1 , x2 )dx1 dx2

(1.23)

( )

f (x1 , x2 ) x1 f (x1 , x2 ) x2

=

f (x1 , x2 ) x1 f (x1 , x2 ) x2

(1.24)

=

(1.25)

1.4.

11

1.4

) , (N, V , = N/V ).

) . ( ), ( ) ( ). , + .

) , N1 , N2 , . . . , Nr , , U , , V . U . N1 , N2 , . . . , Nr , U V . p , Nij , i = 1, . . . , r , Vj Uj , j = 1, . . . , p. :p

Ujj=1 p

= U = UT ()

(1.26)

Vjj=1 p

= V = VT ()T = Ni = Ni , i = 1, . . . , r, ().

(1.27)

Nijj=1

(1.28)

, . . . .

) , ,

12

1. ---- (U, U + dU ). ( ):

S(U ) = kB ln . kB = 1, 38066 1023

(1.29)

JK

1

Boltzmann.

S(U ) , U (S). ( ) , n1 , n2 , . . . , nr ,

S(U, V, n1 , n2 , . . . , nr ),

U (S, V, n1 , n2 , . . . , nr ).

(1.30)

p =

1

,

p = 1,

(1.31)

V , n = (n1 , n2 , . . . , nr )T (U, U + dU ) ( ) (Gibbs)

S(U, V, n) = kB ln = kB

1

ln

1

= kB

p ln p .(1.32)

) ( ) ( ( )) ( ). , U, S, V, n1 , n2 , . . . , nr . (, p = 1) ( ). , (, p = 0) . n p- (p) ( ):

f (x1 , x2 , . . . , xn ) =

1 p

f (x1 , x2 , . . . , xn ).

(1.33)

EULER ( ):n

f (x1 , x2 , . . . , xn ) =i=1

f xi

xi .

(1.34)

1.5.

13

. -. ( -) , . (q ) , q = 0.

1.5

EULER :

U (S, V, n1 , n2 , . . . , nr ) =

U SV,ni

S+

U VS,ni

r

V+i=1

U niS,V,nj=i

ni .(1.35)

: : : :f ) xi

T P i

= = =

U S U V U niV,ni

(1.36) (1.37)S,ni

.S,V,nj=i

(1.38)

(xi ,

. ,

(S, T ), (V, P ), (ni , i ), . , ( ), :

r

U (S, V, ni ) = T S P V +i=1

i ni

(1.39)

r

dU = T dS P dV +i=1

i dni

(1.40)

, )

(

S(U, V, ni ) =

1 T

U+

P T

r

V i=1

i T

ni

(1.41)

14

1. ----

dS =

1 T

dU +

P T

r

dV i=1

i T

dni

(1.42)

1.61.6.1

:

.

dU = 0. r

(1.43)

dU = q + w +i=1

i dni .

(1.44)

q w . (P, V ), r

dU = q P dV +i=1

i dni .

(1.45)

r

U = q + w +i=1

i ni .

(1.46)

, -. . q w ( ). , dU dS U S . U S . r i=1 i dni

dU = q + w

(1.47)

1.6.

15

1.6.2

:

[t, t + dt] ( ) . - , .dST dt

UT ,VT ,nT i

0

(1.48)

, ( ) - ( unconstrained) ( ). , ST (Ul , Vl , nil ; UT , VT , nT ) i

(Gradient) (ST )UT ,VT ,nT = 0 (), ., i ST Ul ST Vl ST nil

(1.49)

i+

j+

k = 0,

(1.50)

(Hessian) ( 2 ST )UT ,VT ,nT 0. i

(1.51)

( ) ( )

2 ST Ul2

0,

2 ST 2 ST Ul2 Vl2

2 ST Ul Vl

2

0

(1.52)

1.6.3

(de S) - (di S)

dS = de S + di S.

(1.53)

di S dt

0,

(1.54)

16

1. ----

-. p l

di Sl dt

0,

di Sl 0,

l = 1, . . . p

(1.55)

de S =

dq T

=

dU dw T

=

dU + P dV T

r i=1

i de ni

.

(1.56)

- di S - .

1.6.4

(T = 0) (S = 0). ( = 1).

1.7

(U, S, T, V, P, i , ni ) (U , S , T , V , P , i , ni ) , , , ST = S + S (+), ((ST )UT ,VT ,nT = 0) ,i

U +U V +V ni + ni

= =

UT nT , i i = 1, . . . , r

(1.57) (1.58) (1.59)

= VT

U + U V + V ni + ni

= = =

0 0 0, i = 1, . . . , r

(1.60) (1.61) (1.62)

T P i

= = =

T P

( ) ( ) ( ).

(1.63) (1.64) (1.65)

i , i = 1, . . . , r

1.7.

17

( ) Lagrange ( ). , dS = de S + di S de S. (1.66) 1.56 ( CLAUSIUS)

T dS q

(1.67)

T S q

(1.68)

,

T S = q . r

U T S P V +i=1

i ni .

(1.69)

r

dU = T dS P dV +i=1

i dni .

(1.70)

, , - ( unconstrained) , U (Sl , Vl , nil ; ST , VT , nT ) ( i l, T )

(Gradient) (U )S,V,ni = 0 (), (Hessian) ( 2 U )S,V,ni 0 ().

(1.71)

(1.72)

U ( ) ( )

2U S 2

0,

2U 2U S 2 V 2

2U SV

2

0.

(1.73)

(dU/dt)S,V,ni 0.

(1.74)

18

1. ----

1.67 . ( ) ! 1.52 ( 2 S < 0) 1.73 ( 2 U > 0), :

V kT = CV T

V (P ) S TV,ni T ,ni

0,

(1.75)

= ni i

0, 0.

(1.76)

(1.77)

T ,V,j

CP = T

S TP,ni

,

(1.78) (1.79)

CP CV > 0. :

q = CV dT.,

(1.80)

CV =

U TV,ni

(1.81)

:

q = CP dT :

(1.82)

=

1 V

V TP,ni

(1.83)

:

T =

1 V

V PT ,ni

(1.84)

:

S =

1 V

V PS,ni

(1.85)

1.8. 19

1.8

, / , (Ueq , Veq , nieq ). Taylor

S(Ueq + U, Veq + V, nieq + ni )

= +

Seq (Ueq , Veq , nieq )

S(U, V, ni ) 1 2 + S(U, V, ni ) 2 + . (1.86)

S(U, V, ni ) =

1 T

1 Teq

U +

P T

Peq Teq

r

V i=1

i T

eqi Teq

ni .

(1.87)

T P i

= = =

Teq Peq eqi , i = 1, , r,(1.88)

S = 0.

2 S(T, V, ni ) =

CV2 Teq

(T )2 1

(< 0)

(1.89)

(V )2 (< 0) (1.90) Teq Veq T i (ni nj ) (< 0), (1.91) nj Teq ij

(U = CV T ). S = 0

S Seq = 1/2 2 S < 0.

(1.92)

, - di S = Seq S = 1/2 2 S > 0, . . 2 S 0,

20

1. ----

. , - ( 2 S ) , , . (Seq ) : Lyapunov

L(T, V, ni ) =

1 2

2 S(T, V, ni ) < 0,

(1.93)

dL(T, V, ni ) dt

=

d dt

2 S(T, V, ni ) 2

> 0.

(1.94)

- .

1.9

Legendre

, . Legendre . . ) (S, P, ni ). .

H(S, P, ni ) = U (P )V

(1.95)

r

dH = T dS + V dP +i=1

i dni

(1.96)

H SP,ni

= T,

H PS,ni

=V

H niS,P,nj

= i .

(1.97)

1.9. LEGENDRE

21

( ( unconstrained) ) .

(H)S,P,ni = 0 (), 2H S 2P,ni

(1.98)

0,

2H P 2S,ni

0,

2H n2 iS,P,nj

0. (1.99)(1.100) (1.101) (1.102)

dH dt S,P,ni

iS = T ddt 0,

CP =

H TP

,

(dH)P = q.

) (T, V, ni ). HELMHOLTZ.

A(T, V, ni ) = U T S

(1.103)

r

dA = SdT P dV +i=1

i dni

(1.104)

A TV,ni

= S,

A VT ,ni

= P

A niT ,V,nj

= i .(1.105)

HELMHOLTZ ( - ) .

(A)T ,V,ni = 0 (), A T 2V,ni 2

(1.106)

0,

A V2 T ,ni

2

0,

A n2 iT ,V,nj

2

0. (1.107)(1.108)

dA dt T ,V,ni

iS = T ddt 0.

) (T, P, ni ). GIBBS.

G(T, P, ni ) = U T S (P )V = H T S = A + P V(1.109)

22

1. ---- Euler Gibbs

r

r

G(T, P, ni ) =i=1

i (T, P )ni =i=1

i ni

(1.110)

r

dG = SdT + V dP +i=1

i dni

(1.111)

G TP,ni

= S,

G PT ,ni

=V

G niT ,P,nj

= i . (1.112)

GIBBS ( - ) .

(G)T ,P,ni = 0 (), 2G T 2P,ni

(1.113)

0,

2G P 2T ,ni

0,

2G n2 iT ,P,nj

0. (1.114)(1.115)

dG dt T ,P,ni

iS = T ddt 0.

) (T, V, i ). .r

(T, V, i ) = A i=1

ni i = A G = P V

(1.116)

r

d = SdT P dV i=1

ni di

(1.117)

TV,i

= S,

VT ,i

= P,

iT ,V,j

= ni .(1.118)

( - ) .

1.10. MAXWELL

23

()T ,V,i = 0 (), T 2V,i 2

(1.119)

0,

V2 T ,i

2

0,

2 iT ,V,j

2

0 (1.120)(1.121)

d dt T ,V,i

iS = T ddt 0.

1.10

MaxwellT V T P S V S PS

= =S

PV

S V S PP

, , , .P

(1.122) (1.123) (1.124) (1.125)

=T

=T

T V V T

1.11

(X1 , X2 , . . . , Xr , Ir+1 , Ir+2 , . . . , Is ),

(1.126)

r , (X1 , X2 , . . . , Xr ) s r , (Ir+1 , Ir+2 , . . . , Is ) : :r s

d =i=1

Ii dXi j=r+1

Xj dIj .

(1.127)

Maxwell

Ii Ij

=

Xj Xi = =

, (j > r i r).

(1.128)

Xi Ij Ii Xj

Xj Ii Ij Xi

, (i, j r). , (i, j > r).

(1.129)

(1.130)

24

1. ----

, -

= 0. (convex)

(1.131)

22 Xi X1 ,...,Xi1 ,Xi+1 ,...,Xr ,Ir+1 ,...,Is

0,

(1.132)

(concave)

22 Ir+j+1 X1 ,...,Xr ,Ir+1 ,...,Ir+j ,Ir+j+2 ,...,Is

0.

(1.133)

(Xi , Ii ):

Ii XiX1 ,...,Xi1 ,Xi+1 ,...,Xr ,Ir+1 ,...,Is

0, [(S, T ), (V, P ), (ni , i )] .(1.134)

1.12

Gibbs-Duhemr

SdT V dP +i=1

ni di = 0.

(1.135)

1.13

) (V, T ), P = f (V, T ) ,

1.13.

25

(i) (ii) (iii) (iv)

CV V S V U V H VT

= T =T

2P T 2 P T V P T P T P TV V

(1.136) (1.137)

= TT

P +VV

(1.138)

= TT

P VT

(1.139) (1.140)

= T (v) CP CV = T

V

1 T2

P TV

/2

P VT

(1.141) (1.142)

= T T V

P T P TV V

= T V 2 /T (vi) H TV

(1.143) (1.144)

= CV + V

) (P, T ), V = f (P, T ) ,

26

1. ----

(i) (ii)

CP P S P U P H PT T T

= =T

T

2V T 2 VP

(1.145) (1.146) (1.147)

= (iii) = = (iv) = = (v) CP CV = = (vi) U TP

T P V V T P T P T V + P V T V V T T P V TV 2 V T / T P T V 2 /T CP P VP

V PT

(1.148) (1.149) (1.150) (1.151)

V PT

(1.152) (1.153) (1.154) (1.155)

= =

T CP P V

) (dS = 0)

1.14.

27

(i) (ii)

T V T PS

= =S

T

PV

CV T T VP

(1.156) (1.157) (1.158) (1.159)T

= (iii) V PS

CP T TV CP CV V CP V PT

= = =

P +

T

VP

2

T V + T S P

CP T 2 2 V T CP

(1.160) (1.161) (1.162)

CP CV P VS

= = =

(iv)

V T CV 2 T 1 2 T V T CV

T

P TV

2

(1.163) (1.164)

1.14

(Gibbs)

C . ()

F =C+2 F C = = = , , .

(1.165)

, , , , , (X/ni )T ,P,nj , (r ) i r , i

r = i

nr i

C

, nr

nr =i=1

nr , r = 1, . . . , . i

(1.166)

R M () ,

F =C+2RM

(1.167)

28

1. ----

1.15

Duhem

, , .

2

2.1

, . .

2.2

( ) . Taylor . , ( 2.1), Hessian , .

(df (x1 , x2 ))

f f = . x1 x2 x2 x1

(2.1)

df , . , -.

df (x1 , x2 ) = 0,29

(2.2)

30

2.

2.1: x0 . , x0 .

f(x)f(x)

f(x 0 )

L(x)=f(x0 )+f(x 0 )(xx0 )

f(x) = (df/dx)L(0)=f(x0 )f(x 0 )x0

0

x0

x

f (x1 , x2 ) = 0.

(2.3)

( -), - . f f . Taylor.

2.3

, , , . [3, 4].

2.4

-

. .

2.5.

31

, , , , . . - . , + .

2.5

2.2: Ei , i = 1, 2, . . . , Vi , i = 1, 2, . . . Ni , i = 1, 2, . . . . .

E , V 2, N 2 2

E , V 1, N 1 1 E , V 3, N 3 3

E , V 4, N 4 4

E , Vi , Ni i

p

Ei = UT ()i=1 p

(2.4)

Vi = VT ()i=1 p

(2.5)

Ni = NT ().i=1

(2.6)

32

2.

. 2.3 . . (), . , (constrained) - (unconstrained). 2.4. 2.3: () ( ) -.

11111 00000 11111 00000 11111 00000 11111 00000 11111 00000

()

()

11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000

2.6

. , N Ei ( )N

U=i=1

Ei .

(2.7)

Ei = Ti + Vi . (2.8)

2.6.

33

2.4: () ( ) .

111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000() ()

: . .

dU = 0.

(2.9)

U = 0, U = 0.

(2.10)

. . , - . , F 2.9. z . , , . , +,

dU = q + w.

(2.11)

q w . q w ( ).

34

2.

U = U U = q + w.

(2.12)

, q, w , . , . , .

U = 0 = q + w, w = q.

(2.13)

, . (q = 0), w = 0.

2.7

(-) . : . , - . ST + ,

dST 0. dt

(2.14)

, , (U, V, N ). .

S(U, V, N )

=

S(U, V, N ),( , )

(2.15) (2.16) (2.17) (2.19)

S(U2 , V, N ) S(U1 + U2 , V1 + V2 , N1 + N2 )

S(U1 , V, N ) U2 U1 , S(U1 , V1 , N1 ) + S(U2 , V2 , N2 ).

( , )(2.18) ( , ). 2.5 2.6. (concave)

2.7.

35

S (cU1 + (1 c)U2 ) cS(U1 ) + (1 c)S(U2 ),

c [0, 1].

(2.20)

2.5: .

S

S(cU 1+(1c)U2 ) >= cS(U 1) + (1c)S(U2 ) S2 S1

U1

U2

U

( 2.7,2.8).

U (S, V, N )

= U (S, V, N ),( , )

(2.21) (2.22) (2.23) (2.25)

U (S2 , V, N ) U (S1 , V, N ) S2 S1 , U (S1 + S2 , V1 + V2 , N1 + N2 ) U (S1 , V1 , N1 ) + U (S2 , V2 , N2 ).

( , ) (2.24) ( , ). (convex) , ( 2.7 2.8).

U (cS1 + (1 c)S2 ) cU (S1 ) + (1 c)U (S2 ),

c [0, 1].

(2.26)

36

2.

2.6: .

S

V

U

.

S = S(U ),

S1 S2 S

= = =

S(U1 ), S(U2 ), S(U ) = cS1 + (1 c)S2 , c [0, 1],

(2.27) (2.28) (2.29)

. U , S1 S2 .

U = U (cS1 + (1 c)S2 ).,

(2.30)

2.7.

37

2.7: .

U

U[cS 1+(1c)S 2] T ,

2.8. -- ,dST dt

43

> 0, dU < 0, dt(2.63)

. , . T < T , dU > 0 . dt , P > P , .

P S = , V T

(2.64)

dST dt

= = =

S dV S dV + V dt V dt S S dV V V dt P P dV 0. T T dt

(2.65) (2.66) (2.67)

(T = T ) P > P dV /dt > 0, . () (). , .

S = , N T dST dt S N S N T dN S dN + , dt N dt S dN , N dt dN 0. + T dt

(2.68)

= = =

(2.69) (2.70) (2.71)

(T = T ), = . > dN /dt < 0 . N , d(/T ) d(/T ) z , dz , dz dz ,

44

2. d(/T )

. , dz z , . . , , ( uctuations) . , , , . , , . . , 1/T , P/T /T . , , . S(U, V, N ) (. 2.53). , , U (S, V, N ), , , (. 2.44), . , .

2.9

, . (T = 0) (S = 0). ( = 1). .

2.10

p . (Sj , Vj , Nj ), . U (S1 , . . . , Sp , V1 , . . . , Vp , N1 , . . . , Np ) (Sj , Vj , Nj )

2.10.

45

p

F1

=j=1 p

Sj ST = 0 Vj VT = 0j=1 p

(2.72)

F2

=

(2.73)

F3

=j=1

Nj NT = 0.

(2.74)

2.11: .

U

Sj Vj

2.11 , () . , . 2.12 () . -.

46

2.

2.12: .

Lagrange ( ). 3

G(Sj , Vj , Nj , i ) = U (Sj , Vj , Nj ) i=1

i Fi .

(2.75)

G U = 1 Sj Sj G U = 2 Vj Vj G U = 3 Nj Nj

= = =

0 (j = 1, . . . , p) 0 (j = 1, . . . , p) 0 (j = 1, . . . , p).

(2.76) (2.77) (2.78)

Tj Pj j

= 1 (j = 1, . . . , p) = 2 (j = 1, . . . , p) = 3 (j = 1, . . . , p),

(2.79) (2.80) (2.81)

2.10.

47

2.13: . , , , . .

S(U) S2 S1

1/T2 1/T 1

T1 = T 2 U2 U 1 U

T1 P1 1

= ... = ... = ...

= = =

Tp Pp p

= = =

( ) ( ) ( ).

( 2.13. +, (U, S, T, V, P, , N ) (U , S , T , V , P , , N ) ,

U +U V +V N +N

= UT = VT = NT ,

(2.82) (2.83) (2.84)

U + U V + V N + N

= = =

0 0 0.

(2.85) (2.86) (2.87)

ST = S + S 0,

(2.88)

48

2.

S + (

P U + V N ) 0, T T T P U V + N 0, S T T T

(2.89) (2.90) (2.91) (2.92)

T S U P V + N 0,

U T S P V + N.

dU = T dS P dV + dN. ,

(2.93)

dU = w + q.

(2.94)

w = P dV + dN, ,

(2.95)

T dS = q.

(2.96)

T S = q.

(2.97)

- + S + S 0. (2.98) T = T , dS = q /T . , q = q

T dS q ( CLAUSIUS),

(2.99)

T S q.

(2.100)

2.99 , T dS = q . , , - ( unconstrained) , U (Sl , Vl , Nl ; ST , VT , NT ) ( l, T ).

2.10.

49

.

, U (S, V ), (Gradient) (U )S,V,N = 0 (), (2.101)

(Hessian) 2 U 0 (), ., 2U S 2 0, 2U 2U S 2 V 2 2U SV2

(2.102) (2.103)

0 .

2U S 2

=V

T S

=V

T 0. CV

(2.104)

2U 2U S 2 V 2

2U SV

2

0,

(2.105)

T (P ) T (P ) 0, S V V S T T P P + S V V S V S SP S V P V T S V T V

(2.106)

0.V

(2.107)

0

(2.108)

S

S

( )

(P, T ) (P, T ) = 0. (S, V ) (V, S) - (V, T )

(2.109)

(P, T )/(V, T ) 0. (V, S)/(V, T ) (P/V )T 0. (S/T )V

(2.110)

(2.111)

50

2.

(kT ) :

V kT =

V P

0.T,N

(2.112)

, (T (S/T )V ), . ...

N

0.T,V

(2.113)

CP = T

S T

,P,N

(2.114) (2.115)

CP CV > 0, :

q = T dS = CV dT.,

(2.116)

CV =

U T

.V,N

(2.117)

:

q = T dS = CP dT :

(2.118)

=

1 V 1 V 1 V

V T V P V P

(2.119)P,N

:

T =

(2.120)T,N

:

S =

(2.121)S,N

2.11.

51

2.11

. ( 2.14 N , T0 , , V1 , V2 , , P1 P2 . ( 2.14 , . SA = S1 + S2 , SB .

SB SA . . SA = SB , q = T S = 0.

w = U =

3 kB (2N )(T T0 ). 2

T . ,

S=

3 kB N ln T + N kB ln 2

V N

+

SB SA

=

3kB N ln T + 2N kB ln

V1 + V2 2N 3 V1 3 kB N ln T0 N kB ln kB N ln T0 N kB ln 2 N 2

V2 N

= 0.

3kB N ln T +

2 ln 3

V1 + V2 2N ln

ln T0

1 ln 3

V1 N1/3

1 ln 3

V2 N

=0

T = ln T0

4V1 V2 (V1 + V2 )2 4V1 V2 (V1 + V2 )2

1/3

T = T0

.

52

2.

2.14: P2 > P1 . .

AT0, P1, V1, n S1

111 000 111 000 111 000T0,P2,V2,n 111 000 111 000 111 000 S2 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000

BT,V/2,P,n S/2

11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00

T,V/2,P,n S/2

()

()

2.12

, , . (uctuations) ( ) ( ). , Legendre. Legendre (/) .

- , (di S/dt), , , , , . (S, U, T, V, i , ni ) (S , U , T , V , i , ni )

2.12.

53

S(Ueq + U, Veq + V, neqi + ni )

= + + +

Seq (Ueq , Veq , neqi ) S(U, V, ni ) 1 2 S(U, V, ni ) 2 .

(2.122)

S(U, V, ni ) =

1 1 T Teq

U +

P Peq T Teq

r

V i=1

eqi i T Teq

ni .(2.123)

S = 0

T P i

=

Teq , i = 1, , r,(2.124)

= Peq = eqi ,

1 2 S(U, V, ni ) = 2

+ +

1 1 2 U T 1 P 2 V T 1 2 ij nj

1 (U )2 U T P + (V )2 V T i i + (ni nj ). T nj T +

U V

1 T P T

= =

1 1 T = 2 , 2 T U V T CV 1 P 1 = , T V T T V T

1 2 S(U, V, ni ) = 2

1 (U )2 2 2 CV Teq

1+

CV CV

1 1 (V )2 V 1+ 2 Teq T Veq V 1 i + 2 ij nj Teq nj

i Teq

(ni nj ).

54 U = CV T

2.

1 2 S(T, V, ni ) = 2

1 CV (T )2 2 2 Teq

1+

CV CV

1 1 (V )2 V 1+ 2 Teq T Veq V 1 i + 2 ij nj Teq nj

i Teq

(ni nj ).

CV r i r). Ij Xi Xi Xj = , (i, j r). Ij Ii Ii Ij = , (i, j > r). Xj Xi

(2.211) (2.212) (2.213)

2.14.

67

, -

= 0. (convex)

(2.214)

2 2 Xi

0,X1 ,...,Xi1 ,Xi+1 ,...,Xr ,Ir+1 ,...,Is

(2.215)

(concave)

2 2 Ir+j+1

0.X1 ,...,Xr ,Ir+1 ,...,Ir+j ,Ir+j+2 ,...,Is

(2.216)

(Xi , Ii ):

Ii Xi

0, [(S, T ), (V, P ), (ni , i )] . (2.217)X1 ,...,Xi1 ,Xi+1 ,...,Xr ,Ir+1 ,...,Is

: (-) () , (-) () . ( (1.52) ). ( (1.73) ). ( ), () . .

68

2.

2.15

Gibbs-Duhem

, , (S, V, Ni ) (T, P, i ) . , U = T S P V + i i Ni ,

(T, P, i ) 0 = U T S + P V i

i Ni .

(2.218)

d(T, P, i ) = 0 = dU T dS SdT + P dV + V dP i

i dNi i

Ni di .(2.219)

dU = T dS P dV +i

i dNi .

SdT V dP +i

Ni di = 0.

(2.220)

Gibbs-Duhem. , . 2.220 . , , , , , . ,

V (N1 , N2 , . . . , ) = V (N1 , N2 , . . . ). Euler

V =i

Ni

V Ni

=T,P,Nj =i i

Ni v i ,

(2.221)

vi . . 2.221 (Ni vi ).

dV =i

dNi

V Ni

=T,P,Nj =i i

dNi vi .

(2.222)

dV =i

dNi vi +i

Ni dvi .

(2.223)

2.16. (2.222, 2.223),

69

Ni dvi = 0.i

(2.224)

vi

dvi =k

vi Nk

dNk .T,P

(2.225)

. 2.224

Nii k

vi Nk

dNkT,P

=k i

Ni

vk Ni

dNk = 0. (2.226)T,P

2V 2V vk vi = = = . Nk Nk Ni Ni Nk Ni

(2.227)

dNk

Nii

vk Ni

= 0.T,P

(2.228)

Gibbs ( ). 2.228

G=k

Nk

G Nk

=T,P k

Nk k ,i

Ni

k Ni

= 0.T,P

(2.229)

Helmholtz (ak ) (hk )

A=k

Nk

A Nk H Nk

=T,P k

Nk a k ,i

Ni

ak Ni hk Ni

= 0.T,P

(2.230)

H=k

Nk

=T,P k

Nk hk ,i

Ni

= 0.T,P

(2.231)

2.16

(w, q, T, P, V, CV , CP , , T , S ) (S, U, A, G).

70

2. 1. (V, T ), P = f (V, T ) ,

(i) (ii) (iii) (iv)

CV V S V U V H V

= TT

=T

=T

=T

= (v) CP CV =

2P T 2 P T V P T T P T T P T T T

(2.232)V

(2.233)

PV

(2.234)

+VV

P V

(2.235)T

V

1 T2

(2.236)

P T

/V

P V2

(2.237)T

= T T V = T V 2 /T (vi) H T = CV + VV

P T P T

(2.238)V

(2.239) (2.240)V

(i) dA = SdT P dV. A T A V(2.241)

S=

, P =V

.T

(2.242)

CV

= T = T

S T 2A . T 2

(2.243)V

(2.244)

2.16.

71

CV V

= TT

3A T 2 V 2 T 2 2P T 2 A V .V

(2.245) (2.246)T

= T = T

(2.247)

(ii) dA = SdT P dV. Maxwell (2.248)

S V

=T

P T

.V

(2.249)

(iii) dU = T dS P dV.(2.250)

U V

= TT

=

T

S V P T

PT

(2.251) (2.252)

P.V

(iv) dH = T dS + V dP.(2.253)

H V

= TT

= T

S V P T

+VT

+VV

P V P V

(2.254)T

.T

(2.255)

72

2.

(v)

CV

S T V (S, V ) = T (T, V ) (S, V )/(T, P ) = T (T, V )/(T, P ) = T = T = TS T P V P S P T T V P T V T P

(2.256) (2.257) (2.258) (2.259)

S T

+TP 2

V T

2

/P

V P

(2.260)T

= CP + T V 2 /(V kT ).

(2.261)

CP CV =

2 T V. kT

(2.262)

...

2. (P, T ), V = f (P, T ) ,

2.16.

73

(i) (ii)

CP P S P U P H P

=T

=T

= (iii) =T

= (iv) =T

= (v) CP CV = = (vi) U T =P

2V T 2 P V T P V V T P T P T V + P V T V V T T P V TV T T V T2

(2.263) (2.264) (2.265)

V P

(2.266)T

(2.267) (2.268) (2.269)

/P

V P

(2.270)T

T V 2 /T V CP P T CP P V P

(2.271) (2.272) (2.273)

=

3. (dS = 0)

74

2.

(i) (ii)

T V T P

= S

=S

= (iii) V P =S

T CV T CP TV CP CV CP V P

P T V V T P

(2.274) (2.275) (2.276)

V P +T

(2.277)T

=

T CP

V T

2

(2.278)P

= T V + CP CV (iv) P VS

2 V 2 T CP

(2.279) (2.280)

= =

T S P V T

T CV

P T

2

(2.281)V

=

2 T 1 2 T V T CV

(2.282)

(i) T V = S

(S/V )T (S/T )V T P = CV T

(2.283)

.V

(2.284)

(iii) V P =S

= = =

(V, S) (P, S) (V, S)/(V, T ) (V, T ) T (P, S)/(P, T ) (P, T )S T V S T P

(2.285) (2.286) (2.287)

V P

T

CV CP

V P

(2.288)T

2.17. ...

75

2.17

(Gibbs)F = C + 2. F C = , = , = .(2.289)

G = n,

(2.290)

( ). , .. ,

Gm = (T, P ).

(2.291)

Gm . C () C

G=i=1

i ni .

(2.292)

nC

n=i=1

ni ,C

(2.293)

Gm

G = = n

i=1

ni i , n

(2.294)

76

2.

i =

ni . n

(2.295)

C

Gm =i=1 C

i i

(2.296)

i = 1.i=1

(2.297)

(C 1) + 2 , (C 1) , . , . , . C

G=r=1 C

Gr ,

(2.298)

Gr =i=1

r nr . i i

(2.299)

C

n =i=1

r

nr , r = 1, . . . , , iC

(2.300)

Gr = m

Gr = nr

i=1

nr r i , r = 1, . . . , . nr i

(2.301)

i r

r = i

nr i . nr

(2.302)

r

Gr m

Gr = r = n

C

r r , r = 1, . . . , , i ii=1

(2.303)

2.17. C

77

r = 1, r = 1, . . . , . ii=1

(2.304)

G ( ) (C 1)+2 . i 1 = 2 = = = i , (2.305) i i i ( 1) . C C( 1) C( 1) . . Gibbs C

F = (C 1) + 2 C( 1) = C + 2,

(2.306)

. , . , , . , C (R), (M ).

F = C + 2 R M.

(2.307)

Gibbs (T, P, 1 , 2 , . . . , C ), (2 + C) . , . Gibbs-Duhem (). F = (2 + C) = C + 2.

. 1) : F = 2 ( ).

2) : F = 1 ( 2.18, 2.19).

78

2.

2.18: .

chemical potential

14 12 10 8 6 4 2 0 -2 -4 7 6 1 5 2 3 T 4 4 3 5 6 2 7 8 1 P

8

. (T, P ) . ; ClausiusClapeyron (T, P ). 1 2

1 (T, P ) = 2 (T, P ),

(2.308)

1 dP 2 2 dP 1 + = + . T P dT T P dT

(2.309)

d = Sm dT + Vm dP,. 1 1 Sm + Vm

(2.310)

Sm Vm dP 2 2 dP = Sm + Vm dT dT,

(2.311)

Clausius-Clapeyron.1 Sm dP S 2 Sm = = m , 2 V1 dT Vm Vm m

(2.312)

2.17.

79

2.19: .

Tc

T

Sm = q/T.

dP q = . dT T Vm

(2.313)

(.. -) , (T, S) (T, V ), ( 2.20).

2.20: Tc .

S

V

Tc

T

Tc

T

80

2.

(T, P ) (T, V ) 2.21.

2.21: . .

P

V 2 2 1,21+2

P = P1

1 1 T T

P2+3

VP2 P1

3 2 31+2 1+3

1,3

2,3 2 1,2

1 T

1 T

3) ( = 3), F = 0. , (Tc , Pc ) . (Tc , Pc ) .

4) x1 x2 , (x1 + x2 = 1) F = 3 . , ,

2.17.

81

2.22: .

CV < 0, CV = dT T2

U T V

=

S 2 U V

/

2S U 2

V

. d

1 T

=

S 2

1+2

1

U

, .. x1 .

5) , F = 2 . , x1A x1B .

8 9 Atkins [9] .

82

2.

2.23: . ( Legendre LS (T, Vm ) = S(U, Vm ) U/T = Am /T . , S(U, Vm ) U , LS (T, Vm ) . Helmholtz, Am (T, Vm ) = T LS , T Vm .)

Am

1 2 1+2

Vm

2.24: . ( Legendre LS (T, P ) = S(U, Vm ) U/T P Vm /T = Gm /T . S(U, Vm ) U Vm , LS (T, P ) . Gibbs, Gm (T, P ) = T LS , T P .)

P

= 1

Gm

= Gm

=

1+2 I II 1+2 2 1+2 1 2 2 1

V

V

P

2.17.

83

2.25: . ( .)

P 2

Tcr1

Gm

Tcr1 Tcr Tc2 Tc1

Tcr 1 Tc2 1 2 2

Tc1

1 Vm T

84

2.

2.26: . ( (bifurcations) (pitchfork) - (saddle-node).)

T < Tc Tcr P T1 Tc T2

2

1T > Tc

T Gm T1 < Tc Tc T2 > Tc

2 1+2

2

1 1+2 1+2

1

Vm

Vm

Vm

T1 < Tc Vm

T2 > Tc

1

1

2

2

T

Tcr

Tc

T

2.18. DUHEM

85

2.18

DUHEM

, , (Ni0 ) . Duhem Gibbs Duhem . , C Ni0 , i = 1, . . . , C , (2 ) C , nr , , r = 1, . . . , . i C

nr = Ni0 , ir=1

i = 1, . . . , C

(2.314)

1 (T, P ) = 2 (T, P ) = = (T, P ), i i i

i = 1, . . . , C

(2.315)

C( 1) . F = C + 2 C( 1) C = 2. (a ) , a.

86

2.

2.19

2.19.

87

2.1: . (U ) - , (H ) - , Helmholtz (A) - , Gibbs (G) - ,

U U (S, V, N ) dU (S, V, N ) dU (S, V, N )= = =

H T S + (P )V + N T dS + (P )dV + dN T dS P dV + dN H(S, P, N ) dH(S, P, N ) dH(S, P, N )= = =

U (P )V d(U + P V ) T dS + V dP + dN

U S V,N U V S,N U N S,V

= = =

T P

H S P,N H P S,N H N S,P

= = =

T V

T V S,N T N S,V

= = =

P N S,V

P S V,N S V,N V S,N

T P S,N T N S,P V N S,P

= = =

V S P,N S P,N P S,N

A A(T, V, N ) dA(T, V, N ) dA(T, V, N )= = =

G U TS d(U T S) SdT P dV + dN G(T, P, N ) dG(T, P, N ) dG(T, P, N )= = =

U T S (P )V d(U T S + P V ) SdT + V dP + dN

A T V,N A V T ,N A N T ,V

= = =

S P

G T P,N G P T ,N G N T ,P

= = =

S V

S V T ,N S N T ,V P N T ,V

= = =

P T T V

V,N V,N T ,N

-

S P T ,N S N T ,P V N T ,P

= = =

V T T P

P,N P,N T ,N

88

2.

3

3.1 P V = nRT,( ) (3.1)

3.1.1

U (T, n) = 3 nRT. 2(3.2)

3.1.2

w = nRT ln(Vf /Vi ).(3.3)

89

90

3.

3.1.3

S = CV ln T + nR ln V n + .(3.4)

S = CP ln T nR ln P + .

(3.5)

3.1.4

CV = 3 nR. 2(3.6)

CP =

5 nR. 2

(3.7)

3.1.

91

CP CV = nR.

(3.8)

3.1.5

- T = Pex. V /CV .(3.9)

3.1.6

P V = .(3.10)

= CP /CV .

(3.11)

92

3.

w = CV Ti [(Vi /Vf )1 1].

(3.12)

3.1.7

S = nR ln(Vf /Vi ).(3.13)

3.1.8

, G(P ) = G(P ) + nRT ln(P/P ).(3.14)

3.1.

93

P

G(P )

= = = =

G(P ) +P P

V dP nRT dP PP

(3.15) (3.16) (3.17) (3.18)

G(P ) +P

dP P P G(P ) + nRT ln(P/P ). G(P ) + nRT

3.1.9

(P ) = (P ) + RT ln(P/P ).(3.19)

. G(P ) = G(P ) + nRT ln(P/P ). (3.20)

G(P ) G(P ) = + RT ln(P/P ). n n

(3.21)

3.1.10

Clausius-Clapeyron ()-d ln P/dT = Hm,. /RT 2 .(3.22)

94

3.

dP dT dP P dT d ln P dT

= q/(T Vm ) = = = Hm,. P/RT 2 Hm,. /RT 2 Hm,. /RT 2 .

(3.23) (3.24) (3.25) (3.26)

P = P exp

Hm,. R

1 1 T T

.

(3.27)

d ln P = Hm,. dT /RT 2 .

(3.28)

(T , P ) (T, P )

ln(P/P ) =

Hm,. R

1 1 T T

.

(3.29)

.

3.1.11

xi Gmix = nRT [xA ln xA + xB ln xB ] .(3.30)

3.1. P ,

95

Gi

= nA A + nB B i i = nA A + RT ln P P + nB B + RT ln P P .

(3.31) (3.32)

PA PB ,

Gf

= nA A + nB B f f = nA A + RT ln PA P + nB B + RT ln PB P .

(3.33) (3.34)

Gmix = Gf Gi

= nA RT ln

PA P

+ nB RT ln

PB P

. (3.35)

PA = A P, Gmix

PB = B P.

(3.36)

= nA RT ln A + nB RT ln B nA nB = nRT [ ln A + ln B ] n n = nRT [A ln A + B ln B ],

(3.37) (3.38) (3.39)

n = nA + nB .

Smix = nR [xA ln xA + xB ln xB ] .

(3.40)

G = S. T

(3.41)

96

3.

G = S. T

(3.42)

Smix = nR [xA ln xA + xB ln xB ] . Smix = Gmix /T.

(3.43) (3.44)

Umix = Hmix = Vmix = 0.

(3.45)

Gmix = Hmix T Smix .

(3.46)

Hmix = 0.

(3.47)

G = V. P G = V. P

(3.48) (3.49)

Gmix = 0 = Vmix . P

(3.50)

G = U T S + P V.

(3.51)

Umix = 0.

(3.52)

1. , .

Gmix = T Smix .

(3.53)

3.1.

97

2. . 3. ;

3.1.12

: 0 =J

J J.

(3.54)

r G m =j

j j .

(3.55)

nB nC nA = = = = , A B C

(3.56)

, J . J .

98

3.

r G =j

nj j j jj

(3.57) (3.58) (3.59) (3.60) (3.61)

r G = r G/ r Gm =j

j j r G/ j j .j

= =

r Gm = 0. ,

(3.62)

aA + bB pP + qQ,

(3.63)

pP + qQ = aA + bB .

(3.64)

r Gm =j

j j . j f Gm,j ,

(3.65)

r Gm =j

(3.66)

f Gm,j j .

.

3.2.

99

r Gm = r Gm + RT ln Q. Q=j

(3.67) (3.68)

Pj P

j

.

r Gm = RT ln K. K=j

(3.69) (3.70)

Pj P

j

.

3.2

3.2.1

van der Waals

P =

nRT n a V nb V

2

.

(3.71)

100

3.

3.2.2

Hess

: 0 =J

J J.

(3.72)

r H

=J

J f HJ .

(3.73)

3.2.3

KirchhoT2

r H(T2 ) = r H(T1 ) +T1

r CP (T )dT.

(3.74)

r C P =J

J CP,J .

(3.75)

3.2.

101

3.2.4

Gibbs r G =J

J f GJ .

(3.76)

r S

=J

J SJ .

(3.77)

3.2.5

Gibbs-Helmholtz(G/T ) T =P

H . T2

(3.78)

G = H T S. G H = S. T T

(3.79) (3.80)

102

3.

T

G T

=P

H T

T H /T 2 P

S T

(3.81)P

CP CP H 2 T T T H = 2. T =

(3.82) (3.83)

(G/T ) T

=P

H . T2

(3.84)

3.2.6

Gibbs P

G(P ) = G(P ) +P

V dP.

(3.85)

G P

= V.T,n

(3.86) (3.87)

dG(P ) = V dP,P

G(P ) = G(P ) +P

V dP.

(3.88)

3.2.

103

3.2.7

Raoult PA = x A PA .

(3.89)

. xA A . PA A . PA . A (l) = (l) + RT ln(xA ). A(3.90)

3.2.8

: A = A xA . (3.91)

A (l) = (l) + RT ln(A ). A

(3.92)

3.2.9

:

T =

RT 2 Hb,m

xB

(3.93)

104

3.

3.2.10

:

ln(xB ) =

Hf,m R

1 1 T T

(3.94)

3.2.11

: , vant HoV = nB RT.(3.95)

3.2.12

nl /ng = d /d.(3.96)

3.2.

105

V = nv = nxl vl + nxg vg v = xl vl + (1 xl )vg xl = (vg v)/(vg vl ) xg = (v vl )/(vg vl ) xl /xg = (vg v)/(v vl ) d = vg v d = v vl nl /ng = d /d

(3.97) (3.98) (3.99) (3.100) (3.101)

(3.102)

3.2.13

vant Ho:d ln K r H = . dT RT 2(3.103)

106

3.

4

107

108

4.

LEGENDRE G(q1 , . . . , qs ) s , , L, qi , i = 1, . . . , r uj = G/qj , j = r + 1, . . . , s. Legendres

L(q1 , . . . , qr , ur+1 , . . . , us ) = G(q) i=r+1

qi ui .

(.1)

,

s

dG =i=1

G dqi = qi

s

ui dqi ,i=1

(.2)

s

dL = dG r

(qj duj + uj dqj )j=r+1 s

(.3)

=i=1

ui dqi j=r+1

qj duj .

Maxwell s

dG =i=1

ui dqi .

(.4)

G

2G 2G = qi qj qj qi109

(.5)

110

. LEGENDRE

ui uj = uij = uji = . qj qi Maxwell.

(.6)

p

L(q1 , . . . , qs ) = p L(q1 , . . . , qs ).

(.1)

s

pp1 L(q1 , . . . , qs ) =i=1

L 1 qi = (qi )

s

i=1

L qi . qi

(.2)

,

pp L(q1 , . . . , qs ) = pL(q1 , . . . , qs ). , = 1s

(.3)

pL(q1 , . . . , qs ) =i=1

L qi . qi

(.4)

Euler.

111

112

.

LAGRANGE L(q1 , . . . , qs ) s , m (s > m)

Fi (q1 , . . . , qs ) = 0, i = 1, . . . , m, m

(.1)

G(q, ) = L(q1 , . . . , qs ) i=1

i Fi (q1 , . . . , qs ),

(.2)

.,

m

G(q, ) =

Li=1

i Fi = 0.

(.3)

i m (.1) (.3).

113

114

. LAGRANGE

.

f (x1 , x2 ) =

f f i+ j = 0. x1 x2 f = 0. x2

(.1)

f = 0, x1

(.2)

, Hessian. , / /, ( ) . f 2 x2 1

| 2 f (x1 , x2 ) I| =

f 2 x1 x2

= 0,f 2 x2 x1 f 2 x2 2

(.3)

I .

2 + p + q = 0,115

(.4)

116

.

p=

f 2 f 2 + , x2 x2 1 2 f 2 x1 x22

(.5)

q=

f 2 f 2 x2 x2 1 2

.

(.6)

1 + 2 = p, 1 2 = q,

(.7) (.8)

() q > 0, p > 0 () q > 0, p < 0. q < 0.

(JACOBIANS)

u = u(x, y),

v = v(x, y).

(.1)

(u, v) (x, y)

=

u x y v x y

u y v y

x x

(.2)

=

u x

y

v y

x

u y

x

v x

(.3)y

dudv (u, v) (x, y)

(u, v) dxdy (x, y) (u, v) = (y, x) (v, u) = (x, y) (v, u) = + (y, x) =

(.4) (.5) (.6) (.7)

117

118

.

u x u y v y v x

=y

(u, y) (x, y) (u, x) (u, x) = (y, x) (x, y) (v, x) (v, x) = (y, x) (x, y) (v, y) (x, y)

(.8) (.9) (.10) (.11)

=x

=x

=y

x(y, z) 1)

x y2)

z

y x

=z

(x, z) (y, z) = +1. (y, z) (x, z)

(.12)

x y

z

y z

x

z x

=y

(x, z) (y, x) (z, y) . . (y, z) (z, x) (x, y)

(.13) (.14) (.15)

(x, z) (x, y) (y, z) . . (y, z) (x, z) (x, y) = 1 =3)

x y

=z

=

(x, z) (w, z) . (w, z) (y, z) (x/w)z (y/w)z

(.16) (.17)

, .. , , , , ... . , O , , . : 1. : (U, V, N ) ( ) 2. : (T, V, N ) ( ) 3. : (T, V, ) ( ) 4. - ((T, P, N ) p , O n

O < O >==1

p O ,

( .1)

n , p n

p = 1.=1

( .2)

119

120

.

(O)2 < (O)2 > = =

< (O < O >)2 > < O2 > (< O >)2 .

( .3) ( .4)

p

S = kB

p ln p .

( .5)

. p Lagrange. n

F = S (=1

p 1),

( .6)

S . .

F p kB (ln p + 1) kB ln p p

= = = =

0 0 ( + kB ) exp[( + kB )/kB ]

( .7) ( .8) ( .9) ( .10)

.

p =

1 , (U, V, N )

( .11)

(U, V, N ) U , V N . E , V N ; , +

ptotal = penv p ,

( .12)

.

121 . ptotal penv . p = ptotal /penv . ( .13) ( )

p (E , V , N ) =

env [(Etotal E ), (Vtotal V ), (Ntotal N )] . total [Etotal , Vtotal , Ntotal ]

( .14)

exp p (E , V , N ) =

1 kB Senv [(Etotal

E ), (Vtotal V ), (Ntotal N )] .

exp

1 kB Stotal [Etotal , Vtotal , Ntotal ]

( .15) (S )

Stotal (Etotal , Vtotal , Ntotal ) = Senv (Etotal U, Vtotal V, Ntotal N )+S(U, V, N ).( .16)

U , V , N , . Taylor , Etotal U , , Vtotal V , , Ntotal N .

Senv [(Etotal E ), (Vtotal V ), (Ntotal N )] Senv [(Etotal U + U E ), (Vtotal V + V V ), (Ntotal N + N N )] Senv [(Etotal U ), (Vtotal V ), (Ntotal N )]Senv (Etotal E ) |Etotal U (U Senv (Vtotal V ) |Vtotal V

= + + + =S E |U (U 1 T

E )

(V V ) N ) E ) +P T S V |V

Senv (Ntotal N ) |Ntotal N (N

Senv [(Etotal U ), (Vtotal V ), (Ntotal N )] + Senv [(Etotal U ), (Vtotal V ), (Ntotal N )] +

(V V ) + T (N

S N |N (N

N )

=

(U E ) +

(V V )

N )

exp p (E , V , N ) =

1 1 kB [( T

(U E ) + exp

P T

(V V )

T (N

N )]( .17)

1 kB S(U, V, N )

122

.

p (E , V , N ) = exp

1 1 (U T S + P V N ) exp (E + P V N ) . kB T kB T( .18)

1. Helmholtz, A(T, V, N ) .

p (E ) = exp

E 1 A(T, V, N ) exp . kB T kB T

( .19)

Z=

exp

E , kB T

( .20)

p = exp

1 A(T, V, N ) Z = 1. kB T

( .21)

p = exp [A(T, V, N )] Z = 1,

( .22)

= 1/kB T .

123

A = = U = =

kB T ln Z 1 ln Z < E >=

( .23) ( .24)

E p

( .25) ( .26) ( .27) ( .28) ( .29)

E eE /Z

= = = < (E U )2 > =

(ln Z)/ (A)/ ln Z kB T 2 T

V,N

(E U )2 e(AE )

( .30) ( .31)

= = = = S P = =

(E U )

(AE ) e U

(E U )e(AE )

( .32) ( .33) ( .34)

U kB T 2 CV kB ln Z + kB T kB T kB T ln Z V ln Z N ln Z TV,N

( .35) ( .36)

T,N

=

( .37)T,V

2. , (T, V, ) .

p (E , N ) = exp

1 1 (U T S N ) exp (E N ) kB T kB T( .38) ( .39)

p (E , N ) = exp [(T, V, )] exp [(E N )]

=

e(E N )

( .40)

124

.

U N

= kB T ln = 1 ln = (ln )/ = ()/ ln T

( .41) ( .42) ( .43) ( .44) ( .45)V,

S P N

= kB ln + kB T = kB T = kB T ln V ln

( .46)T,

( .47)T,V

3. -

, G(T, P, N ) .

p (E , V ) = exp

1 1 (U T S + P V ) exp (E + P V ) kB T kB T( .48)

p (E , V ) = exp [G(T, P, N )] exp [(E + P V )]

( .49)

=

e(E +P V )

( .50)

125

G = = U + PV = =

kB T ln 1 ln (ln )/ (G)/ U kB (1/T )

( .51) ( .52) ( .53) ( .54) ( .55)P/T,N 2

< (E )2 > = = < (E )(V ) > = = < (V )2 > = = =4.

kB T 2 CP 2kB T P V + kB T P 2 V T ( .56) V kB ( .57) (1/T ) P/T,N kB T 2 V kB T P V T V kB (P/T ) 1/T,N kB T V PT,N

( .58) ( .59) ( .60) ( .61)

kB T V T

X0 , X1 , . . . , Xs F0 , F1 , . . . , Fs , Xi , .. (U, 1/T ), (V, P/T ), (N, /T ), X0 , X1 , . . . , Xs

p = exp

1 1 (S F0 X0 Fs Xs ) exp (F0 X0 + + Fs Xs ) kB kB( .62)

W =

e

k1 (F0 X0 ++Fs Xs )B

( .63)

LS [F0 , . . . , Fs ] Legendre LS [F0 , . . . , Fs ] = S F0 X0 Fs Xs ( .64)

p = exp

1 1 LS [F0 , . . . , Fs ] exp (F0 X0 + + Fs Xs ) kB kB( .65)

126

.

W = exp

1 LS [F0 , . . . , Fs ] kB

( .66)

LS [F0 , . . . , Fs ] = kB ln W.

( .67)

< Xj Xk >= kB

Xj Fk

( .68)F0 ...Fk1 Fk+1 ...Fs Xs+1 ...Xt

H . ( ) (E ) Schr dinger o

H = E , < | >= . ( )

( .69) ( .70)

eH = eE

( .71)

eE =< |eH | >,

( .72)

A = = =

ln

eE < |e H | >

( .73) ( .74) ( .75)

ln

ln T r(e H ).

T r < |e H | >.

127 . , . , ( Fermi-Dirac) ( Bose-Einstein).

(-)

)

H H(xi , yi , zi , pxi , pyi , pzi ), i = 1, . . . N.

( .76)

xi , yi , zi , pxi , pyi , pzi N .) , h Planck.

dxN dyN dzN dpxN dpyN dpzN dx1 dy1 dz1 dpx1 dpy1 dpz1 . . . 1/2 1/2 1/2 1/2 1/2 1/2 1/2 h1/2 h1/2 h1/2 h1/2 h1/2 h h h h h h h 1 h3N N dxi dyi dzi dpxi dpyi dpzi i

( .77)

( .78)

)

1 N dxi dyi dzi dpxi dpyi dpzi . h3N i

( .79)

Z=

1 h3N

eH(xi ,yi ,zi ,pxi ,pyi ,pzi ) N dxi dyi dzi dpxi dpyi dpzi i

( .80)

Z =

1 h3

dxdydz

dpx dpy dpz e(px +py +pz )/2m .( .81)

2

2

2

128

.

Z =

V 3/2 [2mkB T ] . h3

( .82)

N

Z =

VN 3N/2 [2mkB T ] . N !h3N

( .83)

U =< E >=

(ln Z ) () P =V

=

3N 3 = N kB T, 2 2 =

( .84)

(ln Z ) V P V = N kB T.

N , V

( .85)

( .86)

, U (xi , yi , zi ), , H = T + U. ( .87) Helmholtz

A = kB T ln Z = kB T ln Q + c(T, V, N ),

( .88)

Q(T, V, N ) =

exp

1 U (xi , yi , zi ) N dxi dyi dzi i kB T

( .89)

c(T, V, N ) . c , , , , . . . , . . A B

U (xi , yi , zi )

= =

(1 )UA (xi , yi , zi ) + UB (xi , yi , zi ), UA + (UB UA ),

( .90) ( .91)

129 . = 0 A = 1 B . .

A() = ( )

U

,

( .92)

2 A() 1 = 2 kB T

U

2

U

2

( .93)

2 A() 1 = 2 kB T

U

U

2

0.

( .94)

. b

A = A(b ) A(a ) =a

U

d.

( .95)

ln Z( ) =0

ln Z() d,

( .96)

Z() =0

exp[U ()/kB T ]dU, 1 U kB T exp[U ()/kB T ]dU,

( .97) ( .98) ( .99)

Z =

0

ln Z 1 Z = . Z

ln Z 1 = kB T

U

exp[U ()/kB T ] Z

dU,

( .100)

130

.

ln Z 1 = kB T ln Z( ) ln Z(0) =

U 0

,

( .101)

1 kB T

U

d.

( .102)

A = kB T ln Z,

( .103)

A = A( ) A(0) =0

U

d.

( .104)

...

. ( .94) . BOGOLIUBOV . Helmholtz. Helmholtz [10, 11, 12]. .92 .

A =

A =

(A+ A )

( .105)

min = 0 max = 1 .

A = kB T ln Z , 1 U (xi , yi , zi ) N dxi dyi dzi , i kB T = U + (U+ U ) Z+ Z

( .106)

Z (T, V, N ) =

exp

( .107)

U+

( .108)

A = kB T (ln Z+ ln Z ) = kB T ln

,

( .109)

131

A

= =

kB T ln kB T ln

eU /kB T e(U+ U )/kB T N i dxi dyi dzi Z eU /kB T Z e(U+ U )/kB T N dxi dyi dzi . i( .110)

,

e(U+ U )/kB T , U . A = kB T ln e(U+ U )/kB T

,

( .111)

BOGOLIUBOV

(H0 ) (H1 )

H = H0 + H1 ,

( .112)

Helmholtz BOGOLIUBOV

A At = A0 + < H1 >0 .

( .113)

H1 H0 .

A T S0 + < H >0 .

( .114)

, BOGOLIUBOV ( .113) . . BOGOLIUBOV ( .94). H0 H1

H = H0 + H1 ,

( .115)

132

.

Ht = H0 + < H1 >0 , < Ht >0 =< H >0 , At () = A0 + < H1 >0 .

( .116)

( ),

( .117)

= 1 BOGOLIUBOV, At A. ( .1)

.1: BOGOLIUBOV, At A, . .92 .95

A < t= 0+ At A0

A0

0

1

: S = kB ln . S S1 S2 1 2 ,

S = S1 + S2 ,

(.1) (.2)

= 1 2 .

S , S = f (), .

f () = f (1 ) + f (2 ). 1 :

(.3)

df () d df (1 ) df () = = d1 d d1 d1 f (2 ) 1 .

(.4)

d = 2 = , d1 1

(.5)

df () df (1 ) = , d 1 d1133

(.6)

134

. BOLTZMANN

df () df (1 ) df (2 ) = = = = kB . dln dln1 dln2

(.7)

1 2. .

df () = kB , dln Boltzmann

(.8)

S = f () = kB ln() + .

(.9)

S 0 = 1, = 0, . ...

- - . () , , . 11 Introduction to Modern Thermodynamics Dilip Kondepudi [13]. - Onsager 1931, - De Donger, Prigogine . - - , Ilya Progogine . , , - , , , , ... . , , , , , 135

136

. -

i , x, , t

T = T (x, t),

P = P (x, t),

i = i (x, t), i = 1, , r.

,

s[T (x, t), i (x, t)] = s(x, t), u[T (x, t), i (x, t)] = u(x, t), i (x, t) i . x

T (x)ds(x) = du(x) i

i (x)di (x).

(.1)

. -

(x, t) =

di s(x, t) 0. dt

(.2)

di S = dt

(x, t)dV 0.V

(.3)

- Fi Ji

=i

Fi Ji .

(.4)

, , , .

Ji =j

Lij Fj .

(.5)

Lij Onsager. ,

=ij

Lij Fi Fj 0.

(.6)

137 Lij Onsager , Lij = Lji . . . 1)

Fq =

1 T (x)

,

Jq = (T (x)) (Jm2 s1 ),

(.7)

( Fourier). 2)

FD =

k (x) T (x)

,

JD = Dk (k (x)) (mol m2 s1 ),

(.8)

Dk ( Fick). 3)

Fe =

E () = , T T

Je =

V E = (Cm2 s1 ), R

(.9)

, E , I , V , (R, ) ( Ohm). 3)

Fr =

Ar , T

Jr = vr =

1 dr (mol m3 s1 ). V dt

(.10)

Ar r, r mole, vr V. -. r 0 = a1 A1 a2 A2 an An + b1 B1 + b2 B2 + + bm Bm ,(.11)

r

dnA1 dnA2 dnAn dnB1 dnB2 dnBm = = = = = = = = dr . a1 a2 an b1 b2 bm

(.12)

(AFFINITY) .11 n m

Ar =i=1

ai Ai i=1

bi Bi ,

(.13)

138

. -

Ai Ai Bi Bi . Gibbs

Ar = r Gm . : 1. Ar = 0 dr /dt = 0. 2. Ar > 0 3. Ar < 0 .

(.14)

, Gibbs . . - , r ,

di S = dt

r

Ar dr 0. T dt

(.15)

Gibbs . : - , Gibbs . r

vr =

1 dr = Rf (r ) Rr (r ), V dt

(.16)

Rf Rr . molL1 s1 .

aA + bB = cC + dD

Rf = kf [A]a [B]b ,

Rr = kr [C]c [D]d .

(.17)

i , i , i = i + RT ln(i ), (.18)

Ar = RT ln(Kr (T )) + RT ln

n i=1 m i=1

a Aii bi Bi

.

(.19)

139

Kr (T )

Kr = kf /kr . ( 3.65 3.69)

(.20)

r Gm =j

j j = Ar ,

(.21)

r Gm = RT ln Kr .

(.22)

Ar = RT ln

Rf (r ) Rr (r )

.

(.23)

, -

1 di S 1 = V dt V

r

Ar dr =R T dt

[Rf (r ) Rr (r )] lnr

Rf (r ) Rr (r )

.

(.24)

.

d 2 S 1 2 S= = 2 dt 2

Fk Jk 0,k

(.25)

2 S Fk Jk Fk Jk .

, s(u, i ), i = 1, , r,

s(u, i )

= =

s u 1 T

r

u +i r i=1

s i

iu

(.26)

u +i=1

i i . T

(.27)

140

. -

2 s(u, ) =

2s u2

r

(u)2 + 2i i=1

2s ui

ui +ij

2s i j

i j .u

(.28) . ,

d[ 2 s(u, i )] = 2 s(u, i ) dt

= + +

2 2

2s u2r

u u i

i=1

2s ui 2s i j

( ui + u i ) i j , u

2ij

(.29)

2 s(u, i )

= +

2 2

1 u Tr

r

u u + 2 i=1

1 i T

ui j i T j i ,(.30) u

i=1

i u T

u i + 2 ij

1 T i T

=

1 u T i u T

u +i

1 i T i j T

i

(.31)

=

u +j

j ,

(.32)

2 s(u, i )

=

2

1 T

u + i

i T

i .

(.33)

2 S = 2

1 T

u + i

i T

i dV. u

(.34)

u = u =

Ju , Ju ,

(.35) (.36)

141

i i

= =

Ji +j

ij vj , ij vj .j

(.37) (.38)

Ji +

ij ith , vi . u, i .

(f J) = f J + J f, Gauss

(.39)

( f J)dV =V

f Jda,

(.40)

f JdV +V V

J f dV =

f Jda,

(.41)

1 2 S= 2 +

1 T

Ju da +V

1 T V i

Ju dV i Ji dV T(.42)

i

i Ji da T Ai T vi dV.

+V i

V da . j ij (j /T ) = (Ai /T ). .

1 2 S 2

=V

V i

1 T

Ju dV i Ji dV T Ai T vi dV 0.

(.43) (.44)

+V i

(.45)

1 2 S= 2...

Fk Jkk

(.46)

142

. -

.1

(X D ) (X L ) - (S, T). [S], [L] , k3r , k3r