ΤΡΑΠΕΖΑ ΘΕΜΑΤΩΝ Β ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ (ΟΡΙΖΟΝΤΙΑ ΒΟΛΗ-ΚΥΚΛΙΚΗ ΚΙΝΗΣΗ-ΚΙΝΗΤΙΚΗ ΘΕΩΡΙΑ-ΘΕΡΜΟΔΥΝΑΜΙΚΗ)
ΚΛΑΣΣΙΚΗ ΘΕΡΜΟΔΥΝΑΜΙΚΗ
156
ΚΛΑΣΙΚΗ ΘΕΡΜΟΔΥΝΑΜΙΚΗ: μια γεωμετρική ερμηνεία Σταύρος Κ. Φαράντος Τμήμα Χημείας, Πανεπιστήμιο Κρήτης, και Ινστιτούτο Ηλεκτρονικής Δομής και Λέιζερ, ΄Ιδρυμα Τεχνολογίας και ΄Ερευνας - Ελλάς, Ηράκλειο 711 10, ΚΡΗΤΗ http://tccc.iesl.forth.gr/education/local.html
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Transcript of ΚΛΑΣΣΙΚΗ ΘΕΡΜΟΔΥΝΑΜΙΚΗ
-
:
. , pi , , - ,
711 10, http://tccc.iesl.forth.gr/education/local.html
-
{diS
dt 0
}
-
i
1 ---- 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 . . . . . . . . . . . . . . . . . . . . . . 131.6 . . . . . . . . . . . . . . . . . . . . 14
1.6.1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6.2 : pi . 151.6.3 . . 161.6.4 . . . . . . . . . . . . . . . . 16
1.7 . . . . . . . . . . . . . 161.8 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9 LEGENDRE . . . . . . . . . . . . . . . . . . 201.10 MAXWELL . . . . . . . . . . . . . . . . . . . . . . . . 231.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.12 GIBBS-DUHEM . . . . . . . . . . . . . . . . . . . . . . . 241.13 . . . . . . . . . . . . . . . . . . . . . 241.14 . . . . . . . . . . . . . . . . . . . . . . . . . 271.15 DUHEM . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 292.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 . . . . . . . . . . . . . . . . . . . . . . . 302.4 - . . . . . . . . . . . . . . . . . . . . . . . 302.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 . . . . . . . . . . . . . . . . . . . . . . . . 322.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.8 -- . . . . . . . . . . . . 39
i
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ii
2.9 . . . . . . . . . . . . . . . . . . 442.10 . . . . . . . . . . . . . 442.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.12 . . . . . . . . . . 522.13 LEGENDRE . . . . . . . . . . . . . . . . . . 562.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.15 Gibbs-Duhem . . . . . . . . . . . . . . . . . . . . . . . . 682.16 . . . . . . . . . . . . . . . . . . . . . 692.17 . . . . . . . . . . . . . . . . . . . . . . . . . 752.18 DUHEM . . . . . . . . . . . . . . . . . . . . . . . . . . 872.19 . . . . . 88
3 913.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.1.1 . . . . . . . . . . . . . . . . . . . . . . 913.1.2 pi . . . . . . . . . . . . 913.1.3 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.1.4 . . . . . . . . . . . . . . . . . . . . . 923.1.5 -pi . . . . . . . . . . . 933.1.6 pi . . . . . . . . . . . . . 933.1.7 pi . . . . . . . . . . . . . . 943.1.8 ,
pi . . . . . . . . . . . . . . . 943.1.9 . . . . . . . . . . . . . . . . . . . . . . . 953.1.10 Clausius-Clapeyron
()- . . . . . . . . . . . . . . . . . . . . . . . 953.1.11 xi
pi . . . . . . . 963.1.12
pi . . . . . . . . . . . . . . . . . . . . 993.2 . . . . . . . . . . . . . . . 101
3.2.1 van der Waals . . . . . . . . . . . . 1013.2.2 Hess pi . . . . . . . 1023.2.3 Kirchhoff . . . . . . . . . . . . . . . . . . . . . 1023.2.4 Gibbs
pi pipi . . . . . . 1033.2.5 Gibbs-Helmholtz . . . . . . . . . . . . . . . . . . 1033.2.6 Gibbs pi pi . . 1043.2.7 Raoult . . . . . . . . . 1053.2.8 . . . . . . . . . . . . . . . . . . . . 1053.2.9 : . 1053.2.10 : . . . . . . . . . . . . 1063.2.11 : pi,
vant Hoff . . . . . . . . . . . . . . . . . . . . . . 1063.2.12 . . . . . . . . . . . . . . . . . . . . . 1063.2.13 vant Hoff: . . . . . . . . . . . . . . . . . . . . . . 107
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iii
4 109
LEGENDRE 111
113
LAGRANGE 115
117
119
121
BOLTZMANN 137
- 139.1 . . 146
149
151
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iv
-
1
- - - -
1.1
pi pi pi - pi pi .
pi pi , pi. pi - pi pi , Avogadro (NA 1023), pi pi pi -. pi pi pi () pi pi pi - pi .
pi pi pi.
pi pi pi pipi , . pi pi Avogadro. pipi pi , , . pi pi pi pi. ; pipi - pi pi pi. pi [1, 2] -
1
-
2 1. ----
pi pi pi .
pi pi pi. pi pi . pi pi , pi pi - , pi pi pi. . pi pi ( ) . pipi ( 1.1), pi, () () pi pi pi. pi pi pi pi . pi pi . - pi pi Euler.
pi pi pi pi pi , pi pi pi pi-, pi pi pi , pi-.
pi . -pi pi pi . pi- 1012 pi pi pi . pi pi pi ( / / / / ) pi. , pi, pi pi , pi . , pi pi pi pi . -pi, .
pi pi pi - pi pi ,. pi . pi pi (O) pi pi pi pi pi (T ), () pi -
-
1.1. 3
1.1: pi Ui, i = 1, 2, . . . , Vi, i = 1, 2, . . . Ni, i = 1, 2, . . . . .
1 1 1, N, V
2, V2 2, N
U3, V3, N3
i , Vi , Ni, V4 4, N4
U
U
U
U
.
O =1
T
T0
O(t)dt =< O > . (1.1)
pi pi. . -
pi pi pi . pi pi - pi pi pi pi pi,. pi pi pi . , pi . pi pi pi pi . pi pi ( - ), . pi pi pi pi pi . pi pi- () ( ).
pi, pi
-
4 1. ----
(, ) pi, pi pi ( 1.2). pi - pi pi pi pi pi pi ( pi) pi pi- pi 1. pi, pi pi pi pi, P , (P = f(V )). pi pi .
pi pi , pi . pi 1.3 pi (pi) . pi pi pi pi pi . pi pi pi pi.
pi pi- . pi pi pi, pi pi (1.2), pi, pipi , pi pi pi pi, pi.
pi pi pi pi - pi pi ; pi pi pi pi pi . pi pi pi pi pi .
pi pipi pi pi , pi. pi - . pi pi pi pi-. pi , pi pi. pi -pi ,pi . pi , pi pi , - pi. pi -pi .
pi. ;
1 pi pi () pipi (differential manifold) pi pi k- (kforms). k = 0 pi pi k = 1 .
-
1.1. 5
1.2: pi pi pi Ei pi i. (, , ) pi. pi pi pi pi pi pi. pi pi pi pi pi pi .
F(Ei)
Ei
pi pi pi . - pi pi pi . pi, pi , , pi pipi pi. pi pi . pi pi pi . pi pi .
pi ; pi. pipi pipi pi Taylor:
F (x) = F (x0) +dF (x)
dx
x=x0(x x0) + 12 d2F (x)dx2x=x0
(x x0)2 + . . .(1.2)
pi , pi pi pi pi.
.
pi ; pi pi pi pi-
-
6 1. ----
1.3: pi pi pi -. (, , ) pi. pipi pi pi pi pi.
pi pi ;
, , pi. pi -pi ; pi pi pi ;
pi E, pi , pi pi pi pi pi pi ;
pi pi pi , pi , .
pipi pi pi - pi , . pi pi. pi pi pi
-
1.2. 7
, .
pi pi pipi -, pi pi., pi pi pi .
pi :
, pi- pi pi pi - pi pi ,
, -
pi,
pi pi -pi !
pi pi , -, pi pi .pi pi pi -pi . pi pi pipi pi. pi pi pi pi .
pi pi pi- pi. pi pi pi pipi . pipi pi pi pi.
1.2
pi pi pi .
-
8 1. ----
pi pi pi .
-
1.3. 9
1.3
: d (1.3)- : [ (1.4)
(Gradient): (1.5)(Hessian): 2 (1.6)
pi (pipi): (1.7)pi (pipi): (1.8)
d~x =
[dx1dx2
](1.9)
d~xT =[dx1 dx2
](1.10)
f(x1, x2) =
fx1fx2
(1.11)f(x1, x2) =
f
x1~i+
f
x2~j (1.12)
2f(x1, x2) =
f2
x21
f2
x1x2
f2
x2x1
f2
x22
(1.13)
TAYLOR
df(x1, x2) = f(x1 + dx1, x2 + dx2) f(x1, x2) (1.14) (f)T (d~x) + 1
2(d~x)T (2f) (d~x) + . . . (1.15)
fx1
dx1 +f
x2dx2 + (1.16)
1
2
2f
x21dx21 +
1
2
2f
x22dx22 +
2f
x1x2dx1dx2 + . . .
pi
-
10 1. ----
f(x1, x2) = f(x1 + x1, x2 + x2) f(x1, x2) (1.17) (f)T (~x) + 1
2(~x)T (2f)(~x) + . . . (1.18)
fx1
x1 +f
x2x2 + (1.19)
1
2
2f
x21x21 +
1
2
2f
x22x22 +
2f
x1x2x1x2 + . . .
pi
pi pi, pipi
f(x1, x2) = f(x1 + x1, x2 + x2) f(x1, x2) (1.20) (f)T (~x) + 1
2(~x)T (2f) (~x) + . . . (1.21)
fx1
x1 +f
x2x2 + (1.22)
1
2
2f
x21x21 +
1
2
2f
x22x22 +
2f
x1x2x1x2 + . . .
pipi ( pi) -
(f(x1, x2)dx1dx2
)=
f(x1, x2)dx1dx2 (1.23)
pi pi pipi ( pi) -
(f(x1, x2)
x1
)=f(x1, x2)
x1(1.24)
(f(x1, x2)
x2
)=f(x1, x2)
x2(1.25)
-
1.4. 11
1.4
) pi pi - , - pi (N,V pipi , = N/V ).
) pi pi - . pi ( - pi), ( pi) ( pi). , + .
) pi pi pi, N1, N2, . . . , Nr, ,U , pi - , V . U . pi pi N1, N2, . . . , Nr, U V . pi p pi- , Nij, i = 1, . . . , r, Vj Uj, j = 1, . . . , p. :
pj=1
Uj = U = UT () (1.26)
pj=1
Vj = V = VT () (1.27)
pj=1
Nij = Ni = NTi , i = 1, . . . , r, (). (1.28)
pi pi , pi . pi pi-. pi pi . pi - pi pi pi .
) pi pi pi , ,
-
12 1. ----
(U,U +dU). ( ):
S(U) = kB ln . (1.29)
kB = 1, 38066 1023 JK1 Boltzmann. S(U) pi - pi, U(S). pi - ( ) pi pi , n1, n2, . . . , nr,
S(U, V, n1, n2, . . . , nr), U(S, V, n1, n2, . . . , nr). (1.30)
p =1
,
p = 1, (1.31)
pi V , ~n = (n1, n2, . . . , nr)T U ( pi) pi pi (Gibbs)
S(U, V, ~n) = kB ln ((U, V, ~n))
= kB(
1
)ln
(1
)= kB
p ln p . (1.32)
) ( ) ( pi pi ( )) ( - ). pi pi , U, S, V, n1, n2, . . . , nr. pi - (, p = 1) ( pi ). , pi (, p = 0) pi . n p- (p) ( ):
f(x1, x2, . . . , xn) =1
pf(x1, x2, . . . , xn). (1.33)
-
1.5. 13
EULER ( ):
f(x1, x2, . . . , xn) =
ni=1
(f
xi
)xj 6=i
xi, (1.34)
pipi pi
f(x1, x2, . . . , xn) =
ni=1
f
xixi. (1.35)
- pi. -.
(pi -) pi pi, . - (q) , [q = 0.
1.5
EULER pi :
U(S, V, n1, n2, . . . , nr) =
(U
S
)V,ni
S+
(U
V
)S,ni
V+
ri=1
(U
ni
)S,V,nj 6=i
ni.
(1.36) pi :
: T =
(U
S
)V,ni
(1.37)
: P =(U
V
)S,ni
(1.38)
: i =
(U
ni
)S,V,nj 6=i
. (1.39)
pi (xi, fxi ) . ,(S, T ), (V,P ), (ni, i), .
, ( ), :
U(S, V, ni) = TS PV +ri=1
ini (1.40)
-
14 1. ----
dU = TdS PdV +ri=1
idni (1.41)
pi, pi (
)
S(U, V, ni) =1
TU +
P
TV
ri=1
i
Tni (1.42)
dS =1
TdU +
P
TdV
ri=1
i
Tdni (1.43)
1.6
1.6.1 :
.
dU = 0. (1.44)
dU = [q + [w +
ri=1
idni. (1.45)
[q [w pi pi. (P, V ),
dU = [q PdV +ri=1
idni. (1.46)
pipi
U = q + w +
ri=1
ini. (1.47)
pi -, pi pi -pi. pi
-
1.6. 15
pi pi pi pi pi.
pi pi pi pi pi pi [q [w (- [). , dU dS U S pi pi pi . U S .
pi ri=1 idni
pi
dU = [q + [w (1.48)
1.6.2 : -pi
[t, t+dt] pi pi ( ) -pi. pi -pi , pi pi.
(dSTdt
)UT ,VT ,nTi
0 (1.49)
pi , ( ) pi pi -pi (unconstrained) ( ). , ST (Ul, Vl, nil;UT , VT , n
Ti ) pi
(Gradient) (ST )UT ,VT ,nTi = 0 (), ., (1.50)
ST
Ul~i+
ST
Vl~j +
ST
nil~k = 0, (1.51)
(Hessian) (2ST )UT ,VT ,nTi 0. (1.52) pi ( pi - ) pi ( )[(
2ST
U2l
) 0,
2ST
U2l
2ST
V 2l(2ST
UlVl
)2 0
](1.53)
-
16 1. ----
1.6.3
pi pi - pi pi pi (deS) -pi (diS) pi
dS = deS + diS. (1.54)
pipi pi diS
dt 0, (1.55)
pi pipi pi pi pi pi -pi. p pi pi l
diSl
dt 0, diSl 0, l = 1, . . . p (1.56)
pi pi pi pi pi
deS =dq
T=dU dw
T=dU + PdV ri=1 ideni
T. (1.57)
pi pi -pi - pi pi diS -pi.
1.6.4
pi pi (T = 0) (S = 0). pi pi pi pi pi ( = 1).
1.7 pi
(U, S, T, V, P, i, ni) (U , S, T , V , P , i, n
i) pi pi,
, pi, ST = S + S (+pi), - ((ST )UT ,VT ,nTi = 0) ,
U + U = UT (1.58)V + V = VT (1.59)ni + n
i = n
Ti , i = 1, . . . , r (1.60)
-
1.7. 17
U + U = 0 (1.61)V + V = 0 (1.62)ni + n
i = 0, i = 1, . . . , r (1.63)
T = T ( pi) (1.64)P = P ( pi) (1.65)i =
i, i = 1, . . . , r (pi ). (1.66)
pi pi - ( ) pi Lagrange ( ).
,dS = deS + diS deS. (1.67)
pi 1.57 pi pipi - ( CLAUSIUS)
TdS [q (1.68) pipi
TS q (1.69) pipi pi ,
TS = q. pi
U TS PV +ri=1
ini. (1.70)
pi
dU = TdS PdV +ri=1
idni. (1.71)
pi pi, , - pi -pi (unconstrained) , U(Sl, Vl, nil;ST , VT , nTi ) ( pipi pi-pi l, T pi )
(Gradient) (U)S,V,ni = 0 (), (1.72)
-
18 1. ----
(Hessian) (2U)S,V,ni 0 (). (1.73) U ( pi pi ) pi ( )[(
2U
S2
) 0,
2U
S22U
V 2(2U
SV
)2 0.
](1.74)
pipi(dU/dt)S,V,ni 0. (1.75)
1.68 pi . pi pipi ( ) pi !
pi 1.53 pi - pi (2S < 0) 1.74 pi (2U > 0), pi :
V kT =
(V
(P ))T,ni
0, (1.76)
CV
T=
(S
T
)V,ni
0, (1.77)(ni
i
)T,V,j
0. (1.78)
pi
CP = T
(S
T
)P,ni
, (1.79)
CP CV > 0. (1.80) pi
:
[q = CV dT. (1.81)
pi,
CV =
(U
T
)V,ni
(1.82)
:
[q = CPdT (1.83)
:
=1
V
(V
T
)P,ni
(1.84)
-
1.8. 19
:
T = 1
V
(V
P
)T,ni
(1.85)
:
S = 1
V
(V
P
)S,ni
(1.86)
1.8 pi pi - pi
pi pi pi , / pi pi, (Ueq, Veq, nieq). pi pi pi pi Taylor
S(Ueq + U, Veq + V, nieq + ni) = Seq(Ueq, Veq, nieq)
+ S(U, V, ni)
+1
22S(U, V, ni)
+ . (1.87) pi pi pi
S(U, V, ni) =
(1
T 1Teq
)U+
(P
T PeqTeq
)V
ri=1
(i
T eqiTeq
)ni.
(1.88) pi
T = Teq
P = Peq
i = eqi, i = 1, , r,(1.89)
pi S = 0. pi
2S(T, V, ni) = [CV
T 2eq
](T )2 (< 0) (1.90)
[
1
TeqVeqT
](V )2 (< 0) (1.91)
ij
[
nj
(i
Teq
)](ninj) (< 0), (1.92)
-
20 1. ----
pi (U = CV T ).pi pi S = 0 pi
S Seq = 1/22S < 0. (1.93) , pi pi -pi -pi pi pi diS = Seq S = 1/22S > 0, -. . 2S 0, pi.
pipi, pi -pi pi pi pi (2S) ,, .
pi (Seq) pi : pi pi pi Lyapunov
L(T, V, ni) =1
22S(T, V, ni) < 0, (1.94)
dL(T, V, ni)
dt=
d
dt
(2S(T, V, ni)
2
)> 0. (1.95)
-pi pi pi- pi.
1.9 Legendre
pi pipi , pi pi. pipi pi- Legendre . - pi - pi pi pi .
) (S, P, ni). .
H(S, P, ni) = U (P )V (1.96)
-
1.9. LEGENDRE 21
dH = TdS + V dP +
ri=1
idni (1.97)
(H
S
)P,ni
= T,
(H
P
)S,ni
= V
(H
ni
)S,P,nj
= i. (1.98)
pi pi ( pi -pi (unconstrained) ) pi .
(H)S,P,ni = 0 (), (1.99)(2H
S2
)P,ni
0,(2H
P 2
)S,ni
0,(2H
n2i
)S,P,nj
0.(1.100)(
dHdt
)S,P,ni
= T diSdt 0, (1.101)
CP =
(H
T
)P
, (1.102)
(dH)P = [q. (1.103)
) (T, V, ni). HELMHOLTZ.
A(T, V, ni) = U TS (1.104)
dA = SdT PdV +ri=1
idni (1.105)
(A
T
)V,ni
= S,(A
V
)T,ni
= P(A
ni
)T,V,nj
= i.
(1.106) pi HELMHOLTZ ( pi -pi ) - pi .
(A)T,V,ni = 0 (), (1.107)(2A
T 2
)V,ni
0,(2A
V 2
)T,ni
0,(2A
n2i
)T,V,nj
0. (1.108)
(dAdt
)T,V,ni
= T diSdt 0. (1.109)
-
22 1. ----
) (T, P, ni). GIBBS.
G(T, P, ni) = U TS (P )V = H TS = A+ PV(1.110)
pi Euler Gibbs pi
G(T, P, ni) =
ri=1
i(T, P )ni =
ri=1
ini (1.111)
dG = SdT + V dP +ri=1
idni (1.112)
(G
T
)P,ni
= S,(G
P
)T,ni
= V
(G
ni
)T,P,nj
= i. (1.113)
pi GIBBS - ( pi -pi ) pi .
(G)T,P,ni = 0 (), (1.114)(2G
T 2
)P,ni
0,(2G
P 2
)T,ni
0,(2G
n2i
)T,P,nj
0. (1.115)
(dGdt
)T,P,ni
= T diSdt 0. (1.116)
) (T, V, i). .
(T, V, i) = Ari=1
nii = AG = PV (1.117)
d = SdT PdV ri=1
nidi (1.118)
(
T
)V,i
= S,(
V
)T,i
= P,(
i
)T,V,j
= ni.(1.119)
-
1.10. MAXWELL 23
pi ( pi -pi ) - pi .
()T,V,i = 0 (), (1.120)(2
T 2
)V,i
0,(2
V 2
)T,i
0,(2
2i
)T,V,j
0 (1.121)
(ddt
)T,V,i
= T diSdt 0. (1.122)
1.10 Maxwell
(T
V
)S
= (P
S
)V
, (1.123)(T
P
)S
=
(V
S
)P
, (1.124)(S
V
)T
=
(P
T
)V
, (1.125)(S
P
)T
= (V
T
)P
. (1.126)
1.11
(X1, X2, . . . , Xr, Ir+1, Ir+2, . . . , Is), (1.127)
r , (X1, X2, . . . , Xr) s r , (Ir+1, Ir+2, . . . , Is) :
pi :
d =
ri=1
IidXi s
j=r+1
XjdIj. (1.128)
Maxwell
Ii
Ij= Xj
Xi, (j > r i r). (1.129)
-
24 1. ----
Xi
Ij=Xj
Ii, (i, j r). (1.130)
Ii
Xj=Ij
Xi, (i, j > r). (1.131)
pi pi pi, pi -pi
= 0. (1.132)
(convex) pi (2
X2i
)X1,...,Xi1,Xi+1,...,Xr,Ir+1,...,Is
0, (1.133)
(concave) pi (2
I2r+j+1
)X1,...,Xr,Ir+1,...,Ir+j ,Ir+j+2,...,Is
0. (1.134)
pi - (Xi, Ii):(
Ii
Xi
)X1,...,Xi1,Xi+1,...,Xr,Ir+1,...,Is
0, [(S, T ), (V,P ), (ni, i)] .(1.135)
1.12 Gibbs-Duhem
SdT V dP +ri=1
nidi = 0. (1.136)
1.13
) (V, T ), pi pi pi P = f(V, T ) - pi ,
-
1.13. 25
(i)
(CV
V
)T
= T
(2P
T 2
)V
(1.137)
(ii)
(S
V
)T
=
(P
T
)V
(1.138)
(iii)
(U
V
)T
= T
(P
T
)V
P (1.139)
(iv)
(H
V
)T
= T
(P
T
)V
+ V
(P
V
)T
(1.140)
= T
(P
T
)V
1T
(1.141)
(v) CP CV = T[(P
T
)V
]2/
(P
V
)T
(1.142)
= TTV
[(P
T
)V
]2(1.143)
= TV 2/T (1.144)
(vi)
(H
T
)V
= CV + V
(P
T
)V
(1.145)
) (P, T ), pi pi V = f(P, T ) - pi pi,
-
26 1. ----
(i)
(CP
P
)T
= T(2V
T 2
)P
(1.146)
(ii)
(S
P
)T
= (V
T
)P
(1.147)
= V (1.148)(iii)
(U
P
)T
= T(V
T
)P
P(V
P
)T
(1.149)
= TV + PV T (1.150)(iv)
(H
P
)T
= V T(V
T
)P
(1.151)
= V TV (1.152)(v) CP CV = T
[(V
T
)P
]2/
(V
P
)T
(1.153)
= TV 2/T (1.154)
(vi)
(U
T
)P
= CP P(V
T
)P
(1.155)
= CP PV (1.156)
) pi (dS = 0)
-
1.14. 27
(i)
(T
V
)S
= TCV
(P
T
)V
(1.157)
(ii)
(T
P
)S
=T
CP
(V
T
)P
(1.158)
=TV
CP(1.159)
(iii)
(V
P
)S
=CV
CP
(V
P
)T
(1.160)
=
(V
P
)T
+T
CP
[(V
T
)P
]2(1.161)
= TV +2V 2T
CP(1.162)
CP
CV=
T
S(1.163)
(iv)
(P
V
)S
=
(P
V
)T
TCV
[(P
T
)V
]2(1.164)
= 1TV
2
2T
T
CV(1.165)
1.14 (Gibbs)
pi pi C . () pi pi
F = C + 2 (1.166)F = ,C = , = .
pi , , pi, , pi , (X/ni)T,P,nj , - (ri ) i r,
ri =nrinr, nr =
Ci=1
nri , r = 1, . . . ,. (1.167)
R - pi M () , pi
F = C + 2RM (1.168)
-
28 1. ----
1.15 Duhem
pipi , , pi - pi .
-
2
2.1
pi, pi pi pi . .
2.2
pi ( pi pi ) pi pi - . pi pi Taylor pi . , pi pi pi ( 2.1), pipipi pi pi pi pi pi Hessian , pi.
(df(x1, x2)) pi f
x1x2=
f
x2x1. (2.1)
df , pi . , pi -.
df(x1, x2) = 0, (2.2)
29
-
30 2.
2.1: pi x0. pi pi pi-, pi pi x0.
f(x)
f(x
x0 x
0 )
0
L(0)=f(x0)f(x 0)x 0
f(x)
L(x)=f(x0)+f(x 0)(xx0)
f(x) = (df/dx)
[f(x1, x2) 6= 0. (2.3)
pi pi pi ( -), -pi . pipi pi - f pipi pipi f . pi pi pi Taylor.
2.3
pi pi , pi pi pi , pi pi , pi - pipi . pi [3, 4].
2.4 -
pi pi . -. pi pi pi
-
2.5. 31
pi , - , , , pi pi - . pipi pi pi . pipi pi-pi pi pi.
, + .
2.5
2.2: pi Ei, i =1, 2, . . . , Vi, i = 1, 2, . . . Ni, i = 1, 2, . . . . .
E1 1 1, N, V
E2, V2 2, N
E3, V3, N3
Ei , Vi , NiE4, V4 4, N
pi pi
pi=1
Ei = UT () (2.4)
pi=1
Vi = VT () (2.5)
pi=1
Ni = NT (). (2.6)
-
32 2.
pi pi - pi pi pi. 2.3 pi pi . . pi pi pi (pi), pi - pi. pi, pi pi (constrained) -pi(unconstrained). pi pi 2.4.
2.3: pi () pi () -pi.
() ()
2.6
pi . pi, N Ei ( pi)
U =
Ni=1
Ei. (2.7)
Ei = Ti + Vi. (2.8)
-
2.6. 33
2.4: pi () pi () -pi.
()
()
: . .
dU = 0. (2.9)
piU = 0, U = 0. (2.10)
pi. pi .
pi , pi pi - pi. - , pi pi F 2.9. z pi pi .
pi, , pi- pi pi pi- pi.
pi , pi - +pi,
dU = [q + [w. (2.11)
[q [w pi pi pi pi. pi pi pi pi pi pi [q [w ( [).
-
34 2.
pipi
U = U U = q + w. (2.12), q, w , pi pi.
, . pi -, pi - .
U = 0 = q + w, w = q. (2.13) pi pi pi pi pi , pi pi . pi pi (q = 0), pipi pi w = 0.
2.7 pi
pi pi pipi pi pi pi (-pi) .
: pi. pi
pi, pi -pi . ST pi + pi, pi
dSTdt 0. (2.14)
pi , , (U, V,N). pi .
S(U, V, N) = S(U, V,N), (2.15)( , ) (2.16)
S(U2, V,N) S(U1, V,N) U2 U1, (2.17)( , )(2.18)
S(U1 + U2, V1 + V2, N1 +N2) S(U1, V1, N1) + S(U2, V2, N2). (2.19)( , pi).
pi pi - 2.5 2.6. pi (concave)
-
2.7. 35
S (cU1 + (1 c)U2) cS(U1) + (1 c)S(U2), c [0, 1]. (2.20)
2.5: .
U 1 U 2
S
U
S(cU 2)
1S
2S
1 ) >= cS(U2 +(1c)U ) + (1c)S(U1
pi pi - pi pi pi pi pi ( 2.7,2.8).
U(S, V, N) = U(S, V,N), (2.21)( , ) (2.22)
U(S2, V,N) U(S1, V,N) S2 S1, (2.23)( , ) (2.24)
U(S1 + S2, V1 + V2, N1 +N2) U(S1, V1, N1) + U(S2, V2, N2). (2.25)( , ).
(convex) , pi (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
U
V
S
pi - pi pi pi .
pi S = S(U),
S1 = S(U1), (2.27)S2 = S(U2), (2.28)S = S(U) = cS1 + (1 c)S2, c [0, 1], (2.29)
. U pi, S1 S2.
U = U(cS1 + (1 c)S2). (2.30)
,
-
2.7. 37
2.7: .
1 +(1c)S 2]
-
38 2.
2.8: .
pi pi pi, - pi ( (1.53) ) pi. , U pi pi, pi U2 pi , pi 2U ,
S
(U +
U
2
)+ S
(U U
2
) S(2U) (2.35)
S
(U +
U
2
)+ S
(U U
2
) 2S(U) 0 (2.36)
limU0
S(U + U2
)+ S
(U U2
) 2S(U)U
=2S
U2 0. (2.37)
pi pi . () , pi pi pi pipi - .
pi pipi -, , , , pi pi pi pi pi pi, (deS), -pi
-
2.8. -- 39
(diS) pi ,
dS = deS + diS, (2.38)
pi
diS
dt 0. (2.39)
p pi pil
diSldt 0, diSl 0, l = 1, . . . p. (2.40)
pi pi pi pi pi
deS =dq
T=dU dw
T=dU + PdV ri=1 ideni
T. (2.41)
pi pi -pi - pi pi diS/dt.
2.8 --
- pi pi, . pipi EULER
U(S, V,N) =
(U
S
)V,N
S +
(U
V
)S,N
V +
(U
N
)S,V
N. (2.42)
pi pi (S, V,N), U pi pi
dU(S, V,N) =
(U
S
)V,N
dS +
(U
V
)S,N
dV +
(U
N
)S,V
dN. (2.43)
pi pipi pi pi. , pi , pi pi .
pi -- - :
-
40 2.
: T =
(U
S
)V,N
(2.44)
: P = (U
V
)S,N
(2.45)
: =
(U
N
)S,V
. (2.46)
, pi (x, f/x) .
U(S, V,N) = TS PV + N. (2.47)
2.9: pi (pi W. Craig Carter, MIT,Dept. of Materials Science and Engineering, http://prue.mit.edu/3.00/
pi pi pi ~F pi pi pi , A. pi pi pi pi dz,
-
2.8. -- 41
P =~F
A(2.48)
= (U/z)SA
(2.49)
= (U
V
)S
, (2.50)
dV = Adz. pi, PV pi, pi pi .
pi pi
S =U
T+P
TV
TN. (2.51)
S pi pi pi
dS =1
TdU +
P
TdV
TdN. (2.52)
pi
(S
U
)V,N
=1
T(2.53)(
S
V
)U,N
=P
T(2.54)(
S
N
)U,V
= T. (2.55)
pi pi ,
ST = S + S = ( pi ), (2.56)
UT = U + U = ( pi pi), (2.57)
VT = V + V = (), (2.58)
NT = N +N = (). (2.59)
pi pipi pi , pi pi- pi, , , ,
-
42 2.
2.10: pi.
U, V, N, S, T, P, M
U, V, N, S, T, P, M
(U, V,N). pi pi, - pi (1/T, P/T,/T ).
pi pi ,(U , V , N , T , P , S, ). pi pi pi
dSTdt
=
(dS
dt+dS
dt
)(2.60)
=
(dS
dU dU
dt+dS
dU
dU
dt
)(2.61)
=
(1
T 1T
)dU
dt 0. (2.62)
pi (dSTdt = 0) - pi , (T = T ). pidU /dt . pi - pi pi pi .
, pi T > T ,
-
2.8. -- 43
pi, dSTdt > 0, pi
dU
dt< 0, (2.63)
. pi pi , pi . T < T , dU
dt > 0 pi pi.
pi pi pi pi pi pi, pi P >P , .
S
V=P
T, (2.64)
dSTdt
=
(S
V dV
dt+S
V
dV
dt
)(2.65)
=
(S
V SV
)dV
dt(2.66)
=
(P
T PT
)dV
dt 0. (2.67)
(T = T ) P > P pipi dV /dt > 0, . pi- () pi ().
, pi pi pi . pi pi pipi
S
N=
T, (2.68)
dSTdt
=
(S
N dN
dt+S
N
dN
dt
), (2.69)
=
(S
N SN
)dN
dt, (2.70)
=
(
T +
T
)dN
dt 0. (2.71)
(T = T ), pi pipi = . > dN /dt < 0 pi pi pi pi .
N , pi - pi z, d(/T )dz , d(/T )dz dz, pi
-
44 2.
. pi, d(/T )dz pi z, pi . - pi.
, pi -, pi (fluctuations).
pi, pi pi -pi , , pi . pi, pi pi pi, pi - pi pi.
. pipi pi pi , - 1/T , P/T /T ., pi , . pi S(U, V,N) pi - pi pi (. 2.53). pi, pi , U(S, V,N), - pi , pi , (. 2.44), pi pi . pipi , pi pi .
2.9
, . pi - pi (T = 0) (S = 0). pi pi pi pi pi- ( = 1). pi .
2.10 pi
pi p pipi . pi pi (Sj , Vj , Nj), pi . pi U(S1, . . . , Sp, V1, . . . , Vp, N1, . . . , Np) pi - (Sj , Vj , Nj) pi pi
-
2.10. 45
F1 =
pj=1
Sj ST = 0 (2.72)
F2 =
pj=1
Vj VT = 0 (2.73)
F3 =
pj=1
Nj NT = 0. (2.74)
2.11: pi.
U
Sj
Vj
U
2.11 - pi pi pi, pi (-pi) pi pi .
, pi -pi pi. 2.12 - pi pi (pi) pi pi . pi pi -pi.
-
46 2.
2.12: pi.
pi -pi Lagrange ( ). -
G(Sj , Vj , Nj , i) = U(Sj , Vj , Nj)3i=1
iFi. (2.75)
G
Sj=U
Sj 1 = 0 (j = 1, . . . , p) (2.76)
G
Vj=U
Vj 2 = 0 (j = 1, . . . , p) (2.77)
G
Nj=
U
Nj 3 = 0 (j = 1, . . . , p). (2.78)
pi
Tj = 1 (j = 1, . . . , p) (2.79)Pj = 2 (j = 1, . . . , p) (2.80)j = 3 (j = 1, . . . , p), (2.81)
-
2.10. 47
2.13: pi pi - . pi pi , , pi pi pi, . .
U
1/T
1/T2
1
T = T 21
U
S
2
1
1
S
S(U)
2U
pi pi pi
T1 = . . . = Tp = ( pi)P1 = . . . = Pp = ( pi)1 = . . . = p = ( pi).
pi ( 2.13. +pi,
(U, S, T, V, P, ,N) (U , S, T , V , P , , N ) pi pi - pi,
U + U = UT (2.82)V + V = VT (2.83)N +N = NT , (2.84)
U + U = 0 (2.85)V + V = 0 (2.86)N + N = 0. (2.87)
pi pi
ST = S + S 0, (2.88)
-
48 2.
S + (U
T +P
T V
T N ) 0, (2.89)
S UT P
T V +
T N 0, (2.90)
T S U P V + N 0, (2.91) pi
U T S P V + N. (2.92) pi
dU = TdS PdV + dN. (2.93)
,
dU = [w + [q. (2.94)
[w = PdV + dN, (2.95)
pi , pi
TdS = [q. (2.96)
pipi TS = q. (2.97)
-pi pi + pi pipi
S + S 0. (2.98) pi T = T
pi pi, pi - dS = [q/T . pipi, pi [q = [q pipi pi
TdS [q ( CLAUSIUS), (2.99)
pipi TS q. (2.100)
2.99 pi , TdS = [q. pi pi, , -
pi -pi (unconstrained) - , U(Sl, Vl, Nl;ST , VT , NT ) ( pipi pipi- l, T pi ).
-
2.10. 49
pi pipi pi pi.
pi , U(S, V ),
pipi(Gradient) (U)S,V,N = 0 (), (2.101)
(Hessian) 2U 0 (), ., (2.102)[(2U
S2
) 0,
2U
S22U
V 2(2U
SV
)2 0]. (2.103)
pi pi pi (2U
S2
)V
=
(T
S
)V
=T
CV 0. (2.104)
2U
S22U
V 2(2U
SV
)2 0, (2.105)
T
S
(P )V
TV
(P )S
0, (2.106)
(T
S
)V
(P
V
)S
+
(T
V
)S
(P
S
)V
0. (2.107)
pipi pi (PS
)V
(TS
)V(
PV
)S
(TV
)S
0 (2.108) ( )
(P, T )
(S, V )= (P, T )
(V, S) 0. (2.109)
- (V, T )
(P, T )/(V, T )(V, S)/(V, T )
0. (2.110)
(P/V )T(S/T )V
0. (2.111)
-
50 2.
pi (kT ) pi pi pi-pi pi pi pi - pi :
V kT = (V
P
)T,N
0. (2.112)
pi pi pi , (T (S/T )V ), pi.
...
pi (N
)T,V
0. (2.113)
pi pi pi
CP = T
(S
T
)P,N
, (2.114)
CP CV > 0, (2.115) pi
:
[q = TdS = CV dT. (2.116)
pi,
CV =
(U
T
)V,N
. (2.117)
:
[q = TdS = CP dT (2.118)
:
=1
V
(V
T
)P,N
(2.119)
:
T = 1V
(V
P
)T,N
(2.120)
:
S = 1V
(V
P
)S,N
(2.121)
-
2.11. 51
2.11
pi . ( 2.14 pi N , T0, , V1, V2, pi pi pi pi, P1 P2. ( 2.14pi pi , pi pi pi pi .
pi SA = S1 + S2 , SB. pi pi pi pi pi
SB SA. pi pi pi - pi pi pi. pipi SA = SB pi pi , q = TS = 0. pi - pi
w = U =3
2kB(2N)(T T0).
T pi pi ., pi
S =3
2kBN lnT +NkB ln
(V
N
)+
SB SA = 3kBN lnT + 2NkB ln(V1 + V2
2N
)
3
2kBN lnT0 NkB ln
(V1N
) 3
2kBN lnT0 NkB ln
(V2N
)= 0.
pi
3kBN
[lnT +
2
3ln
(V1 + V2
2N
) lnT0 1
3ln
(V1N
) 1
3ln
(V2N
)]= 0
lnT
T0= ln
(4V1V2
(V1 + V2)2
)1/3T = T0
(4V1V2
(V1 + V2)2
)1/3.
-
52 2.
2.14: pi pi pi piP2 > P1. pi pi pi pi .
() ()
T0, P1, V1, n T0,P2,V2,n T,V/2,P,n T,V/2,P,n
S1 S2 S/2 S/2
A B
2.12
pi - , - pi, . pi - pi (fluctuations) pi ( ) pi pi pi ( - ). pi pi , pi - Legendre. pi Legendre pi (/) .
pi - pi pi pi -pi , (diS/dt), pipi , , , , pi pi pi pi pi . - (S,U, T, V, i, ni) pi (S, U , T , V , i, ni) pi pipi pi
-
2.12. 53
S(Ueq + U, Veq + V, neqi + ni) = Seq(Ueq, Veq, neqi)
+ S(U, V, ni)
+1
22S(U, V, ni)
+ . (2.122)
pi
S(U, V, ni) =
(1
T 1Teq
)U +
(P
T PeqTeq
)V
ri=1
(iT eqiTeq
)ni.
(2.123) pi S = 0 pi
T = Teq,
P = Peq
i = eqi, i = 1, , r,(2.124)
pi pi pi
1
22S(U, V, ni) = +
1
2
[
U
(1
T
)+
U
(1
T
)](U)2
+1
2
[
V
(P
T
)+
V
(P
T
)](V )2
12
ij
[
nj
(iT
)+
nj
(iT
)](nin
j).
pi
U
(1
T
)= 1
T 2
(T
U
)V
= 1T 2CV
,
V
(P
T
)=
1
T
(P
V
)T
= 1TV T
,
1
22S(U, V, ni) = 1
2
(U)2
CV T 2eq
(1 +
CVC V
) 1
2
1
TeqT
(V )2
Veq
(1 +
V
V
) 1
2
ij
[
nj
(iTeq
)+
nj
(iTeq
)](nin
j).
-
54 2.
pi U = CV T
1
22S(T, V, ni) = 1
2
CV (T )2
T 2eq
(1 +
CVC V
) 1
2
1
TeqT
(V )2
Veq
(1 +
V
V
) 1
2
ij
[
nj
(iTeq
)+
nj
(iTeq
)](nin
j).
pi pi
CV
-
2.12. 55
pi .
pi S = 0 pi
S Seq = 1/22S < 0. (2.132)
, pi pi pi -pi pi pi diS = SeqS = 1/22S > 0, . . 2S 0, .
(Seq) pi : pi pi pi Lyapunov
L(U, V, ni) =1
22S(U, V, ni) < 0, (2.133)
dL(U, V, ni)
dt=
d
dt
(2S(U, V, ni)
2
)> 0. (2.134)
-pi pi pi
1
2
d(2S)
dt=
V
(
1
T
) ~JudV (2.135)
V
i
(iT
) ~JidV (2.136)
+
V
i
(AiT
)vidV, (2.137)
pi Ju, Ji, vi , . Ai ith .
-
56 2.
2.13 Legendre
pi, . pi pi , ; pi , pi pi.
pi Legendre [5, 6, 7] - - pi . pi pi pi Legendre pi .
Legendre pi 2.15. y = f(x) Legendre -
L(df/dx) = f(x(df/dx)) dfdxx(df/dx). (2.138)
pi f(x) . 2.15api pi pi pipi pi. pi, pi pi pi (x, f(x)) pi pi x pi y, L, . (
(dfdx
)x, L) ( 2.15b). pi pi
(df
dx
)x
=f(x) Lx 0 , (2.139)
Legendre
L = f(x) dfdxx. (2.140)
. pi f(x) - pi . pi pi pi x df/dx, x(df/dx).
pi pi Legendre - pi pi ( 2.15c). pi , . () pi pi pi () .
-
2.13. LEGENDRE 57
2.15: Legendre.
L(f)
f=df/dx
L=f(x)fxf=df/dx
a b c
f(x1)f=df/dx
x2 x2x1
f(x)
Lm
xmxmx10 0
L2L2
Lm
L1L1
fxx
f(x)
2.16: () .
f(x)
x1 x1
f(x)
x
f(x1) + f(x1)(xx1)
f(x1) + f(x1)(xx1)
f(x) >= f(x1) + f(xx1)f(x)
-
58 2.
Legendre pi - y. , x pi pi cy1 + (1 c)y2,
L[cy1 + (1 c)y2] = f(x) x[cy1 + (1 c)y2] (2.144)= f(x) + cf(x) cf(x)
x[cy1 + (1 c)y2] (2.145)= c[f(x) xy1] + (1 c)[f(x) xy2] (2.146) c[f(x1) x1y1] + (1 c)[f(x2) x2y2] (2.147)= cL(y1) + (1 c)L(y2). (2.148)
pi f(x, z) Legendre pi x pi z, - L(y, z) pi pi z. ,
L(y, [cz1 + (1 c)z2]) = f(x, [cz1 + (1 c)z2]) yx (2.149) cf(x, z1) + (1 c)f(x, z2)
cyx (1 c)yx (2.150)= c[f(x, z1) yx] +
(1 c)[f(x, z2) yx] (2.151)= cL(y, z1) + (1 c)L(y, z2). (2.152)
...
pi pi f(x) = ln(x)., f = df/dx = 1/x Legendre
L(f ) = f(x) xf = ln(1/f ) 1 = [1 + ln(f )]. L(f ) pi f .
pi pi . pipi pi pipipi Legendre.
pi pi [6] [5], - 5, 131. pi -pi Legendre Stephen Boyd and Lieven Vandenberghe [8], Convex Optimization "http://www.stanford.edu/ boyd/cvxbook.html".
Legendre
-
2.13. LEGENDRE 59
1. (S, P,N). pi.
H(S, P,N) = U (P )V, (2.153)
dH = TdS + V dP + dN, (2.154)(H
S
)P,N
= T,
(H
P
)S,N
= V,
(H
N
)S,P
= . (2.155)
Maxwell(T
P
)S,N
=
(V
S
)P,N
. (2.156)
(T
N
)S,P
=
(
S
)P,N
. (2.157)
(V
N
)S,P
=
(
P
)S,N
. (2.158)
pi pipi .
H(S, P,N) = U + PV, (2.159)
dH = dU+d(PV ) = dU+(PdV+V dP ) = (TdSPdV+dN)+(PdV+V dP ),(2.160)
dH = TdS + V dP + dN. (2.161)
pi pi
dH =
(H
S
)P,N
dS +
(H
P
)S,N
dP +
(H
N
)S,P
dN. (2.162)
(H
S
)P,N
= T,
(H
P
)S,N
= V,
(H
N
)S,P
= . (2.163)
-
60 2.
pi dH
2H
SP=
2H
PS, (2.164)
(T
P
)S,N
=
(V
S
)P,N
. (2.165)
2H
PN=
2H
NP, (2.166)
(V
N
)S,P
=
(
P
)S,N
. (2.167)
2H
SN=
2H
NS, (2.168)
(T
N
)S,P
=
(
S
)P,N
. (2.169)
...
dU = [q, (P =) dH = [q, PV .
dU = [q PdV = [q. (2.170)
H = U + PV (2.171)dH = dU + d(PV ) (2.172)
= dU + PdV + V dP (2.173)= dU + PdV (2.174)= [q. (2.175)
-
2.13. LEGENDRE 61
...
Joule-Thomson pi Joule-Thomson pi.
2.17: Joule-Thomson .
P1, V1, T 1, U1
P2, V2, T 2, U2
(q = 0) pi (P1, P2). pi, pi pi pi
U = U2 U1 = w = P2(V2 0) P1(0 V1) = P1V1 P2V2, (2.176)
U2 + P2V2 = U1 + P1V1, (2.177)
H2 = H1. (2.178)
Joule-Thomson
JT =
(T
P
)H
. (2.179)
-
62 2.
(T
P
)H
=(T,H)
(P,H)(2.180)
=(T,H)/(P, T )
(P,H)/(P, T )(2.181)
= (H/P )T(H/T )P
(2.182)
=T(VT
)P V
CP(2.183)
=V (T 1)
CP. (2.184)
pi pi (S
P
)H
= (H/P )S(H/S)P
= VT< 0. (2.185)
...
2. (T, V,N). Helmholtz.
A(T, V,N) = U TS, (2.186)
dA = SdT PdV + dN, (2.187)(A
T
)V,N
= S,(A
V
)T,N
= P,(A
N
)T,V
= (2.188)
Maxwell (S
V
)T,N
=
(P
T
)V,N
, etc. (2.189)
pi pipi .
-
2.13. LEGENDRE 63
(dV = 0) (T =) Clausius pi A.,
TdS [q = dU + PdV, (2.190)dU TdS 0, (2.191)
d(U TS) = dA 0. (2.192) Helmholtz -pi pi- pi. (
dA
dt
)T,V,ni
= T diSdt 0. (2.193)
3. (T, P,N). Gibbs.
G(T, P,N) = (T, P )N = N (2.194)
dG = SdT + V dP + dN, (2.195)(G
T
)P,N
= S,(G
P
)T,N
= V,
(G
N
)T,P
= . (2.196)
Gibbs .
Maxwell (S
P
)T,N
= (V
T
)P,N
, etc. (2.197)
pi pipi .
-
64 2.
(P =) (T =) - Clausius pi GibbsG.,
TdS [q = dU + PdV, (2.198)dU TdS + PdV 0, (2.199)
d(U TS + PV ) = dG 0. (2.200) pi Gibbs -pi pi- pi. (
dG
dt
)T,P,ni
= T diSdt 0. (2.201)
pipi PV , pi.. ,
[q = dU + PdV dwe. (2.202)
pi pipi pi
d(U TS + PV ) = dG +dwe. (2.203)
pi pi pi dwe = dwe.
dwe dG. (2.204) -PV pi pi pi Gibbs.
4. (T, V, ). .
(T, V, ) = AN = AG = PV, (2.205)d = SdT PdV Nd, (2.206)(
T
)V,
= S,(
V
)T,
= P,(
)T,V
= N. (2.207)
-
2.14. 65
pi pipi .
pi pi Legendre pi- pi pi pi .
2.14
pi, . pi pi pi . pi pi :
U(S, V,N, z) = U0(S, V,N) +mgz, (2.208)
pi m , g pi pi , z pi pi pi . U0 pi., pipi pi
-
66 2.
1. fdl : () x (pi)
2. pi pi d : ( ) x (pi)
3. piAdQ : ( ) x ()
4. pi piBdI : ( pi) x ( pi)
5. ...
pi .
(X1, X2, . . . , Xr, Ir+1, Ir+2, . . . , Is), (2.209)
r , (X1, X2, . . . , Xr) s r - , (Ir+1, Ir+2, . . . , Is) :
pi :
d =
ri=1
IidXi s
j=r+1
XjdIj . (2.210)
Maxwell
IiIj
= XjXi
, (j > r i r). (2.211)
XiIj
=XjIi
, (i, j r). (2.212)
IiXj
=IjXi
, (i, j > r). (2.213)
-
2.14. 67
pi pi pi, pi -pi
= 0. (2.214)
(convex) pi (2
X2i
)X1,...,Xi1,Xi+1,...,Xr,Ir+1,...,Is
0, (2.215)
(concave) pi (2
I2r+j+1
)X1,...,Xr,Ir+1,...,Ir+j ,Ir+j+2,...,Is
0. (2.216)
pi - (Xi, Ii):(
IiXi
)X1,...,Xi1,Xi+1,...,Xr,Ir+1,...,Is
0, [(S, T ), (V,P ), (ni, i)] . (2.217)
pi - pi:
pi pi (pi-)pipi () pi pi , pi pi (pi-)pipi() pi pi pi.
pi pi pi pipi ( (1.53) ). pi pi ( (1.74) ).
pi pi pi pi - ( pi ), () . .
-
68 2.
2.15 Gibbs-Duhem
, , pi pipi - (S, V,Ni) (T,P, i) . ,pi U = TS PV +i iNi,
(T, P, i) 0 = U TS + PV i
iNi. (2.218)
d(T, P, i) = 0 = dU TdS SdT + PdV + V dP i
idNi i
Nidi.
(2.219)pi
dU = TdS PdV +i
idNi.
SdT V dP +
i
Nidi = 0. (2.220)
Gibbs-Duhem. , pi -
. 2.220 pi pi pipi. pi pi pi- pi pi, , pi,pi, , .
,V (N1, N2, . . . , ) = V (N1, N2, . . . ).
Euler pi
V =i
Ni
(V
Ni
)T,P,Nj 6=i
=i
Nivi, (2.221)
vi . . 2.221
(Nivi).
dV =i
dNi
(V
Ni
)T,P,Nj 6=i
=i
dNivi. (2.222)
pi
dV =i
dNivi +i
Nidvi. (2.223)
-
2.16. 69
pi (2.222, 2.223), pi i
Nidvi = 0. (2.224)
vi
dvi =k
(viNk
)T,P
dNk. (2.225)
pi . 2.224
i
Ni
(k
(viNk
)T,P
dNk
)=k
(i
Ni
(vkNi
)T,P
)dNk = 0. (2.226)
pi
viNk
=2V
NkNi=
2V
NiNk=vkNi
. (2.227)
pi dNk pi -pi
i
Ni
(vkNi
)T,P
= 0. (2.228)
pi Gibbs (pi pi ). pi pi pi 2.228
G =k
Nk
(G
Nk
)T,P
=k
Nkk,i
Ni
(kNi
)T,P
= 0. (2.229)
pi Helmholtz (ak) pi (hk)
A =k
Nk
(A
Nk
)T,P
=k
Nkak,i
Ni
(akNi
)T,P
= 0. (2.230)
H =k
Nk
(H
Nk
)T,P
=k
Nkhk,i
Ni
(hkNi
)T,P
= 0. (2.231)
2.16
pi pi pi - (w, q, T, P, V, CV , CP , , T , S ) pi pi -pi (S,U,A,G).
-
70 2.
1. (V, T ), pi pi pi P = f(V, T ) - pi ,
(i)
(CVV
)T
= T
(2P
T 2
)V
(2.232)
(ii)
(S
V
)T
=
(P
T
)V
(2.233)
(iii)
(U
V
)T
= T
(P
T
)V
P (2.234)
(iv)
(H
V
)T
= T
(P
T
)V
+ V
(P
V
)T
(2.235)
= T
(P
T
)V
1T
(2.236)
(v) CP CV = T[(
P
T
)V
]2/
(P
V
)T
(2.237)
= TTV
[(P
T
)V
]2(2.238)
= TV 2/T (2.239)
(vi)
(H
T
)V
= CV + V
(P
T
)V
(2.240)
(i)
dA = SdT PdV. (2.241)
S = (A
T
)V
, P = (A
V
)T
. (2.242)
CV = T
(S
T
)V
(2.243)
= T 2A
T 2. (2.244)
-
2.16. 71
(CVV
)T
= T 3A
T 2V(2.245)
= T 2
T 2
(A
V
)T
(2.246)
= T
(2P
T 2
)V
. (2.247)
(ii)
dA = SdT PdV. (2.248)
Maxwell (S
V
)T
=
(P
T
)V
. (2.249)
(iii)
dU = TdS PdV. (2.250)
(U
V
)T
= T
(S
V
)T
P (2.251)
= T
(P
T
)V
P. (2.252)
(iv)
dH = TdS + V dP. (2.253)
(H
V
)T
= T
(S
V
)T
+ V
(P
V
)T
(2.254)
= T
(P
T
)V
+ V
(P
V
)T
. (2.255)
-
72 2.
(v)
CV = T
(S
T
)V
(2.256)
= T(S, V )
(T, V )(2.257)
= T(S, V )/(T, P )
(T, V )/(T, P )(2.258)
= T
(ST
)P
(VP
)T ( SP )T (VT )P(VP
)T
(2.259)
= T
(S
T
)P
+ T
[(V
T
)P
]2/
(V
P
)T
(2.260)
= CP + T2V 2/(V kT ). (2.261)
CP CV = 2
kTTV. (2.262)
...
2. (P, T ), pi pi V = f(P, T ) - pi pi,
-
2.16. 73
(i)
(CPP
)T
= T(2V
T 2
)P
(2.263)
(ii)
(S
P
)T
= (V
T
)P
(2.264)
= V (2.265)(iii)
(U
P
)T
= T(V
T
)P
P(V
P
)T
(2.266)
= TV + PV T (2.267)(iv)
(H
P
)T
= V T(V
T
)P
(2.268)
= V TV (2.269)
(v) CP CV = T[(
V
T
)P
]2/
(V
P
)T
(2.270)
= TV 2/T (2.271)
(vi)
(U
T
)P
= CP P(V
T
)P
(2.272)
= CP PV (2.273)
3. pi (dS = 0)
-
74 2.
(i)
(T
V
)S
= TCV
(P
T
)V
(2.274)
(ii)
(T
P
)S
=T
CP
(V
T
)P
(2.275)
=TV
CP(2.276)
(iii)
(V
P
)S
=CVCP
(V
P
)T
(2.277)
=
(V
P
)T
+T
CP
[(V
T
)P
]2(2.278)
= TV + 2V 2T
CP(2.279)
CPCV
=TS
(2.280)
(iv)
(P
V
)S
=
(P
V
)T
TCV
[(P
T
)V
]2(2.281)
= 1TV
2
2T
T
CV(2.282)
(i) (T
V
)S
= (S/V )T(S/T )V
(2.283)
= TCV
(P
T
)V
. (2.284)
(iii) (V
P
)S
=(V, S)
(P, S)(2.285)
= T(V, S)/(V, T )
(P, S)/(P, T )
(V, T )
(P, T )(2.286)
=
(ST
)V(
ST
)P
(V
P
)T
(2.287)
=CVCP
(V
P
)T
(2.288)
-
2.17. 75
...
2.17 (Gibbs)
F = C + 2. (2.289)F = ,C = , = .
- Gibbs
G = n, (2.290)
pi ( ). pi , pi.. pi,
Gm =G
n= (T, P ). (2.291)
piGm . - C - pi ()
G =
Ci=1
ini. (2.292)
pipi - n
n =
Ci=1
ni, (2.293)
Gm =G
n=
Ci=1
nini, (2.294)
-
76 2.
i =nin. (2.295)
Gm =
Ci=1
ii (2.296)
Ci=1
i = 1. (2.297)
pi pipi pi pi (C 1) + 2 - , (C 1) , pi.
pi pi pi pi pi, - pi pi . , pi pipi - .
C
G =
r=1
Gr, (2.298)
Gr =
Ci=1
rinri . (2.299)
pi pipi
nr =
Ci=1
nri , r = 1, . . . ,, (2.300)
Grm =Gr
nr=
Ci=1
nrinrri , r = 1, . . . ,. (2.301)
i r
ri =nrinr. (2.302)
r
Grm =Gr
nr=
Ci=1
riri , r = 1, . . . ,, (2.303)
-
2.17. 77
Ci=1
ri = 1, r = 1, . . . ,. (2.304)
G ( ) pi (C1)+2 pi . ipi pi pi
1i = 2i = = i = i, (2.305)
( 1) . C C( 1) - pi pi pi C(1) - pipi.
. Gibbs pi pi pi pi pi C pi
F = (C 1) + 2 C( 1) = C + 2, (2.306)
. , pipi pi . pi pipi, pi- pi, . pi, C pipi pi - (R), pi pi (M ).
F = C + 2RM. (2.307)
pi pi Gibbs (T, P, 1, 2, . . . , C), (2 + C) . pi pi - , -pi. Gibbs-Duhem pi pipi pi (pipi). F = (2 + C) = C + 2.
pi pi.
1) : F = 2 ( pi ).
2) : F = 1 ( 2.18, 2.19).
-
78 2.
2.18: pi pi pi .
1
2
(T, P)
T
P
1 = 2
dp(t)/dt
pi pi pi pi . (T, P ) pi pi pi pi.
pi ; pi Clausius-Clapeyron pi pi (T, P ).
pi pi 1 2
1(T, P (T )) = 2(T, P (T )), (2.308)
pi1
T+1
P
dP
dT=2
T+2
P
dP
dT. (2.309)
d = SmdT + VmdP, (2.310)Sm Vm pi -. pi pi
S1m + V 1mdP
dT= S2m + V 2m
dP
dT,(2.311)
-
2.17. 79
2.19: .
TTc
Clausius-Clapeyron.
dP
dT=S2m S1mV 2m V 1m
=SmVm
, (2.312)
pi -pi Sm = q/T.
dP
dT=
q
TVm. (2.313)
pi (pi.. -) pi pi , - pi (T, S) (T, V ), ( 2.20).
pi (T, P ) (T, V ) pi- 2.21.
3) pi ( = 3), F = 0. , pi - (Tc, Pc) pi pi . (Tc, Pc) pi .
-
80 2.
2.20: pi - Tc pi.
S
T
V
TTc Tc
4) x1 x2, (x1 + x2 =1) F = 3 . pi , , pi , pi.. x1.
5) , F = 2 . - pi , x1A x1B.
8 9 Atkins [9] .
-
2.17. 81
2.21: pi -. pi pi.
P
T
2
V
T
1
2
1,2
1
T
P
T
V
1
22,3
3 1,3
1
21,21+2
1+3
2+3
1+2
P = P1
P1P23
-
82 2.
2.22: pi . CV < 0, CV =
(UT
)V
= ( SU )2V /( 2SU2)V . pi d ( 1T ) =dTT 2
U
S 2
1
1+2
-
2.17. 83
2.23: pi . ( Legen-dre pi LS(T, Vm) = S(U, Vm) U/T =Am/T . , S(U, Vm) pi U pi , LS(T, Vm) - pi pi . Helmholtz, Am(T, Vm) = TLS , pi T pi Vm .)
Vm
Am
1
2
1+2
2.24: pi . ( Le-gendre pi LS(T, P ) = S(U, Vm) U/T PVm/T = Gm/T . S(U, Vm) pi U Vm , LS(T, P ) pi pi. Gibbs, Gm(T, P ) = TLS , pi T P .)
V V P
GmGm
P = = =
1
2
2 1
1+2
1
21+2
III
1+2
-
84 2.
2.25: pi . ( pi -pi.)
Vm
GmP
T
Tc2 Tc1
1
2
1
21
2
TcrTcr1 Tcr1
Tcr
Tc2
Tc1
-
2.17. 85
2.26: pi . ( pi pi (bifurcations) - pi (pitchfork) - (saddle-node).)
Vm Vm
GmT
T1 < Tc
Vm
TcT2 > Tc
P
2 2 1 1
1+2 1+2 1+2
TcTcr
2
1
2
T T
T1 < Tc T2 > Tc
1
2
Tcr
1
TcT1T2
T < Tc
T > Tc
Vm
-
86 2.
2.18 DUHEM
pipi , , pi (N0i ) pi .
Duhem pi Gibbs Duhem .
, C N0i - pi , i = 1, . . . , C, - pi (2 ) C pi pi ,nri , pi , r = 1, . . . ,. pipi C
r=1
nri = N0i , i = 1, . . . , C (2.314)
pipi pi pipi
1i (T, P ) = 2i (T, P ) = = i (T, P ), i = 1, . . . , C (2.315)
C( 1) .pi
F = C + 2 C( 1) C = 2. pi pi pipi pi (a) pi pipi , pi pi a.
-
2.19. 87
2.19 -
-
88 2.
2.1: . (U ) - , pi (H ) - , Helmholtz (A) - , Gibbs (G) - , -
U H
U(S, V,N) = TS + (P )V + N H(S, P,N) = U (P )VdU(S, V,N) = TdS + (P )dV + dN dH(S, P,N) = d(U + PV )dU(S, V,N) = TdS PdV + dN dH(S, P,N) = TdS + V dP + dN
(US
)V,N
= T(HS
)P,N
= T(UV
)S,N
= P (HP
)S,N
= V(UN
)S,V
= (HN
)S,P
=
(TV
)S,N
= (PS
)V,N
(TP
)S,N
=(VS
)P,N(
TN
)S,V
=(S
)V,N
(TN
)S,P
=(S
)P,N
( PN
)S,V
=(V
)S,N
(VN
)S,P
=(P
)S,N
A G
A(T, V,N) = U TS G(T, P,N) = U TS (P )VdA(T, V,N) = d(U TS) dG(T, P,N) = d(U TS + PV )dA(T, V,N) = SdT PdV + dN dG(T, P,N) = SdT + V dP + dN
(AT
)V,N
= S (GT
)P,N
= S(AV
)T,N
= P (GP
)T,N
= V(AN
)T,V
= (GN
)T,P
=
(SV
)T,N
=(PT
)V,N
-(SP
)T,N
=(VT
)P,N
( SN
)T,V
=(T
)V,N
-(SN
)T,P
=(T
)P,N
( PN
)T,V
=(V
)T,N
(VN
)T,P
=(P
)T,N
-
3
3.1
PV = nRT, ( ) (3.1)
3.1.1
U(T, n) =3
2nRT. (3.2)
3.1.2 pi
w = nRT ln(Vf/Vi). (3.3)
89
-
90 3.
3.1.3 pi
S = CV lnT + nR ln
(V
n
)+ . (3.4)
S = CP lnT nR lnP + . (3.5)
3.1.4
CV =3
2nR. (3.6)
CP =5
2nR. (3.7)
-
3.1. 91
CP CV = nR. (3.8)
3.1.5 -pi
T = Pex.V/CV . (3.9)
3.1.6 pi
PV = . (3.10)
= CP /CV . (3.11)
-
92 3.
w = CV Ti[(Vi/Vf )1 1]. (3.12)
3.1.7 pi
S = nR ln(Vf/Vi). (3.13)
3.1.8 , pi
G(P ) = G(P ) + nRT ln(P/P ). (3.14)
-
3.1. 93
G(P ) = G(P ) + PP V dP (3.15)
= G(P ) + PP
nRT
PdP (3.16)
= G(P ) + nRT PP
dP
P(3.17)
= G(P ) + nRT ln(P/P ). (3.18)
3.1.9
(P ) = (P ) +RT ln(P/P ). (3.19)
pi pipi -
.G(P ) = G(P ) + nRT ln(P/P ). (3.20)
G(P )
n=G(P )n
+RT ln(P/P ). (3.21)
3.1.10 Clausius-Clapeyron ()-
d lnP/dT = Hm,./RT2. (3.22)
-
94 3.
dP
dT= q/(TVm) (3.23)
= Hm,.P/RT2 (3.24)
dP
PdT= Hm,./RT
2 (3.25)
d lnP
dT= Hm,./RT
2. (3.26)
P = P exp
[
Hm,.R
(1
T 1T
)]. (3.27)
d lnP = Hm,.dT/RT2. (3.28)
pi (T , P ) (T, P ) pi -pi pi
ln(P/P ) = Hm,.
R
(1
T 1T
). (3.29)
pi .
3.1.11 xi pi
Gmix = nRT [xA lnxA + xB lnxB ] . (3.30)
-
3.1. 95
pi P ,
Gi = nAAi + nB
Bi (3.31)
= nA
[A +RT ln
(P
P
)]+ nB
[B +RT ln
(P
P
)]. (3.32)
pi PA PB,
Gf = nAAf + nB
Bf (3.33)
= nA
[A +RT ln
(PAP
)]+ nB
[B +RT ln
(PBP
)]. (3.34)
Gmix = Gf Gi = nA[RT ln
(PAP
)]+ nB
[RT ln
(PBP
)]. (3.35)
pi pi
PA = AP, PB = BP. (3.36)
Gmix = nART lnA + nBRT lnB (3.37)= nRT [
nAn
lnA +nBn
lnB ] (3.38)
= nRT [A lnA + B lnB ], (3.39)
pi n = nA + nB .
Smix = nR [xA lnxA + xB lnxB ] . (3.40)
G
T= S. (3.41)
-
96 3.
G
T= S. (3.42)
pi pipi pi
Smix = nR [xA lnxA + xB lnxB ] . (3.43)
Smix = Gmix/T. (3.44)
Umix = Hmix = Vmix = 0. (3.45)
Gmix = Hmix TSmix. (3.46)
Hmix = 0. (3.47)
G
P= V. (3.48)
G
P= V. (3.49)
GmixP
= 0 = Vmix. (3.50)
piG = U TS + PV. (3.51)
Umix = 0. (3.52)
1. pi pi pi, - .
Gmix = TSmix. (3.53)
-
3.1. 97
2. pi pi pi pi pi pipi pi .
3. pipi ;
3.1.12 pi
: 0 =J
JJ. (3.54)
rGm =j
jj . (3.55)
pi pi
pi pipi pi
nAA
=nBB
=nCC
= = , (3.56)
, pi pi - pi pi - pi J . J pi .
pi
-
98 3.
rG =j
njj (3.57)
rG =j
jj (3.58)
rG/ =j
jj (3.59)
rGm = rG/ (3.60)=
j
jj . (3.61)
pi
rGm = 0. (3.62)
pi,
aA+ bB pP + qQ, (3.63)
pi
pP + qQ = aA + bB . (3.64)
rGm =
j
jj . (3.65)
rGm =
j
jfGm,j , (3.66)
pi fGm,j j.
pi
pi pipi .
-
3.2. 99
rGm = rGm +RT lnQ. (3.67)
Q =j
(PjP
)j. (3.68)
rGm = RT lnK. (3.69)
K =j
(PjP
)jpi. (3.70)
3.2
3.2.1 van der Waals
P =nRT
V nb a( nV
)2. (3.71)
-
100 3.
3.2.2 Hess pi
: 0 =J
JJ. (3.72)
rH =
J
JfHJ . (3.73)
3.2.3 Kirchhoff
rH(T2) = rH(T1) +
T2T1
rCP (T )dT. (3.74)
rCP =J
JCP,J . (3.75)
-
3.2. 101
3.2.4 Gibbs pi pipi
rG =
J
JfGJ . (3.76)
rS =
J
JSJ . (3.77)
3.2.5 Gibbs-Helmholtz((G/T )
T
)P
= HT 2. (3.78)
G = H TS. (3.79)G
T=H
T S. (3.80)
-
102 3.
T
(G
T
)P
=
[(H
T
)P
T H]/T 2
(S
T
)P
(3.81)
=CPT HT 2 CP
T(3.82)
= HT 2. (3.83)
pi ((G/T )
T
)P
= HT 2
. (3.84)
3.2.6 Gibbs pi pi
G(P ) = G(P ) + PP V dP. (3.85)
(G
P
)T,n
= V. (3.86)
dG(P ) = V dP, (3.87)
G(P ) = G(P ) + PP V dP. (3.88)
-
3.2. 103
3.2.7 Raoult
PA = xAPA. (3.89)
pi .xA A .PA pi A .P A .
A(l) = A(l) +RT ln(xA). (3.90)
3.2.8
: A = AxA. (3.91)
A(l) = A(l) +RT ln(A). (3.92)
3.2.9 :
T =
(RT 2
Hb,m
)xB (3.93)
-
104 3.
3.2.10 :
ln(xB) = Hf,mR
(1
T 1T
)(3.94)
3.2.11 : pi, vant Hoff
V = nBRT. (3.95)
3.2.12
nl/ng = d/d. (3.96)
-
3.2. 105
V = nv = nxlvl + nxgvg (3.97)
v = xlvl + (1 xl)vg (3.98)xl = (vg v)/(vg vl) (3.99)xg = (v vl)/(vg vl) (3.100)xl/xg = (vg v)/(v vl) (3.101)
d = vg vd = v vlnl/ng = d
/d (3.102)
3.2.13 vant Hoff:
d lnK
dT=
rH
RT 2. (3.103)
-
106 3.
-
4
107
-
108 4.
-
LEGENDRE
G(q1, . . . , qs) s , pi , L, pi qi, i = 1, . . . , r uj = G/qj , j =r + 1, . . . , s. pi Legendre
L(q1, . . . , qr, ur+1, . . . , us) = G(~q)s
i=r+1
qiui. (.1)
pi,
dG =
si=1
G
qidqi =
si=1
uidqi, (.2)
dL = dGs
j=r+1
(qjduj + ujdqj) (.3)
=
ri=1
uidqi s
j=r+1
qjduj .
Maxwell pi
dG =
si=1
uidqi. (.4)
pi pi G pi
2G
qiqj=
2G
qjqi(.5)
109
-
110 . LEGENDRE
uiqj
= uij = uji =ujqi
. (.6)
Maxwell.
-
p pi
L(q1, . . . , qs) = pL(q1, . . . , qs). (.1)
pi pi pi-
pp1L(q1, . . . , qs) =si=1
L
(qi)qi =
1
si=1
L
qiqi. (.2)
pi,
ppL(q1, . . . , qs) = pL(q1, . . . , qs). (.3)
, pi = 1
pL(q1, . . . , qs) =
si=1
L
qiqi. (.4)
Euler.
111
-
112 .
-
LAGRANGE
L(q1, . . . , qs) s , pi- pi m (s > m)
Fi(q1, . . . , qs) = 0, i = 1, . . . ,m, (.1)
G(~q, ~) = L(q1, . . . , qs)mi=1
iFi(q1, . . . , qs), (.2)
.,
G(~q, ~) = Lmi=1
iFi = 0. (.3)
i m pi pi pi (.1) (.3).
113
-
114 . LAGRANGE
-
pi pi pi pi pi pi . -
f(x1, x2) = fx1
~i+f
x2~j = 0. (.1)
f
x1= 0,
f
x2= 0. (.2)
, pipi Hessian. , / /, ( ) . pi
|2f(x1, x2) I| =
f2
x21 f2x1x2
f2
x2x1
f2
x22
= 0, (.3)pi I pi.
pi
2 + p+ q = 0, (.4)
115
-
116 .
pip =
f2
x21+f2
x22, (.5)
q =f2
x21
f2
x22(
f2
x1x2
)2. (.6)
pi1 + 2 = p, (.7)
12 = q, (.8)
pi () q > 0, p > 0 () q > 0, p < 0. q < 0.
-
(JACOBIANS)
u = u(x, y), v = v(x, y). (.1)
(u, v)
(x, y)=
(ux
)y
(uy
)x(
vx
)y
(vy
)x
(.2)=
(u
x
)y
(v
y
)x
(u
y
)x
(v
x
)y
(.3)
dudv =(u, v)
(x, y)dxdy (.4)
(u, v)
(x, y)= (u, v)
(y, x)(.5)
= (v, u)(x, y)
(.6)
= +(v, u)
(y, x)(.7)
pi
117
-
118 .
(u
x
)y
=(u, y)
(x, y)(.8)(
u
y
)x
=(u, x)
(y, x)= (u, x)
(x, y)(.9)(
v
y
)x
=(v, x)
(y, x)= (v, x)
(x, y)(.10)(
v
x
)y
=(v, y)
(x, y)(.11)
pi x(y, z)1) (
x
y
)z
(y
x
)z
=(x, z)
(y, z)
(y, z)
(x, z)= +1. (.12)
2)
(x
y
)z
(y
z
)x
(z
x
)y
=(x, z)
(y, z).(y, x)
(z, x).(z, y)
(x, y)(.13)
=(x, z)
(y, z).(x, y)(x, z) .
(y, z)(x, y)
(.14)
= 1 (.15)
3) pi
(x
y
)z
=(x, z)
(w, z).(w, z)
(y, z)(.16)
=(x/w)z(y/w)z
(.17)
-
pi pi pi -pi , pi.. pi , , pi , pi , ... - pi pi . pi pi, O,pi pi pi , pi pi .
pi pi :
1. : (U, V,N) ( )
2. : (T, V,N) ( )
3. : (T, V, ) ( )
4. - (T, P,N)
p pi - , pi pi O
O < O >=n=1
pO , (.1)
pi n , pi p pi pi
n=1
p = 1. (.2)
119
-
120 .
pi pi pi pi
(O)2 < (O)2 > = < (O < O >)2 > (.3)= < O2 > (< O >)2. (.4)
pi p
S = kB
p ln p . (.5)
pi pi pi . - pi pi pi p pi pi pipi Lagrange.
F = S (n=1
p 1), (.6)
pipi S. pi pi pi pi pi pi. pi
F
p= 0 (.7)
kB(ln p + 1) = 0 (.8)kB ln p = (+ kB) (.9)
p = exp[(+ kB)/kB ] (.10)
pi. pi pi pi pi
p =1
(U, V,N), (.11)
pi (U, V,N) pi - U , V N .
pi E , V N ; pi pi pi pi pi, pi+pi
ptotal = penvp , (.12)
. pi - pi pi pi
-
121
pi. ptotal pi- penv pi pi.
p = ptotal/penv. (.13) pi ( pi) - pi
p(E , V , N) =env[(Etotal E), (Vtotal V), (Ntotal N)]
total[Etotal, Vtotal, Ntotal]. (.14)
pi pi
p(E , V , N) =exp
[1kBSenv[(Etotal E), (Vtotal V), (Ntotal N)]
]exp
[1kBStotal[Etotal, Vtotal, Ntotal]
] .(.15)
pi pi pi (S) pi
Stotal(Etotal, Vtotal, Ntotal) = Senv(EtotalU, VtotalV,NtotalN)+S(U, V,N).(.16)
U , V , N , - pi . pi pipi pi pi Taylor pi , 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
(EtotalE) |EtotalU (U E) +
Senv(VtotalV) |VtotalV (V V) +
Senv(NtotalN) |NtotalN (N N) =
Senv[(Etotal U), (Vtotal V ), (Ntotal N)] + SE |U (U E) + SV |V (V V) + SN |N (N N) =
Senv[(Etotal U), (Vtotal V ), (Ntotal N)] + 1T (U E) + PT (V V) T (N N)pi pi
p(E , V , N) =exp
[1kB
[( 1T (U E) + PT (V V) T (N N)]]
exp[
1kBS(U, V,N)
] (.17)
-
122 .
p(E , V , N) = exp
[1
kBT(U TS + PV N)
]exp
[ 1kBT
(E + PV N)].
(.18)
pi pi
1. Helmholtz, A(T, V,N) pi pi - .
p(E) = exp
[1
kBTA(T, V,N)
]exp
[ EkBT
]. (.19)
Z =
exp
[ EkBT
], (.20)
pi pi pi - pi
p = exp
[1
kBTA(T, V,N)
]Z = 1. (.21)
pi
p = exp [A(T, V,N)]Z = 1, (.22)
pi = 1/kBT .
-
123
pi pi pi
A = kBT lnZ (.23)= 1 lnZ (.24)
U = < E >=
Ep (.25)
=
EeE/Z (.26)
= (lnZ)/ (.27)= (A)/ (.28)
= kBT2
( lnZ
T
)V,N
(.29)
< (E U)2 > =
(E U)2e(AE) (.30)
=
(E U)
e(AE) (.31)
=
(E U)e(AE) U
(.32)
= U
(.33)
= kBT2CV (.34)
S = kB lnZ + kBT
( lnZ
T
)V,N
(.35)
P = kBT
( lnZ
V
)T,N
(.36)
= kBT( lnZ
N
)T,V
(.37)
2. , (T, V, ) .
p(E , N) = exp
[1
kBT(U TS N)
]exp
[ 1kBT
(E N)]
(.38)p(E , N) = exp [(T, V, )] exp [(E N)] (.39)
=
e(EN) (.40)
-
124 .
= kBT ln (.41)= 1 ln (.42)
U N = (ln )/ (.43)= ()/ (.44)
S = kB ln + kBT
( ln
T
)V,
(.45)
P = kBT
( ln
V
)T,
(.46)
N = kBT
( ln
)T,V
(.47)
3. -
, G(T, P,N) .
p(E , V) = exp
[1
kBT(U TS + PV )
]exp
[ 1kBT
(E + PV)
](.48)
p(E , V) = exp [G(T, P,N)] exp [(E + PV)] (.49)
=
e(E+PV) (.50)
-
125
G = kBT ln (.51)= 1 ln (.52)
U + PV = (ln )/ (.53)= (G)/ (.54)
< (E)2 > = kB
(U
(1/T )
)P/T,N
(.55)
= kBT2CP 2kBT 2PV + kBTP 2V T (.56)
< (E)(V) > = kB(
V
(1/T )
)P/T,N
(.57)
= kBT2V kBTPV T (.58)
< (V)2 > = kB
(V
(P/T )
)1/T,N
(.59)
= kBT(V
P
)T,N
(.60)
= kBTV T (.61)
4. pi pi X0, X1, . . . , Xs pi pi pi pi F0, F1, . . . , Fs, pi Xi -pi pi, pi.. (U, 1/T ), (V, P/T ), (N,/T ), pi- pi X0 , X1 , . . . , Xs
p = exp
[ 1kB
(S F0X0 FsXs)]
exp
[ 1kB
(F0X0 + + FsXs)]
(.62)
W =
e 1kB (F0X0++FsXs) (.63)
LS [F0, . . . , Fs] Legendre pi pi
LS [F0, . . . , Fs] = S F0X0 FsXs (.64)
p = exp
( 1kB
LS [F0, . . . , Fs]
)exp
[ 1kB
(F0X0 + + FsXs)]
(.65)
-
126 .
W = exp
(1
kBLS [F0, . . . , Fs]
)(.66)
LS [F0, . . . , Fs] = kB lnW. (.67)
pi pi pi
< XjXk >= kB(XjFk
)F0...Fk1Fk+1...FsXs+1...Xt
(.68)
pi pi H. ( ) (E ) pi Schrodinger
H = E , (.69)
< | >= . (.70)pi ( )
eH = eE (.71)
eE =< |eH | >, (.72)
pi
A = ln
eE (.73)
= ln
< |eH | > (.74)
= lnTr(eH). (.75)
Tr pi < |eH | >.
-
127
pi - . pi, pipi pi . , pipi - pi ( Fermi-Dirac) pi ( Bose-Einstein).
(-)
pi pi -
) pi -
H H(xi, yi, zi, pxi, pyi, pzi), i = 1, . . . N. (.76)xi, yi, zi, pxi, pyi, pzi N .
) , pi h Planck.
dx1h1/2
dy1h1/2
dz1h1/2
dpx1h1/2
dpy1h1/2
dpz1h1/2
. . .dxNh1/2
dyNh1/2
dzNh1/2
dpxNh1/2
dpyNh1/2
dpzNh1/2
(.77)
1
h3NNi dxidyidzidpxidpyidpzi (.78)
) pi pi pi
1
h3NNi dxidyidzidpxidpyidpzi. (.79)
pi
Z =1
h3N
eH(xi,yi,zi,pxi,pyi,pzi)Ni dxidyidzidpxidpyidpzi (.80)
pi pi pi
Z =1
h3
dxdydz
dpxdpydpze
(p2x+p2y+p2z)/2m.
(.81)
-
128 .
Z =V
h3[2pimkBT ]
3/2. (.82)
N
Z =V N
N !h3N[2pimkBT ]
3N/2. (.83)
pipi pi pi pi de Broglie.
=h
2pimkBT, (.84)
Z =V N
N !3N. (.85)
de Broglie pi pi- , (V/N)1/3, pi pi pipi -. (V/N)1/3 , (Maxwell-Boltzmann), (V/N)1/3 , , pi Fermi-Dirac Bose-Einstein.
pi pi
U =< E >=
((lnZ)
()
)V
=3N
2=
3
2NkBT, (.86)
P =
((lnZ)
V
)
=N
V, (.87)
PV = NkBT. (.88)
pipi pi , U(xi, yi, zi), -,
H = T + U. (.89)
Helmholtz pi
A = kBT lnZ = kBT lnQ+ c(T, V,N), (.90)
piQ(T, V,N) =
exp
[ 1kBT
U(xi, yi, zi)
]Ni dxidyidzi (.91)
-
129
c(T, V,N) pi pi- pi . pi c , , pipi - , pi, pi .
pipi pi- . pi - pi pi. pi -pi pi pi , .
pi . pi- A B
U(xi, yi, zi) = (1 )UA(xi, yi, zi) + UB(xi, yi, zi), (.92)= UA + (UB UA), (.93)
. = 0 A = 1 B. .
pi pi pi-
A()
=
U
, (.94)
(pi pi)
2A()
2= 1
kBT
[(U
)2
(
U
)2](.95)
2A()
2= 1
kBT
[(U
U
)2] 0. (.96)
pi pi .
pi pi pipi pi -
A = A(b)A(a) = ba
U
d. (.97)
-
130 .
pi
lnZ() =
0
lnZ()
d, (.98)
Z() =
0
exp[U()/kBT ]dU, (.99)
Z
=
0
( 1kBT
U
)exp[U()/kBT ]dU, (.100)
lnZ
=
1
Z
Z
. (.101)
lnZ
= 1
kBT
U
(exp[U()/kBT ]
Z
)dU, (.102)
lnZ
= 1
kBT
U
, (.103)
lnZ() lnZ(0) = 1kBT
0
U
d. (.104)
piA = kBT lnZ, (.105)
A = A()A(0) =
0
U
d. (.106)
...
pi pi pi- pi pi pi pi- .
(.96) pi pi . pi BOGOLIUBOVpi pi pi. pi Helmholtz. - pi Helmholtz pi [10, 11, 12].
pi pi pi .94 pi pi -pi pipi pi pi .
A =
A =
(A+ A) (.107)
-
131
pi pi min = 0 pi max = 1 .
A = kBT lnZ, (.108)
Z(T, V,N) =
exp
[ 1kBT
U(xi, yi, zi)
]Ni dxidyidzi, (.109)
U+ = U + (U+ U) (.110)
A = kBT (lnZ+ lnZ) = kBT ln(Z+Z
), (.111)
pi
A = kBT ln[
eU/kBT e(U+U)/kBT
ZNi dxidyidzi
]
= kBT ln[ (
eU/kBT
Z
)e(U+U)/kBTNi dxidyidzi
].
(.112)
pi ,
e(U+U)/kBT , pi - pi pi U.
A = kBT ln[e(U+U)/kBT
], (.113)
BOGOLIUBOV
pi pi - (H0) (H1)
H = H0 +H1, (.114)
Helmholtz pi BOGOLIUBOV
A At = A0+ < H1 >0 . (.115) H1 pi pi H0. pi
A TS0+ < H >0 . (.116)
-
132 .
pi, pi BOGOLIUBOV (- .115) pi pi . pi - pi .
pi BOGOLIUBOV - pi pi (.96). - H0 H1 pi
H = H0 + H1, (.117)
Ht = H0 + < H1 >0, (.118)
(pi pi),< Ht >0=< H >0,
At() = A0 + < H1 >0 . (.119)
= 1 pi BOGOLIUBOV, At A. pi pi ( .1)
-
133
.1: pi BOGOLIUBOV, At A, pi pi . .94 .97
A
A
0 1
0 01 < = + t
At
A0
-
134 .
-
: S = kB ln
pi pi pi .
pi S pi pi S1 S2 1 2 , pi
S = S1 + S2, (.1)
= 12. (.2) pi S , S = f(),
pi pi pi .
f() = f(1) + f(2). (.3) pi 1 :
df()
d1=df()
d
d
d1=df(1)
d1(.4)
f(2) pi 1.pi
d
d1= 2 =
1, (.5)
pi df()
d
1=df(1)
d1, (.6)
135
-
136 . BOLTZMANN
pi
df()
dln=df(1)
dln1=df(2)
dln2= = kB . (.7)
pi pi pi pi pi 1 2. pi pi pi pi .
df()
dln= kB , (.8)
Boltzmann
S = f() = kB ln() + . (.9)
S 0 = 1, =0, pi pi .
...
-
-
-pi pi pi pi pi pi pi . () pi , pi, pi. pi pi 11 Introduction to Modern Thermodynami-cs Dilip Kondepudi [13].
-pi pi pi - pi pi pi pipi Onsager 1931, pi - pi pi - pi pi pi De Donger, Prigogine . - - , pi - pi Ilya Progogine pi.
pi pi, pi, -pi pi -pi pi pi , pi, pi pi pi , pi, - ... pi. , , pi pi , , - pi , pi
137
-
138 . -
i , ~x, , t
T = T (~x, t), P = P (~x, t), i = i(~x, t), i = 1, , r.
pi pi, pi
s[T (~x, t), i(~x, t)] = s(~x, t),
u[T (~x, t), i(~x, t)] = u(~x, t),
pi i(~x, t) pi i - . pi pi ~x
T (~x)ds(~x) = du(~x)i
i(~x)di(~x). (.1)
pi pi pi .
-pi pi pi
(~x, t) =dis(~x, t)
dt 0. (.2)
pi pi
diS
dt=
V
(~x, t)dV 0. (.3)
-pi pi- pi pi Fi - Ji
=i
FiJi. (.4)
, pi pi pi, pi , .
Ji =j
LijFj . (.5)
Lij - Onsager. pi, pi pi pi pi
=ij
LijFiFj 0. (.6)
-
139
pi pi Lij pi Onsager pi pi pi, Lij = Lji. pi .
.1)
Fq = (
1
T (~x)
), Jq = (T (~x)) (Jm2s1), (.7)
pi ( Fourier).2)
FD = (k(~x)
T (~x)
), JD = Dk(k(~x)) (mol m2s1), (.8)
pi Dk ( Fick).3)
Fe =()T
=E
T, Je =
V
R=E
(Cm2s1), (.9)
pi , E pi, I , V , (R, ) ( Ohm).
3)
Fr =ArT, Jr = vr =
1
V
drdt
(mol m3s1). (.10)
Ar r, r pi pi mole, vr V .
pipi pi pi - pi -pi.
r
0 = a1A1 a2A2 anAn + b1B1 + b2B2 + + bmBm, (.11)
pi pi pi r pi pi pi
dnA1a1 =
dnA2a2 = =
dnAnan =
dnB1b1
=dnB2b2
= = dnBmbm
= dr. (.12)
(AFFINITY) .11 -
Ar =
ni=1
aiAi mi=1
biBi , (.13)
-
140 . -
pi Ai Ai Bi pi Bi. Gibbs pi
Ar = rGm. (.14)pi pi pi :
1. pi Ar = 0 pi dr/dt = 0.
2. Ar > 0
3. Ar < 0 .
pi, pi pi Gibbs. - .
pi -pi , r, pipi pi
diS
dt=r
ArT
drdt 0. (.15)
pi pi Gibbs . : -pi pi, Gibbs pi pi .
r pi pi
vr =1
V
drdt
= Rf (r)Rr(r), (.16)
pi Rf pi Rr . molL1s1.
pi
aA+ bB = cC + dD
pi
Rf = kf [A]a[B]b, Rr = kr[C]
c[D]d. (.17)
i , i,
i = i +RT ln(i), (.18)
pi
Ar = RT ln(Kr(T )) +RT ln
(ni=1
aiAim
i=1 biBi
). (.19)
-
141
Kr(T ) pi
Kr = kf/kr. (.20)
pi ( 3.65 3.69)
rGm =
j
jj = Ar , (.21)
rG
m = RT lnKr. (.22)
pi pi
Ar = RT ln
(Rf (r)
Rr(r)
). (.23)
pi, pi pi - -pi pi
1
V
diS
dt=
1
V
r
ArT
drdt
= Rr
[Rf (r)Rr(r)] ln(Rf (r)
Rr(r)
). (.24)
pi pi pi pi . pi pi -
pi pi pi
1
22S =
d
dt
2S
2=k
FkJk 0, (.25)
pi 2S pi Fk Jk pi Fk Jk.
pi pi pi
, s(u, i), i =1, , r, pi pi
s(u, i) =
(s
u
)i
u+
ri=1
(s
i
)u
i (.26)
=
(1
T
)u+
ri=1
(iT
)i. (.27)
-
142 . -
pi pi pi
2s(u, ) =
(2s
u2
)i
(u)2 + 2
ri=1
(2s
ui
)ui +
ij
(2s
ij
)u
ij .
(.28) pi pi . pi-, pi pi pi pi pi
d[2s(u, i)]
dt= 2s(u, i) = 2
(2s
u2
)i
uu
+ 2
ri=1
(2s
ui
)(ui + ui)
+ 2ij
(2s
ij
)u
ij , (.29)
2s(u, i) = 2
(
u
1
T
)uu+ 2
ri=1
(
i
1
T
)ui
+ 2
ri=1
(
u
iT
)ui + 2
ij
(
i
jT
)u
ji,(.30)
pi
(1
T
)=
(
u
1
T
)u+
i
(
i
1
T
)i (.31)
(iT
)=
(
u
iT
)u+
j
(
j
iT
)j , (.32)
pi pi
2s(u, i) = 2
[
(1
T
)u+
i
(iT
)i
]. (.33)
pi pi - pi
2S = 2
[
(1
T
)u+
i
(iT
)u
i
]dV. (.34)
pi - pi
u = ~Ju , (.35)u = ~Ju, (.36)
-
143
i = ~Ji +j
ijvj , (.37)
i = ~Ji +j
ijvj . (.38)
ij ith -, vi . pi u, ipi pi .
pi
(f ~J) = f ~J + ~Jf, (.39) Gauss
V
( ~fJ)dV =
f ~Jd~a, (.40)
V
f ~JdV +V
~JfdV =
f ~Jd~a, (.41)
pi 1
22S =
(1
T
) ~Jud~a+
V
(
1
T
) ~JudV
+
i
(iT
) ~Jid~a
V
i
(iT
) ~JidV
+
V
i
(AiT
)vidV. (.42)
pi pi pi V d~a pi. pi pi
j ij(j/T ) = (Ai/T ).
pi pi .
1
22S =
V
(
1
T
) ~JudV (.43)
V
i
(iT
) ~JidV (.44)
+
V
i
(AiT
)vidV (.45)
1
22S =
k
Fk ~Jk 0. (.46)
...
-
144 . -
.1 -
pi (XD) - (XL) pi - (S, T). - [S], [L] pi , k3r pi , k3r
-
.1. 145
pi pi
d
dt= k1r+ k2f 2k2r (.66)
d
dt= k1f k1r + k2f k2r(2 + 2) k3f (2 2) (.67)
, .62 - .64, pi pi pi pi .
k1f = 0.5 (.68)k1r = 0.01
k2f = 0.5
k2r = 0.2
k3f = 1.5
k3r = 0.001
[S] = 0.5 (.69)[T] = [S]
[P] = 0
[XD]0 = 0.0
[XL]0 = [XD]0 +
-
146 . -
.1: - [.47 - .51], pi . pi [2-6] pi pi [.47 - .51].
-
[1] L. D. Landau and E. M. Lifshitz, Statistical Physics, Chapter II, PergamonPress, 1970. 1
[2] David Chandler, Introduction to Modern Statistical Mechanics, Oxford Univ.Press, 1987. 1
[3] N.M. Hugenholz, C-algebras and statistical mechanics Operator Algebrasand Applications, (Proc. Symp. Pure Math. 38) part2, ed. R.V. Kadison(Providence, RI: American Mathematical Society) pp. 407-465, 1982. 30
[4] D. Ruelle, Statistical Mechanics, Rigorous Results, London: W.A. Benja-min, 1969. 30
[5] Herbert B. Callen, Thermodynamics and Introduction to Thermostatistics,Second Edition, John Wiley and Sons, Inc., 1985. 56, 58
[6] L. Galgani and A. Scotti, On Subadditivity and Convexity Properties ofThermodynamics Functions, Pure and Appl. Chem. 22, 229, 1970. 56, 58
[7] Robert A. Alberty, Legendre Transforms in Chemical Thermodynamics,Pure and Appl. Chem. 69, 2221-2230, 1997. 56
[8] Stephen Boyd and Lieven Vandenberghe, Convex Optimization, CambridgeUniversity Press, 2004. 58
[9] P. W. Atkins, Physical Chemistry, Part I, Crete University Press, 2000. 79
[10] Thomas Simonson, Free Energy Calculations, in Computational Bioche-mistry and Biophysics, edited by Oren M. Becker, Alexander D. MacKerell,Jr., Benoit Roux, and Masakatsu Watanabe Marcel Dekker, Inc., pp. 169197, 2001. 132
[11] Daan Frenkel and Berend Smit, Understanding Molecular Simulations:From Algorithms to Applications, Academic Press, San Diego, CA USA,1996. 132
[12] Christophe Chipot and Adrew Pohorille (Eds), Free Enegy Calculations:Theory and Applications in Chemistry and Biology, Springer - Verlag,Berlin Heidelberg, 2007. 132
147
-
148
[13] Dilip Kondepudi, Introduction to Modern Thermodynamics, John Wiley& Sons, Chichester, England, 2008. 139
-
, 4, 126 -
, 15 , 1, 13, 26, 34, 91, 17, 21, 22, 115 , 103pi, 13, 17, 48, 130, 11, 33, 119pi, 4, 13, 26, 91pi , 9 , 103, 9 , 27 , 142 pi, 47 , 13, 39, 40, 93 , 14, 15,
34, 18, 19, 139, 104, 3, 31, 91, 92, 2, 12, 17, 21, 22, 48 , 21, 22, 62, 92, 2, 12pi, 6, 12, 34, 90pi, 20, 58, 14, 33, 89 pi, 3 , 11, 32
, 19, 55, 104, 23, 12 , 27, 94 pi, 75, 93, 126 , 102 , 7, 89 , 60, 92 -
, 18pi , 16, 47pi , 16, 47pi , 16, 47 , 104 pi, 1 , 126 , 127, 11, 33, 119 , 67, 76 , 34, 67, 130, 15 pipi, 17 -
, 130 pi, 27 pi, 95 , 68, 20, 56pi, 4, 13, 91 , 4-pi, 45
149
-
150
, 12 , 34, 31, 33, 34 , 11, 31 , 27 , 9 , 128 , 11 , 2, 12, 111, 11, 75pi, 13, 39, 40pi, 51pipi , 9pi, 11pi, 45pi , 14, 33pi , 103pi, 12, 11 , 11 -, 124 , 122 , 123 , 119 pi, 122 , 3, 14, 50 pi,
18 pi pi-
, 18 pi , 16, 40 , 13, 109 , 4, 9, 109 , 16, 44pi pi, 120, 111 , 131 , 13, 4, 14, 33, 50, 90 pi pi,
18
pi ,18
, 24, 13, 39, 40Avogadro, 1Bogoliubov, 130Boltzmann, 12, 133Bose-Einstein, 127Clausius-Clapeyron, 78, 93Clausius, 17, 48Duhem-Jougeut, 54Duhem, 28, 85Euler, 2, 39, 111Fermi-Dirac, 127Gibbs-Duhem, 24, 68Gibbs-Helmholtz, 101Gibbs, 12, 21, 27, 63, 75, 82, 100, 102Gradient, 9, 15, 17Helmholtz, 21, 62, 82, 122, 128, 130Hessian, 9, 15, 17, 29, 115Hess, 99Jacobians, 116Joule-Thomson, 61Kirchhoff, 100Lagrange, 17, 46, 113, 120Legendre, 20, 56, 109, 125Lyapunov, 20, 55Maxwell, 23, 66, 71, 109Planck, 127Raoult, 102Schrodinger, 126Taylor, 5, 9, 29, 121concave, 24, 34, 67convex, 24, 35, 67fluctuations, 44unconstrained, 15, 21, 22, 48van der Waals, 99vant Hoff, 104
Perieq'omena- : :
LEGENDRE MAXWELL GIBBS-DUHEM DUHEM
- LEGENDRE Gibbs-Duhem DUHEM
- , Clausius-Clapeyron ()- xi
van der Waals Hess Kirchhoff Gibbs Gibbs-Helmholtz Gibbs Raoult : : : , van't Hoff van't Hoff:
LEGENDRE LAGRANGE BOLTZMANN -
Bibliograf'iaEuret'hrio
