2013 25 9

i 1 1 1.1................................... 1 1.2........................... 2 1.3..................................... 3 1.4..................................... 4 2 6 2.1................................. 6 2.2......................... 7 2.3................................. 9 2.4.................................. 11 2.5........................... 13 2.6................... 17 3 18 3.1...................................... 18 3.2................................. 19 3.3................................... 21 3.4.............................. 23 3.5................................. 24 4 27 4.1.................................. 27 4.2........................... 28 4.3............................ 31 4.4............................ 32 4.5......................... 33 4.6................................. 36 4.7.............................. 36 4.8..................................... 37

4.8.1........................ 37 5 39 5.1......................... 39 5.2......................... 41 5.3...................... 47 5.4............................. 49 5.5..................................... 50 5.5.1................................. 50 5.5.2........................ 51 6 52 6.1........................................ 52 6.2.............................. 53 6.3 :............. 54 6.4...................... 55 6.5................................ 59 7 63 7.1................................ 63 7.2........... 64 7.3 Ω...................... 65 7.4..................................... 66 7.5......................... 67 7.6........................... 68 7.7....................................... 68 7.8..................................... 70 7.9..................................... 71 7.10..................................... 74 7.10.1.............. 74 7.10.2.................. 75 7.10.3............................ 75 7.10.4............................... 76 7.10.5....................... 78 7.10.6..................... 79 ii

8 81 8.1................... 81 8.2............................ 82 8.3..................................... 83 8.4..................................... 84 9 86 9.1...................................... 86 9.2................................ 86 9.3...................................... 87 9.4..................................... 90 9.5............................ 92 9.6................................. 93 9.7................................. 94 9.8.................................... 96 9.9.................................. 97 9.10.................... 97 9.11.................................. 98 9.12................................. 99 9.13................................. 100 9.14..................................... 101 9.14.1.......................... 101 iii

1 1 1.1 3 PC V P

1.2. 1. 1.2 20 A B A B A B A B A B A B A B A B A C B C B C A A 1742 t 1 0 C 100 C 100 100 C 0 C 1724 0 F 2

1.3. 1. 96 F 96 T K 0K 273.16K 273.15 T = t + 273.15 K 1.3 1662 V P V = P (1.3-1 1787 1802 V 0 t ( V = V 0 1 + t 273.15 (1.3-2 t = 273.15 T = t + 273.15 0 T 0 = 273.15K V T = V 0 T 0 = (1.3-3 T P V T 0, P 0, V 0 P 0 T V 1 = T T 0 V 0 (1.3-4 3

1.4. 1. T P V 1 V = P 0 P V 1 (1.3-5 P V T = P 0V 0 T 0 = (1.3-6 1811 P 0 = 1.01325Pa 0 (T 0 = 273.15K V 0 = 22.414l/mol P 0 V 0 /T 0 = R = 8.314472J/K mol n P V = nrt (1.3-7 P V = nrt P V T n f(p, V, T, n = 0 N A = 6.02214179(30 10 23 mol 1 R k B = 1.380658(12 10 23 J K 1 1.4 3 4 3 n V P 4

1.4. 1. P V V 1 V 2 V 1 + V 2 20 80 100 5

6 2 2.1 18 19 ( ( τ r τ e τ e τ r

2.2. 2. 2.2 S F e P e = F e /S dx F e dx > 0 dv = Sdx d W = F e dx = P e Sdx = P e dv (2.2-1 dv < 0 dv > 0 6 P P e d W = P dv (2.2-2 V 1 V 2 (> V 1 P P dv P W = P V 2 V 1 dv = 7

2.2. 2. P (V 2 V 1 T 1 = P V 1 /nr T 2 = P V 2 /nr(> T 1 2 n T V 1 V 2 (> V 1 P e P P e T P = nrt/v d W = P dv = nrt dv V (2.2-3 W = V2 V 1 3 P dv = nrt V2 V 1 dv V = nrt ln(v 2/V 1 (2.2-4 n T P 0 V 0 T P V (i (ii P 0 V 0 V V P 0 P (i 2 W i = nrt ln(v/v 0 (ii P 0 1 W ii = P 0 (V V 0 P 0 = nrt/v 0 W ii = nrt (V/V 0 1 W i W i W i W ii (i (ii 8

2.3. 2. 4 V V 1 V V 1 V 1 n T P e = 0 W = V V 1 P e dv = 0. 5 m h P dv mg h W = mgh 6 n T 0, P 0, V 0 T, P, V W W = V V 0 P dv 2.3 3,4,5 9

2.3. 2. A B W A B A,B A B C A B W A B B C W B C A C W A C W A C = W A B + W B C (2.3-1 B B B U W i f = U(f U(i (2.3-2 i (initial f (final U U U(f = U(i + W i f (2.3-3 A x B y W A B A B x, y F W A B = F (x, y C z (2.3-1 F (x, z = F (x, y + F (y, z x F (x, z/ x = F (x, y/ x = f(x x y z x f(x x F (x, y = f(xdx + c(y = U(x + c(y f(x x U(x 10

2.4. 2. c x y A B y = x W A A = F (x, x = 0 0 = U(x + c(x c(x U(x W A B = F (x, y = U(y U(x 3 Q d W du, d Q d Q + d W = du + = (2.3-4 Q U n T V n V, T U = U(V, T, n dt dv du du = ( U ( U dt + dv T V,n V T,n 2.4 K 11

2.4. 2. n V (W = 0 Q Q = U U T V U = U(V, T T 2.3 U = ( U/ T V,n T C V = Q/ T C V = ( U T V,n (2.4-1 V n U T c V c V = C V /n P Q T V V = ( V/ T P,n T U = ( U/ T V,n T + ( U/ V T.n V Q W = P V Q + W = U C P = Q U + P V = T T ( U [( U = + T V,n V [ ( U = C V + P + V T,n ]( V + P T ]( V T,n T P,n P,n (2.4-2 C P U P Q = U + P V = (U + P V H = U + P V (2.4-3 Q = H C P = ( H T P,n (2.4-4 H 12

2.5. 2. 2.5 ( 2.2 4 1845 1806 V V 1 (Q = 0 ( U = Q + W = W (W = 0 2.2 4 U(T, V 1 = U(T, V (2.5-1 U V U = U(T C V = ( U/ T V,n ( U/ V T,n = 0 P V = nrt ( V/ T P,n = ( (nrt/p / T P,n = nr/p (2.4-2 C P = C V + nr c P = c V + R (2.5-2 c P = 5R/2 c P = 7R/2 c V c P U(T, V, n = nc V T (2.5-3 H(T, P, n = nc P T (2.5-4 13

2.5. 2. c V = 3R/2 2 c V = 5R/2 3 c V = 7R/2 ( P n V 1 V 2 (> V 1 2.1 1 W = P (V 2 V 1 T 1 = P V 1 /nr U 1 = nc V T 1 T 2 = P V 2 /nr U 2 = nc V T 2 Q = W + U 2 U 1 = P (V 2 V 1 + nc V (T 2 T 1 = nr(t 2 T 1 + nc V (T 2 T 1 c P = Q n(t 2 T 1 = c V + R c P = c V + R Q = nc P (T 2 T 1 = H 2 H 1 2 n T V 1 V 2 (> V 1 2.1 2 W = nrt ln(v 2 /V 1 Q = W = nrt ln(v 2 /V 1 3 n T P 0 V 0 T P V (i 14

2.5. 2. (ii(a P 0 V 0 V (b V P 0 P U = 0 Q W (i 2 W i = nrt ln(v/v 0 Q i = W i = nrt ln(v/v 0 (ii 2.1 3 W ii = nrt (V/V 0 1 Q ii = W ii = nrt (V/V 0 1 Q i Q ii 1 a V 0 V P 0 T = P 0 V 0 /nr T 1 = P 0 V/nR Q a = nc P (T 1 T = H 1 (V, T 1, n H 0 (V 0, T, n c P b P 0 P T 0 T V W = 0 ( Q b = U 2 (V, T, n U 1 (V, T 1, n = nc V (T T 1 c V Q ii = Q a + Q b = nc P (T 1 T + nc V (T T 1 = n(c P c V (T 1 T = nrt (T 1 /T 1 = nrt (V/V 0 1 c P = c V + R 4 n m h 2.2 5 W = mgh U U = W = mgh n T = U/nc V = mgh/nc V 15

2.5. 2. J mgh cal c T =4.1855J/cal kg( 1 cal/g K 1kg h 5 n T 0, P 0, V 0 T, P, V (d Q = 0 dt dv d W = P dv du = nc V dt d Q = 0 P dv = nc V dt dt dv P V = nrt dt = P dv = RT nc V c V V dv dt T = R dv c V V c P = c V + R R/c V = c P /c V 1 = γ 1 γ = c P /c V > 1 (T 0, V 0 (T, V ln(t/t 0 = (γ 1 ln(v/v 0 T V γ 1 = T 0 V γ 1 0 = γ P V = nrt P V γ = P 0 V γ 0 = V γ P V = nrt = P V T γ /P γ 1 = T γ 0 /P γ 1 0 = 16

2.6. 2. 2.6 ( A B ( A B A P A T A B P B T B A V A B V B 0 VB W = P A dv P B dv = P A V A P B V B (2.6-1 V A 0 Q = 0 P A V A P B V B = U B U A U A + P A V A = U B + P B V B (2.6-2 (2.4-3 H = U + P V (2.6-3 H A = H B H 5.3 17

18 3 3.1 T H Q H > 0 W > 0 T L Q L > 0 ( U = 0 Q H Q L W Q H Q L W = 0 Q H = W + Q L (3.1-1 W Q H W η = W Q H = Q H Q L Q H = 1 Q L Q H (3.1-2

3.2. 3. T L Q L W Q H = Q L + W T H ϵ L = Q L W = Q L Q H Q L (3.1-3 W T L Q L T H Q H = Q L + W ϵ H = Q H W = Q H Q H Q L (3.1-4 3.2 (Q H = W Q L = 0 19

3.2. 3. (1 T H Q H W W = Q H T L Q L Q L + W = Q L + Q H Q L (2 T H Q H W Q L = Q H W T L 20

3.3. 3. Q L ( W = Q H Q L 1 2.1 V 1 V 2 2 V 1 V 2 V 1 + V 2 V 1 3.3 ( 21

3.3. 3. Q H Q L W = Q H Q L η R = W/Q H = 1 Q L /Q H Q H W (> W Q L (< Q L W (< W Q L (> Q L Q H = W +Q L Q L Q L > 0 W W = Q L Q L > 0 22

3.4. 3. 3.4 (1820 T H T L n 1: A B T H 2: B C T H T L 3: C D T L 4: D A T L T H P V P V 1 Q H 2.5 2 Q H = nrt H ln(v B /V A 2 4 3 Q L 2.5 2 : Q L = nrt L ln(v D /V C η C = W = 1 Q L = 1 T L ln(v C /V D Q H Q H T H ln(v B /V A (3.4-1 A,B,C,D A D 3 A A,B,C,D 2.5 4 T V γ 1 = 2 T H V γ 1 B 4 T L V γ 1 D 23 = T L V γ 1 C = T H V γ 1 A (3.4-2

3.5. 3. V B V D = V C V A V B V A = V C V D (3.4-3 (3.4-1 η C = 1 Q L Q H = 1 T L T H (3.4-4 T H T L T 3.5 (Otto ( 1: A B T 1 T H V 1 2: B C T H T 2 V 1 V 2 3: C D T 2 T L V 2 4: D A T L T 1 V 2 V 1 24

3.5. 3. 1 (W 1 = 0 T 1 T H U 1 = nc V (T H T 1 Q H = U 1 = nc V (T H T 1 T V γ 1 = 2 T H V γ 1 1 = T 2 V γ 1 2 T H /T 2 = (V 2 /V 1 γ 1 = (T 1 /T L 4 T L V γ 1 2 = T 1 V γ 1 1 (3.5-1 3 Q L = nc V (T 2 T L η = 1 Q L Q H = 1 T 2 T L T H T 1 = 1 T 2 T H = 1 T L T 1 (3.5-2 T 2 T L η C = 1 (T L /T H 1 3 (Joule ( 1: A B P 1 T 1 T H 2: B C T H T 2 P 1 P 2 3: C D P 2 T 2 T L 4: D A T L T 1 P 2 P 1 1 c p Q H = nc p (T H T 1 T γ /P γ 1 = 2 T γ γ 1 H /P1 = T γ 2 /P γ 1 2 4 T γ γ 1 L /P2 = T γ 1 /P γ 1 1 (3.5-3 25

3.5. 3. T H T L = T 1 T 2 3 Q L = nc p (T 2 T L η = 1 Q L Q H = 1 T 2 T L T H T 1 = 1 T 2 T H = 1 T L T 1 (3.5-4 T 2 T L η C = 1 (T L /T H 1 3 26

27 4 4.1 (3.4-4 Q L Q H = T L T H, Q H T H + Q L T L = 0 (4.1-1 Q L T i T i Q i Q i < 0 Q i = 0 (4.1-2 T i i d Q d Q T = 0 (4.1-3 T e T e T T O I A II O T e d Q A O,I d Q O T + d Q A,II T = 0 (4.1-4

4.2. 4. II II O A d Q V 1 V 2 V 2 V 1 A O,I d Q T = A O,II d Q T (4.1-5 O A d Q T O A A O d Q T = S(A S(O (4.1-6 S O A S(A S(O ds = d Q T (4.1-7 ds 4.2 O A S(O S(A (4.1-6 T H Q H T L 28

4.2. 4. Q L η η C 3.3 η = 1 Q L Q H Q H T H < η C = 1 T L T H (4.2-1 + Q L T L < 0 (4.2-2 Q L Q i T i i Q i T i < 0 (4.2-3 O A A O (4.2-3 d Q A d Q O = + T e T e O, A, d Q T e < 0 (4.2-4 A O, d Q T e < O A, d Q T = A O, d Q T = S(A S(O (4.2-5 T e T T T e T e d Q ds ds > d Q T e (4.2-6 29

4.2. 4. [ ] d Q = 0 (ds = 0 (ds > 0 1: W Q = W T H Q (4.1-7 S H = Q/T H S = Q/T H Q Q 2: : T H T L Q Q S H = Q/T H Q S L = Q/T L ( 1 S = S H + S L = Q 1 T L T H T H > T L 30

4.3. 4. Q > 0 Q < 0 3: C(T T 1 T 2 dt d Q d Q = C(T dt S(T 2 S(T 1 = T2 T 1 d Q T = T2 C(T T 1 T dt C V S(T 2, V S(T 1, V = T2 C V (T T 1 T C P S(T 2, P S(T 1, P = T2 C P (T T 1 T dt dt 4.3 T H T L Q W Q W S H = Q/T H, S L = (Q W /T L S t = S H + S L = Q T H + Q W T L 0 (4.3-1 W ( 1 T L T H Q = W max (4.3-2 31

4.4. 4. W max S t = 0 3.3 W max η max = W max Q = 1 T L T H = η C (4.3-3 η = W/Q < 1 T H /T L = η C 4.4 n P V = nrt U = nc V T T 0 V 0 O S 0 (T 0, V 0, n T V A S(T, V, n O A O B:(T 0, V A d Q = du +P e dv OB T 0 (du = 0 d Q = P dv = (nrt 0 /V dv S(B S(O = V V 0 d Q V = nr dv T 0 V 0 V = nr ln V (4.4-1 V 0 BA dv = 0 :d Q = du = nc V dt S(A S(B = T T 0 d Q T T = dt nc V T 0 T = nc V ln T (4.4-2 T 0 S(T, V, n S(T 0, V 0, n = S(A S(O = nc V ln T T 0 + nr ln V V 0 (4.4-3 n T 0 1 v 0 = V 0 /n s(t 0, v 0 n S 0 (T 0, V 0, n = ns(t 0, v 0 S(T, V, n = nc V ln T T 0 + nr ln V nv 0 + ns(t 0, v 0 (4.4-4 32

4.5. 4. n n P V = nrt P 0 V 0 = nrt 0 c P = c V + R S(T, P, n = nc P ln T T 0 nr ln P P 0 + ns(t 0, P 0 (4.4-5 P n 1 3.5 S t 2 T R T S T S T R T R T R C (1 Q (2 S R (3 S S (4 S t ( : x = 1 T S /T R x < x < 1 4.5 n A A P T V A = n A RT/P n B B P T V B = n B RT/P n A + n B V = V A + V B T P A V P A = n A RT/V B P B = n B RT/V P = P A + P B 33

4.5. 4. (1801 A B (4.4-4 ( S A (T, V A, n A = n A c A,V ln T + R ln V A + s A (T 0, v 0A, T 0 n A v 0A ( S B (T, V B, n B = n B c B,V ln T + R ln V B + s B (T 0, v 0B T 0 n B v 0B (4.5-1 c A,V c B,V A B A B V B B V A Q = 0 V B V A A B A V A P A B V P B V A P A + P B V V A P B P A = n A RT/V A A A B V L A A B R B B A 34

4.5. 4. S A+B (T, 2V, n A + n B = S A (T, V, n A + S B (T, V, n B = n A (c A,V ln T V ( + R ln + s A (T 0, v 0A + n B c B,V ln T V + R ln + s B (T 0, v 0B T 0 n A v 0A T 0 n B v 0B (4.5-2 R A R L A V L B L B R V A B ds = 0/T = 0 (4.5-2 (4.5-1 (4.5-2 S = S A+B (S A + S B = n A R ln V V A + n B R ln V V B > 0 (4.5-3 V = V A + V B S = n A R ln n A + n B n A + n B R ln n A + n B n B = nr(x A ln x A + x B ln x B > 0 (4.5-4 n = n A + n B x A = n A /n x B = n B /n = 1 x A V 1 T n V 2 V 1 n 1 V 2 n n 1 4.2 35

4.6. 4. 4.6 4.2 3 S(A S(O = A O C P (T dt (4.6-1 T (4.4-5 C P (T = AT n S(T = T 0 AT n T dt = A T 0 T n 1 dt = A n T n + S(0 (4.6-2 n > 0 S(0 C = γt C T 3 S(0 S(0 = 0 (1905 4.7 A B du d Q d W du = d Q + d W (4.7-1 dv ds du = T ds P dv (4.7-2 36

4.8. 4. T A B P A B du (dt ds dp dv (4.7-2 T ds = du + P dv, ds = 1 T du + P dv (4.7-3 T S ( S U V ( S V U = 1 T = P T (4.7-4 (4.7-5 U V S N, U, V W (N, U, V S(N, U, V = k B ln W (N, U, V (4.7-6 k B = R/N A = 1.3806505(24 10 23 J K 1 (R = 8.314472(15 J mol 1 K 1 N A = 6.0221415(10 10 23 mol 1 ( (4.7-6 T = 0 (W = 1 S(T = 0 = 0 4.8 4.8.1 37

4.8. 4. : Q = 0 (ds = 0 (T = 0 (S = 0 S 0 T = 0 S = 0 S 0 38

39 5 5.1 d Q + d W = du, ds d Q T e (5.1-1 d Q T e d W T e T T e T T e d W = du + d Q du + T e ds = du + d(t S = d(u T S (5.1-2 ds T T e T e T d W F = U T S (5.1-3 F A

5.1. 5. U T S F d W df d W df = F F (5.1-4 df W max (dv = 0 d W = P e dv (d W = 0 df 0 F T e P e T = T e P = P e d W = P e dv = d(p e V = d(p V (5.1-2 d (W W = d W P dv d(u T S + P V (5.1-5 G = U T S + P V (5.1-6 d (W W = d W P dv dg (5.1-7 G ( G 40

5.2. 5. (d W = d W = P dv G dg 0 G 2.6 H = U + P V d Q d Q = du d W = du + P e dv = d(u + P V = dh (5.1-8 G = U T S + P V H G = H T S (5.1-9 F G 5.2 n V P T U S H F G n, V, T n, P, T n 4.7 du dv ds du = T ds P dv (5.2-1 41

5.2. 5. S V S V U = U(S, V ds, dv U(S, V du(s, V = ( U ( U ds + dv (5.2-2 S V V S (5.2-1 (5.2-2 U S V ( U S V = T, ( U = P (5.2-3 V S U V, S T, P U U F,G,H F = U T S df = [(U + du (T + dt (S + ds] [U T S] = du T ds SdT dt ds = du T ds SdT (5.2-4 (dt (ds (5.2-1 df = (T ds P dv T ds SdT = SdT P dv (5.2-5 :F = F (T, V F df (T, V = (5.2-5 (5.2-6 ( F T V ( F ( F dt + dv (5.2-6 T V V T = S, ( F V T = P (5.2-7 F 42

5.2. 5. G = F + P V dg = SdT + V dp (5.2-8 T P dg(t, P = ( G dt + T P ( G dp (5.2-9 P T ( G T P = S, ( G P T = V (5.2-10 H = U + P V dh = T ds + V dp (5.2-11 S P dh(s, P = ( H ds + S P ( H dp (5.2-12 P S ( H S P = T, ( H = V (5.2-13 P S U, F, G, H S T V P x ϕ(x x X = ϕ(x/ x Φ(X Φ(X = ϕ(x xx (5.2-14 dφ = dϕ d(xx = ( ϕ/ xdx xdx Xdx = xdx Φ(X/ X = x ϕ(x Φ(X 43

5.2. 5. (5.2-3 T P S V x, y f(x, y f(x, y x y = y f(x, y x (5.2-15 f ( T = 2 U(S, V V S V S = 2 U(S, V S V ( P = S V (5.2-16 (S = ( T ( P =, U(S, V V S S V ( P ( S =, F (T, V T V V T ( V ( S = G(T, P T P P T ( T ( V =, H(S, P (5.2-17 P S S P (S = (5.2-1 du = T ds P dv (5.2-18 T V S, V S T V T = du (5.2-18 dv ( U V T ( S = T P (5.2-19 V T 44

5.2. 5. ( S/ V T = ( P/ T V ( U V T ( P = T T V P (5.2-20 1: (5.2-18 2: P V = nrt (5.2-20 ( P/ T V = nr/v = P/T (5.2-20 ( U/ V T = 0 3: ( S C V = T T ( S C P = T T V P C P C V = T V α2 κ T ( U = T ( H = T V P = T 2 F (T, V, 2 T = T 2 G(T, P, 2 T α = 1 ( V : (> 0 V T P κ T = 1 ( V : (> 0 V P T : C P = T ( S/ T P (T, P (T, V C P = T ( S T P = T [( S T V + ( S ( V ] ( P ( V = C V + T V T T P T V T P 45

5.2. 5. ( S/ V P = ( P/ T V ( V ( P ( P ( V = = T P ( T V V T T P V P T α κ T 4: κ T κ S κ T = 1 ( V, κ S = 1 ( V V P T V P S C P C V (iii κ T κ S = ( V/ P T ( V/ P S = κ T κ S = (V, T (P, T (V, S (P, S κ T κ S = C P C V T α2 (C P /V > 0 = (P, S (P, T (V, S (V, T = ( S/ T P ( S/ T V = C P C V κ T = 1 V ( V P T = 1 V [( V ( V ( S ] + = κ S 1 ( V [ ( V ] P S S P P T V S P T P (V, P ( V ( V (T, P = = T P S P (S, P ( S = V α C P (T, P T P T κ T κ S = T α2 (C P /V > 0 46

5.3. 5. 5.3 2.6 A B ( A B A P A, T A B P B, T B A B A V A B V B U A + P A V A = U B + P B V B, H A = H B (5.3-1 H 2.6 A B P A > P B T A T B S P dh = T ds + V dp (5.3-2 T P ds = ( S ( S dt + dp = C ( P V T P P T T dt dp (5.3-3 T P dh = C P dt + { ( V } V T dp (5.3-4 T P (dh = 0 ( T = 1 { ( V } T V P H C P T P P V = nrt ( V T P = nr P = V T (5.3-5 (5.3-6 ( T/ P H = 0 47

5.3. 5. 6 ( P + n2 α (V nβ = nrt (5.3-7 V 2 P = ( T ( nr = P + n2 α 2n2 α (V nβ V P V 2 V 3 = nrt V nβ 2n2 α (V nβ (5.3-8 V 3 C P ( T P H ( V ( V [ ( T = T V = T V T P T P V nβ ( V ( nβ [ = T 2α ( V nβ 2 ] T P V nβ Rβ V ( V ( nβ = (T T inv T P V nβ T inv = 2α ( Rβ 1 nβ V 2 2α Rβ 2nα(V nβ ] RV 3 (5.3-9 (5.3-10 V nβ nα/rt V 1 ( V/ T P T T inv ( T/ P H < 0 (dp < 0 (dt > 0 B T A T inv ( T/ P H > 0 B 5.1: (K (K He 23.6 4.2 H 2 195 20.4 N 2 621 77.3 O 2 764 90.1 CO 2 2050 194.7 CO 2 N 2 H 2 He 48

5.4. 5. 5.4 ( x, y z = z(x, y x y dx dy z dz dz = ( z dx + x y ( z dy (5.4-1 y x (i y dy = 0 dz dz 1 = ( z dx ( z ( x lim x y dz 0 dz = x y z y ( x = z y 1 ( z x (ii z dz = 0 dy dy 0 = ( z ( dx lim + x y dx 0 dy ( z = y x y ( z x y ( z/ y x (i ( x y z ( y z x ( x + y z ( z y x (5.4-2 ( z + 1 = 0 (5.4-3 x y (iii x y z w = w(x, y (5.4-1 dx w dx dz ( z lim dx 0 dx = + x y ( z y lim x dx 0 dy dx (5.4-4 w = ( z = x w ( z + x y ( z y x ( y x w (5.4-5 49

5.5. 5. z(x, y w(x, y x y (z, w ( z (x, y = x y (w, z = (x, y ( w y x ( z y x = (z, w (y, x ( w = x y ( z ( x z y y x ( w ( x w y y x ( z (z, y (x, y = x y ( z y x 0 1 = ( z x y (i,(ii,(iii (i (ii (iii (z, y (x, y = 1, (x, y (z, y (x, z (y, x (z, y (y, z (z, x (x, y = 1, (z, w (x, w = (z, w (x, y (x, w (x, y (z, w = (w, z 5.5 5.5.1 T = 0 α = 1 ( V ( : V T P ( V/ T P = ( S/ P T 50

5.5. 5. 5.5.2 V M H M U(S, M du = T ds + H dm (1 M F (T, M = U(S, M T S df = SdT + H dm (2 H G(T, H = U(S, M T S H M dg = SdT M dh 51

52 6 6.1 T P (101325Pa 0 (273.15K 100 (373.15K 0.00603 (610.6Pa, 0.01 (273.16K 218.3 (22.12MPa 374.15 (647.30K 0.324g/cm 3 Fe MnO Fe 3 O 4 SrTiO 3 BaTiO 3 CuZn CCl 4 C 7 F 16 Al

6.2. 6. 6.2 n ( P 1 C P,1 d Q dt d Q = C P,1 dt T (P ( ( 2 P T (P 1 U 1 (T (P, P, n V 1 (T (P, P, n U 2 (T (P, P, n V 2 (T (P, P, n U 2 U 1 P (V 2 V 1 (latent heatl L = (U 2 U 1 + P (V 2 V 1 = H 2 H 1 (6.2-1 H = U + P V ( ( d Q ds d Q = T ds T 1 2 L = T (S 2 S 1 (6.2-2 53

6.3. : 6. L (6.2-1 (6.2-2 U 1 T S 1 + P V 1 = U 2 T S 2 + P V 2, G 1 (T, P, n = G 2 (T, P, n (6.2-3 (6.2-3 T P T P 6.3 : (6.2-3 T P (6.2-3 dt T + dt P + dp G 1 (T + dt, P + dp, n = G 2 (T + dt, P + dp, n (6.3-1 (6.3-1 dt dp (6.2-3 ( G1 dt + T P ( G1 dp = P T ( G2 dt + T P ( G2 dp (6.3-2 P T ( G/ T P = S, ( G/ P T = V (S 2 S 1 dt = (V 2 V 1 dp, dp dt = S 2 S 1 V 2 V 1 = L T (V 2 V 1 (6.3-3 T P L V = V 2 V 1 1 2 C p1 C p2 L dl(t, P (T dt = C p2 C p1 + L T L (V 2 V 1 [( V2 T P ( V1 ] T P (6.3-4 54

6.4. 6. L(T, P = H 2 (T, P H 1 (T, P P = P (T dl(t, P (T dt = [( H2 + T P ( H2 dp ] P T dt [( H1 + T P ( H1 dp ] P T dt C p H C p = ( H/ T P S P ( H P T = ( H + P S ( H ( S S P P T ( V = V + T T P dl(t, P (T dt { [( V2 = C p2 C p1 + (V 2 V 1 + T T = C p2 C p1 + L T L [( V2 (V 2 V 1 T P P ( V1 ]} dp T P dt ( V1 ] T P (6.3-3 L = T (S 2 S 1 2 V 2 V 1 V 1 (6.3-4 L (C p1 > C p2 dl/dt < 0 6.4 P V = nrt T P V (n/v 0 n/v ( P + n2 a V 2 (V nb = nrt (6.4-1 55

6.4. 6. a b P V P = nrt V nb n2 a (6.4-2 V 2 V nb V nb P V T c ( P ( 2 P = = 0 (6.4-3 V T c V 2 T c T c 1 P c T c V c a, b P c = a 27b, T 2 c = 8a 27bR, V c = 3nb. a = 27(RT c 2 64P c, b = RT c 8P c = V c 3n T c ( P/ V T > 0 V 1 ( V/ P T P 1 2 1 2 56

6.4. 6. G(T, P ( G P T G(T, P = G(T, P 0 + = V (T, P (6.4-4 P P 0 V (T, P dp (6.4-5 P V P P G P G G 1 (unstable dp < 0 G G u P dp > 0 G G 2 P G G G 1 G 2 P (T P (T P (T V 1 P (T V 2 2 P (T V 1 V 2 P V 1,V 2 V V 1 V P V 1 V V V 2 V V 2 V 1, V 2 T V T c V 1 V 2 V 1 V 2 miscibility gap ( V = V 2 V 1 T c ( V = 0 T c T V T P 57

6.4. 6. 2 (5.2-20 U(T, V, n = n2 a V + n2 a V 0 + U(T, V 0, n (1 c V U(T, V, n = nc V T + g(v = nc V T n2 a V + U 0 (2 U 0 (5.2-20 ( U V T ( S = T V = T T T ( P P = T T ( nrt V nb n2 a V 2 V P nrt V nb + n2 a V 2 = n2 a V 2 (1 c V = n 1 ( U/ T V (1 c V = n 1 ( U/ T V0 V 0 (2 3 S(T, V = C V ln T T 0 + nr ln V nb V 0 nb + S(T 0, V 0 T V F (T, V = C V T n2 a [ V T C V ln T + nr ln V nb ] T 0 V 0 nb + S(T 0, V 0 du = T ds P dv du + P dv ds = = 1 T T dt = C V T + nr dv V nb ( C V dt + n2 a 4 V 2 dv + 1 T 58 ( nrt V nb n2 a V 2 dv

6.5. 6. V 1 V 2 L = T S = nrt ln V 1 nb V 2 nb V 1 V 2 5 dp/dt dp dt = S 2 S 1 V 2 V 1 = nr ln(v 2 nb ln(v 1 nb V 2 V 1 V 1 V 2 V c = 3nb [ dp dt = lim V2 V c nr V 2 V c V c nb V ] 1 V c 1 = R V c nb V 2 V 1 2ba P = P/P c, T = T/Tc, Ṽ = V/V c P = 8 T Ṽ 1 3 Ṽ 2 P c = T c = Ṽc = 1 d P /d T = (T c /P c dp/dt = 4 P = 1 + p, T = 1 + t, Ṽ = 1 + v p v p = 4t 6vt 3 2 v3 T c (t < 0 v = 0 p v p = 4t t < 0 v 6.5 G(T, P V 59

6.5. 6. T F (T, V P = nrt/v P/ V T = 2 F/ V 2 T < 0 F V n V x V n < V < V x P/ T = 2 F/ V 2 T > 0 F V n V x T c F V 1 F V 2 V 1 V 2 V F (T, V V F (T, V V 1 V 2 x 1 x xf (T, V 1 + (1 xf (T, V 2 F F ( 5 V 1 n V 1 n/v 1 V 2 n V 2 n/v 2 1 V 1 /n 2 V 2 /n V V = xv 1 + (1 xv 2 1 x = (V V 2 /(V 1 V 2 n 1 xn 2 (1 xn nx (V 1 /n + n(1 x (V 2 /n = V V V 2 V 1 60

6.5. 6. P = nrt V nb n2 a (6.5-1 V 2 (1 P F P = F (T, V / V V c = 3nb F (T, V = F (T, V c nrt ln V nb ( n 2 V c nb a V n2 a V c (6.5-2 (2 T c = 8a/27bR V c T = T/T c Ṽ = V/V c F = F (T, V F (T, V c = nrt T ln 3Ṽ 1 9 ( 1 c 2 8 Ṽ 1 (6.5-3 T = 0.9 Ṽ (3 f = F + 3 2 ( 3 [ T (Ṽ 4 1 = (1 + t ln (1 + 3 2 v 3 ] 2 v 9v2 8(1 + v (6.5-4 f v = Ṽ 1 t = T 1( t = 0.1, 0., 0.1 (1 6.4 2 (2 T = 0.9 F 0.4 0.2 0-0.2-0.4-0.6-0.8 0.5 1 1.5 2 2.5 3 3.5 4 4.5 V 61 f 0.15 0.1 0.05 0-0.05 t=0.1 t=0.0 t= 0.1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 v

6.5. 6. (3 t > 0 t < 0 v = 0 v 1 v 2 62

63 7 T P n 7.1 dn dn d Q + d W + µ e dn = du (7.1-1 µ e µ µ e du = T ds P dv + µdn (7.1-2 S V n ( U µ(s, V, n = (7.1-3 n S,V F = U T S, G = U T S + P V, H = U + P V (7.1-4

7.2. 7. dn df = SdT P dv + µdn, dg = SdT + V dp + µdn, dh = T ds + V dp + µdn (7.1-5 µ(t, V, n = ( F, µ(t, P, n = n T,V ( G, µ(s, P, n = n T,P ( H n S,P (7.1-6 7.2 G T P n G n T P n λ G λ G(T, P, λn = λg(t, P, n (7.2-1 λ λ = 1 (7.1 5 = G(T, P, λn λ = n λ=1 G(T, P, λn λn = nµ(t, P, n, = G(T, P, n λ=1 (7.2-2 ( G/ n T,P = µ µ n T P G(T, P, n = nµ(t, P (7.2-3 64

7.3. Ω 7. (7.1-5 (7.2-3 = dg = d(nµ = ndµ + µdn, = SdT + V dp + µdn, dµ = sdt + vdp (7.2-4 s = S/n v = V/n T, P, µ µ T, P : µ(t, P = c P T T [c P ln(t/t 0 R ln(p/p 0 + s(t 0, P 0 ] v = V/n µ(t, v = c P T T [c V ln(t/t 0 + R ln(v/v 0 + s(t 0, v 0 ] 7.3 Ω F G µ T, V, µ Ω T, V, n F (T, V, n Ω(T, V, µ = F (T, V, n µn (7.3-1 7.2-3 µn = G F G = P V Ω(T, V, µ = P (T, µv (7.3-2 65

7.4. 7. ( Ω µ T,V = n(t, V, µ (7.3-3 7.1-5 1 T, P, µ T, P, n G T, P, µ G nµ G = nµ µ T P 2 T, V, µ Ω = P V (7.3-3 P µ ( ( P V µ T,V ( P µ T = V ( P µ T = n V = 1 v = n v dµ = sdt + vdp dp dµ 7.4 c i(= 1, 2, c n i G i µ i G = G(T, P, n 1, n 2,, n c (7.4-1 µ i = G(T, P, n 1, n 2,, n c n i (7.4-2 G n i (i = 1,, c G(T, P, λn 1, λn 2,, λn c = λg(t, P, n 1, n 2,, n c 66

7.5. 7. λ λ = 1 G(T, P, n 1, n 2,, n c = c n i µ i (T, P, n 1, n 2,, n c (7.4-3 i=1 G G i = n i µ i 7.5 T P V 1 n (1 i i(= 1, 2, c V 2 n (2 i i(= 1, 2, c dn i 1 n (1 i dn i 2 n (2 i + dn i G G = G (1 + G (2 dg (1 = i G (1 i ( dn i = n i i µ (1 i dn i, dg (2 = i G (2 i (dn i = n i i µ (2 i dn i dg = i (µ (2 i µ (1 i dn i (7.5-1 µ (1,2 i 1,2 i dg dg 0 ( 1 µ (1 i 2 µ (2 i dn i > 0 1 2 i µ (1 i µ (2 i dn i < 0 1 2 i µ (1 i < µ (2 i µ (1 i = µ (2 i µ (1 i > µ (2 i i i 67

7.6. 7. 7.6 µ (1, µ (2 T P n n (1 1 n (2 = n n (1 2 G(T, P, n = G (1 (T, P, n (1 + G (2 (T, P, n (2 dn 1 2 dg = G(1 G(2 ( dn + n (1 n (dn = (2 (µ(2 µ (1 dn (7.6-1 µ (1 µ (2 dn > 0 dg < 0 1 2 2 µ (1 µ (2 dn < 0 dg < 0 2 1 1 µ (1 (T, P = µ (2 (T, P (7.6-2 (6.2-3 G (1 (T, P, n = G (2 (T, P, n µ (1 (T, P < µ (2 (T, P µ (1 (T, P = µ (2 (T, P µ (1 (T, P > µ (2 (T, P 7.7 T P µ (T, P = µ (T, P T T P T P T 68

7.7. 7. µ (T, P = µ (T, P = µ (T, P T P T P (component c p (phase (freedomf α T (α P (α x (α i = n (α i / c j=1 n(α j c c j=1 x(α j = 1 (c 1 (c + 1 p p(c + 1 T (1 = T (2 = = T (α = = T (p (7.7-1 (p 1 P (1 = P (2 = = P (α = = P (p (7.7-2 (p 1 i µ (1 i = µ (2 i = = µ (α i = = µ (p i, (i = 1, 2,, c (7.7-3 c c(p 1 (c + 2(p 1 f = p(c + 1 (p 1(c + 2 = c p + 2 (7.7-4 (c = 1 (p = 3 (f = 0 69

7.8. 7. (p = 2 f = 1 (p = 1 f = 2 T P 7.8 A B A n A B n B T P x A = n A /(n A + n B x B = n B /(n A + n B P A = x A P, P B = x B P A T P A 7.2 µ A (T, P A = µ 0 A(T, P + RT ln(p A /P = µ 0 A(T, P + RT ln x A (7.8-1 µ 0 A (T, P T P A B µ B (T, P B = µ 0 B(T, P + RT ln x B (7.8-2 7.4 G(T, P, n A, n B = n A µ A (T, P A + n B µ B (T, P B = n A µ 0 A(T, P + n B µ 0 B(T, P + RT (n A ln x A + n B ln x B = G A (T, P + G B (T, P T S (7.8-3 A B 4 S = nr(x A ln x A + x B ln x B (7.8-4 x B 1 ln x A = ln(1 x B x B µ A µ 0 A(T, P RT x B, µ B = µ 0 B(T, P + RT ln x B (7.8-5 70

7.9. 7. 7.9 2H 2 + O 2 = 2H 2 O (7.9-1 (ingredients X i (products Y p ν i ν p ν i X i = p i ν p Y p (7.9-2 (7.9-1 X 1 = H 2, X 2 = O 2, Y 1 = H 2 O ν 1 = 2, ν 2 = 1, ν 1 = 2 δn i δn p δr δn i = ν i δr, δn p = ν p δr (7.9-3 δg = G r δr = G δn r = ( n r=i,p r i µ i ν i + p µ p ν p δr = Aδr (7.9-4 A = i µ i ν i p µ p ν p (7.9-5 ν i µ i = p i ν p µ p (7.9-6 A = 0 (4.5-4 x r = n r /n (7.8-1 µ r (x r : T, P = µ 0 r(t, P + RT ln x r, (r = i, p (7.9-7 71

7.9. 7. (7.8-1(7.8-2 n = i n i + p n p µ 0 (7.9-6 p (x p ν p i (x = K(P, T (7.9-8 i ν i { i K(P, T = exp ν iµ 0 i (T, P p ν pµ 0 p(t, P } RT (7.9-9 7.2 µ 0 (P, T = RT ln(p/p 0 + f(t (7.9-10 K(P, T = e ( i ν i p νp ln(p/p 0 K(P 0, T = ( P P 0 νk(p0, T (7.9-11 ν = p ν p i ν i (7.9-12 ν < 0 T P 0 (P > P 0 (7.9-11 ν < 0 (7.9-8 ν > 0 72

7.9. 7. (7.9-9 ln K(T, P T = 1 [ ( µ 0 } ν RT 2 i {T i µ 0 i T P p i ν p {T ( µ 0 }] p µ 0 p T P (7.9-13 i, h i = T s 0 i µ 0 i, Q = p ν ph p (T, P i ν ih i (T, P ln K(T, P T = Q RT 2 (7.9-14 Q > 0 Q > 0 Q < 0 Q Q > 0 (7.9-14 K K (7.9-8 Q < 0 ln K K (7.9-8 A ν A = 1, ν A + A A + + e = 1, ν e = 1 A A + 7.2 c P 1 h µ 0 (T, P = h T [ 5 2 R ln(t/t 0 R ln(p/p 0 + s(t 0, P 0 ] c P = 5R/2 ( I = 73

7.10. 7. h A + + h e h A ν i µ 0 i (T, P ν p µ 0 p(t, P = µ 0 A (µ 0 A + + µ0 e p i = I + RT [ 5 2 ln(t/t 0 ln(p/p 0 + s(t 0, P 0 /R] (7.9-15 ( K(P, T = A 0 T 5/2 P 1 exp I RT (7.9-16 A 0 T 0, P 0 A n A A + n A + e n e n A + = n e n = n A +n A + x = n e /n n + n e = n(1 + x A A + e (1 x/(1 + x, x/(1 + x, x/(1 + x P = n(1 + xrt [x/(1 + x] 2 (1 x/(1 + x = K(P, T (7.9-17 x 2 ( 1 x = A(T 3/2 /n exp I RT (7.9-18 A (x 1 n 0 x 1 7.10 7.10.1 (7.4-3 SdT V dp + i n i dµ i (T, P, x i = 0 (7.10-1 i n i n = i n i x i = n i /n ( : 74

7.10. 7. 7.10.2 A,B A B n A n B B x = n B /(n A + n B g(x = G n A + n B = (1 xµ A (T, P, x + xµ B (T, P, x (7.10-2 A B µ A (x = g(x x g(x x, µ B(x = g(x + (1 x g(x x (7.10-3 x g x = 0 y µ A (x x = 1 µ B (x : (7.10-1 (dt = dp = 0 (1 x µ A(T, P, x x + x µ B(T, P, x x = 0 (7.10-4 (7.10-2 x (7.10-4 g x = µ B(x µ A (x (7.10-5 (7.10-2 (7.10-5 (7.10-3 XY X = x g(x X = x Y = g(x + g(x (X x (7.10-6 x X = 0 Y = µ A (x, X = 1 Y = µ B (x 7.10.3 7.7 (c = 2 (p = 1 f = c p + 2 = 3 T P µ B (p = 2 f = 2 T P µ B 75

7.10. 7. x T x ( g (1 (T, x 2( g (2 (T, x T x (1 1 x (2 µ (1 A (T, x(1 = µ (2 A (T, x(2, µ (1 B (T, x(1 = µ (2 B (T, x(2 (7.10-7 x (1 x (2 g (1 g (2 : (7.10-3 µ (1 A (T, x(1 = g (1 (T, x (1 x (1 g(1 (T, x (1 x (1 = µ (2 A (T, x(2 = g (2 (T, x (2 x (2 g(2 (T, x (2 x (2, µ (1 B (T, x(1 = g (1 + (1 x (1 g(1 x (1 = µ(2 B (T, x(2 = g (2 + (1 x (2 g(2 (7.10-8 x (2 g (1 (x (1 / x (1 = g (2 (x (2 / x (2 (7.10-8 g (1 (T, x (1 g (2 (T, x (2 x (1 x (2 = g(1 (T, x (1 x (1 = g(2 (T, x (2 x (2 (7.10-9 (x (1, g (1 (x (1 (x (2, g (2 (x (2 (p = 3 T, µ B (T x x 7.10.4 A B (7.8-5 µ A,L µ 0 A,L(T, P RT x L, µ B,L = µ 0 B,L(T, P + RT ln x L (7.10-10 76

7.10. 7. x L B µ 0 A,L,µ0 B,L A B (7.10-10 L S (1 A µ A,L (T, x L = µ A,S (T, x S T A δt = (x S x L RT T A L A (7.10-11 L A A x S = 0 (2 µ B,L (T, x L = µ B,S (T, x S k x S x L = k(t, P = e µ B/RT (7.10-12 : T B,L B = T B (s 0 B,L s0 B,S B µ B (T, P = µ 0 B,L (T, P µ0 B,S (T, P L B(T/T B 1 (1 µ A,L (T, x L = µ A,S (T, x S (7.10-10 µ 0 A,L(T RT x L = µ 0 A,S(T RT x S (7.10-13 A T A µ 0 A,L (T A = µ 0 A,S (T A δt = T T A (7.10-12 µ 0 A,L(T A + µ0 A,L T δt RT x L = µ 0 A,S(T A + µ0 A,S T δt RT x S (7.10-14 s = µ/ T δt = RT (x L x S s S s L = RT T A L (x S x L (7.10-15 L = T (s L s S 77

7.10. 7. (2 B µ 0 B,L(T + RT ln x L = µ 0 B,S(T + RT ln x S (7.10-16 T B T B δt = T T B µ 0 B,L(T B + µ0 B,L T δt + RT ln x L = µ 0 B,S(T B + µ0 B,S T δt + RT ln x S (7.10-17 s B L B 7.10.5 n T V (1 P n V ρ = n/v P (T, V, n = P (T, ρ ( P/ V T,n ( P/ n T,V (2 µ ( µ/ V T,n ( µ/ n T,V (3 ( µ/ V T,n ( P/ n T,V (4 µ n ( n/ µ T,V κ T = 1 V ( n µ T,V ( V P T,n = n2 κ T V 78

7.10. 7. 7.10.6 CaCl 2 f dl d W = fdl T µ dn dl du = d Q + d W + µdn = T ds + fdl + µdn (7.10-18 µ H µ L (1 l 1 n 1 f 1 µ H l 2 n 2 (2 n 2 l 3 (3 µ L n 1 l 4 (4 n 1 l 1 f 1 (1 W = (µ H µ L (n 2 n 1 (7.10-19 (2 η η (3 79

7.10. 7. ( (1 (7.10-18 du = T ds + fdl + µdn (7.10-20 (2 du = ds = 0 W = fdl W = µdn = µ H (n 2 n 1 + µ L (n 1 n 2 = (µ H µ L (n 2 n 1 > 0 (2,(4 n η = µ H > µ L 1 W µ H (n 2 n 1 = µ H µ L µ H = 1 µ L µ H (7.10-21 (3 d Q < T ds T ds > d Q = du f dl µdn f W = f dl < µdn = W 80

81 8 8.1 V H M Maxwell roth = i + D t, (8.1-1 rote = B t, (8.1-2 divb = 0 (8.1-3 ρ = 0 D = ϵ 0 E divd = ρ (8.1-4 M(r, t δt δm(r, t B(r, t/ t H E i(r, t δt (i Edr (8.1-5 ( ( d W = δt (i Edr = δt (roth Edr = δt div(h Edr δt (H rotedr = δt (H E ds + δt (H B dr (8.1-6 t

8.2. 8. δt δb(r, t d W = (H dbdr (8.1-7 du = T ds P dv + (H dbdr (8.1-8 P dv H B = µ 0 H + M M du = T ds + (H dmdr (8.1-9 8.2 U S B M F U F = U T S (8.2-1 df (T, M = SdT + (H dmdr (8.2-2 G = F (H(r M(rdr (8.2-3 dg(t, H = SdT (M dhdr (8.2-4 82

8.3. 8. M r F = f(t, M(rdr, G = g(t, H(rdr (8.2-5 g(t, H(r = f(t, M(r H M (8.2-6 f M = H, g H = M (8.2-7 8.3 P V = nrt M = χ(t H (8.3-1 H M χ χ(t = C T (8.3-2 f f M = H = 1 χ M (8.3-3 f(t, M = f 0 (T + 1 2χ M 2 (8.3-4 83

8.4. 8. 8.4 f M f(t, M = f 0 (T + a 2 M 2 + b 4 M 4 (8.4-1 8.2-7 H = f(t, M M = (a + bm 2 M (8.4-2 H M M H = 0 M = 0 a b M = a/b (8.4-3 a T c a T c 1/2 a = α(t T c (8.4-4 M = (α/b(t c T (8.4-5 f s(t = f T = f 0 T + α 2 M 2 { s0 (T for T > T c = s 0 (T α2 (T (8.4-6 2b c T for T < T c 84

8.4. 8. T c c p (T = T s T { c0 (T for T > T c = c 0 (T + α2 2b T for T < T c (8.4-7 c 0 = T s 0 / T T c H T c H = α(t T c M + bm 3 (8.4-8 M χ(t = C T T c (8.4-9 T c C 85

86 9 9.1 x(t S Ṡ = ds/dt x ẋ = dx/dt Ṡ ẋ ẋ ẋ = 0 x S ẋ ẋ ds dt = 1 ( dx 2 0 (9.1-1 L dt L S x S x ds/dt = ( S/ x(dx/dt (9.1-1 dx dt = L S (9.1-2 x (9.1-2 x 9.2

9.3. 9. a i n i U i S i T i P i µ i n i µ i = G i = U i T i S i + P i a 3 (9.2-1 n i dµ i + µ i dn i = du i T i ds i S i dt i + a 3 dp i (9.2-2 (7.2-4 n i dµ i = S i dt i + a 3 dp i (9.2-3 µ i dn i = du i T i ds i, ds i = 1 T i du i µ i T i dn i (9.2-4 ( Si U i n i = 1 T i, ( Si n i U i = µ i T i (9.2-5 S tot = i S i, U tot = i U i, n tot = i n i (9.2-6 9.3 1 U 1 2 U 2 1 du 1 2 du 2 = du 1 87

9.3. 9. (9.1-2 U 1 du 1 dt = L ( S1 (U 1 + S 2(U 2 du 2 U 1 U 2 du 1 = L ( 1 1 = J1 U (9.3-1 T 1 T 2 L (8.2-5 (9.3-1 X U 1 = S 1 / U 1 = 1/T 1 (9.3-2 1 (9.3-1 1 T 1 T 2 1 J1 U = L(X 2 U X1 U (9.3-3 1 2 du 1 + du 2 = 0 ds tot dt = S 1 du 1 U 1 dt + S 2 du 2 U 2 dt = ( 1 T 1 1 T 2 du 1 dt = L( 1 T 1 1 T 2 2 0 (9.3-4 i U i i 1 i + 1 i i + 1 (9.3-1 J U i = L i,i+1 ( 1 T i 1 T i+1 (9.3-5 L i,i+1 i i+1 i 1 i Ji 1 U = L i 1,i (T 1 1 i 1 Ti i U i du i dt = J U i 1 J U i (9.3-6 88

9.3. 9. x = ai u(x = U i /a 3 a 0 J U i a 2 j u (x = J U i /a 2 (9.3-6 u(x, t t = U(x, t/a3 t = a 3 a 2 (j u (x a j u (x = a 1 j u(x x ( a = j u(x x (9.3-7 (9.3-5 j u (x = J U i /a 2 ( j u = Ji U /a 2 = a 2 1 L(x T (x + a 1 T (x T (x = κ(x x = L(x ( 1 = L(x x T (x T 2 (x T (x x (9.3-8 L = L/a (9.3-8 j u κ = L/T 2 (> 0 (9.3-9 (9.3-8 T/ x κ X U (9.3-8 = 1/T (9.3-1 j u (x = L x( 1 T (9.3-10 r j u (r, t = κ T (r, t (9.3-11 u t + j u = 0 (9.3-12 C u(r = CT (r T t = D T 2 T (r (9.3-13 D T = κ/c (9.3-14 89

9.4. 9. S U δs = ( S/ UδU = δu/t δs = δu/t j s = j u /T s/ t = 1 T u t = 1 T ( j u = T 1 j u = (j u /T + j u (T 1 (9.3-15 s t + j s = j u 1 T (9.3-16 j s j u = κ T = L (1/T ( 1 ( T 2 j u = L( 1/T 2 = κ 0 (9.3-17 T T T 9.4 1 n 1 2 n 2 1 2 n = n 1 + n 2 δn 1 2 n 2 δn 2 = δn 1 ( S1 δs tot = S 2 δn 1 = µ 1 µ 2 δn 1 (9.4-1 n 1 n 2 T (8.2-5 (8.1-2 n 1 dn 1 dt = L T (µ 1 µ 2 = J n 1 (9.4-2 7 1 X n = S 1 / n 1 = µ/t 90

9.4. 9. 1 2 J n 1 = L(X n 2 X n 1 (9.4-3 L T 2 µ 2 µ 1 1 n 1 µ 1 n 1 µ n 1 λ λ L n 1 i n i i 1 i + 1 i i + 1 J n i = L i,i+1 T (µ i+1 µ i = L i,i+1 (X n i+1 X n i (9.4-4 L i,i+1 i i + 1 i (i-1 (i+1 dn i dt = J n i 1 J n i (9.4-5 a 0 x = ai c(x = n i /a 3 j c (x, t = J n i /a 2 j c (x, t = a 2 J n i = L(x a 2 T µ(x [µ(x + a µ(x] = L(x T x = D c(x c x (9.4-6 L = L/a D c (x = L(x µ T c (9.4-7 m 2 s 1 c µ/ c > 0 L T D c > 0 j c X c = µ/t (8.4-6 j c (x = L ( µ (9.4-8 x T 91

9.5. 9. c(x c t = a 3 n(x, t t (9.4-6 (9.4-9 = a 3 a 2 (j c (x a j c (x = j c x c t = x ( D c c x (9.4-9 (9.4-10 (1 (8.4-9 c(r, t t + j c (r, t = 0 (9.4-11 (8.4-6 j c (r, t = D c c(r, t (9.4-12 (8.4-10 c(r, t t = D c 2 c(r, t (9.4-13 D c 7.2 v c v = 1/c c = n/a 3 µ(t, c = RT ln c + A(T (9.4-14 (9.4-7 D c (x = RL/c L c c D c : 9.5 q x ρ e (x = qc(x E ϕ(x = Ex ρ e (xϕ(x = qc(xϕ(x 92

9.6. 9. µ µ 0 µ(x = µ 0 (c + qϕ(x (9.5-1 ρ e j e j c j e = qj c (9.4-6 j c (x, t = L(x µ(x T x (9.5-2 c ρ e ϕ j e (x, t = qj c = ql(x µ(x T x = L(x ϕ(x q2 T x = q2 L(x E T = σe (9.5-3 j e E D c σ = q 2 L T = q2 µ 0 / c (9.5-4 µ 0 = RT ln c σ = cq2 D c RT (9.5-5 9.6 x i x = (x 1, x 2, x (8.1-2 dx dt = L xs 93

9.7. 9. dx i dt = j L ij S x j (9.6-1 L x ds dt = i S x i dx i dt = ij L ij S x i S x j (9.6-2 L X (X LX L ij = L ji (8.6-1 9.7 8.3 8.4 1 2 1 2 1 2 U 1 U 2 n 1 n 2 U 1 + U 2 = U = n 1 + n 2 = n = S t (U 1, n 1 = S 1 (U 1, n 1 + S 2 (U U 1, n n 1 (9.7-1 S t = S 1 S 2 = µ 1 + µ 2 = X1 n X2 n n 1 n 1 n 2 T 1 T 2 S t = S 1 S 2 = 1 1 = X1 U X2 U (9.7-2 U 1 U 1 U 2 T 1 T 2 X n = S/ n = µ/t X U = S/ U = 1/T (9.6-1 1 n 1 U 1 dn 1 dt = L nn (X1 n X2 n + L nu (X1 U X2 U = J1 n du 1 = dt L Un (X1 n X2 n + L UU (X1 U X2 U = J1 U (9.7-3 94

9.7. 9. i (i+1 Ji n = L nn (Xi+1 n Xi n + L nu (Xi+1 U Xi U, Ji U = L Un (Xi+1 n Xi n + L UU (Xi+1 U Xi U (9.7-4 i X n i = µ i T i, X U i = 1 T i (9.7-5 i n i U i (i-1 (i+1 dn i dt = J n i 1 J n i, du i dt = J U i 1 J U i, (9.7-6 a a 2 j c,u = J n,u /a 2 c = n/a 3 c(x, t t = a 3 n(x, t t = j c(x x = a 3 a 2 [j c (x a j c (x] = a 1 j c(x x ( a (9.7-7 u = U/a 3 u(x, t t = a 3 U(x, t t = j u(x x = a 3 a 2 [j u (x a j u (x] = a 1 j u(x x ( a (9.7-8 8.3 8.4 j c j u j c (x = Ji n /a 2 = a 2 [ L nn (X n (x + a X n (x + L nu (X U (x + a X U (x] X n = L cc x + L X U cu x = L ( cc µ 1 + L cu, x T x( T j u (x = Ji U /a 2 = a 2 [ L Un (X n (x + a X n (x + L UU (X U (x + a X U (x] X n = L uc x + L X U uu x = L ( uc µ 1 + L uu (9.7-9 x T x( T 95

9.8. 9. X n = µ/t X U = 1/T L cc = L nn /a L cu = L nu /a L uu = L UU /a j c j u div= c(r, t + j c (r, t = 0 t u(r, t + j u (r, t = 0 (9.7-10 t j c = L cc µ T j u = L uc µ T + L cu 1 T + L uu 1 T (9.7-11 L cu = L uc 9.8 q c(r, t ρ(r, t = qc j c (r, t j e (r, t = qj c (r, t (9.7-10 ρ/ t + j e = 0 (9.8-1 E r ϕ(r E = ϕ(r µ µ = µ 0 (c + qϕ(r 9.7-11 j e q = L cc (µ 0 + qϕ + L cu 1 T T j u = L uc (µ 0 + qϕ + L uu 1 T T (9.8-2 T = 0, µ 0 (c = 0 E = ϕ j e = σe (9.8-2 σ = q2 L cc T (9.8-3 96

9.9. 9. µ 0 (c 8.4 j c = j e q = L cc T µ 0 = L cc T µ 0 c (9.8-4 c D c = (L cc /T ( µ 0 / c j c = D c c L cc D c = σ q 2 µ 0 (c, T c (9.8-5 D c σ µ 0 (c c D c 9.9 (j e = 0 (8.8-1 j e = 0 (µ/t = (L cu /L cc (1/T (8.8-1 j u = L ccl uu L cu L uc L cc 1 T = κ T (9.9-1 κ κ L ccl uu L cu L uc L cc T 2 (9.9-2 (8.3-9 (L uc 9.10 σ κ (9.8-2 L cc = σt/q 2 j e = q σt [ 1 ( 1 q 2 T µ µ + L cu 1 ] ( = σ T L cc T q [ (µ 0 + qϕ + S T ] (9.10-1a j u = L [ uc je 1 ] ( 1 cu ( L cc q L + L uu = (T S + µ 0 + qϕ j e T T q κ T (9.10-1b 97

9.11. 9. (9.10-1b (9.10-1a ( µ/t j e (1/T L uc = L cu S S 1 T (L uc L cc µ = 1 T (L uc L cc µ 0 (c qϕ (9.10-2 j s 8.3 j u T (8.2-4 j s = T 1 [j u (µ 0 + qϕj e /q] (9.10-3 j e ( T = 0 (8.10-1b j s j s = S j e q (9.10-4 j c = j e /q S (9.10-2 L uc /L cc µ = µ 0 + qϕ T (8.10-1a [µ 0 + q(ϕ + (S/qT ] S/q T K j e j q = T j s (8.10-1b j q = Πj e (9.10-5 Π = T S/q (9.10-6 9.11 ρ/ t = j e = 0 (9.10-1b j e = 0 (9.10-1a u t = j u = j e q (T S + µ 0 + qϕ + (κ T = j ( e q q σ j e T S + (κ T = j2 e σ T j e q ds dt T + (κ T (9.11-1 98

9.12. 9. u t = (κ T + j2 e σ τj e T (9.11-2 S T τ T q ds dt (9.11-3 (9.11-2 ( j e T ( 1854 9.12 2 1834 A, B T = 0 8.10 j s = (S/qj e j e A B AB js B js A A B Π AB T q (S B S A = Π B Π A (9.12-1 T (j B S j A S = Π AB j e (9.12-2 Π = T S/q (8.10-6 99

9.13. 9. 9.13 A, B 2 1 2 A 1 2 0 3 1821 1 T 1,T 2 (j e = 0 (9.10-1a (µ 0 + qϕ = S T 0 3 dl 3 (µ 0 + qϕ dl = 3 0 0 3 0 S T dl (9.13-1 µ 0 dl = µ 0 (3 µ 0 (0 (9.13-2 0 3 A T 0 = T 3 µ 0 q(ϕ(3 ϕ(0 = qv AB (9.13-3 3 0 S T dl = = 2 1 1 0 [S T ] A dl + (S T B dl 2 1 2 1 [S T ] B dl + (S T A dl = 3 2 T2 [S T ] A dl T 1 (S B S A dt (9.13-4 0 3 T = 0 1 T 1,T 2 0,3 V AB V AB = T2 T 1 Π AB dt (9.13-5 T Π AB 100

9.14. 9. 9.14 9.14.1 A,B Ludwig-Soret 8.2 a i (8.2-4 ds = du µ A dn A µ B dn B (9.14-1 ρ = m A n A + m B n B ds = du µdn A (9.14-2 (8.2-4 m A, m B µ = µ A m A m B µ B (9.14-3 (1 (8.10-1 j A A j A = α µ β T α, β j u = (µ + βt α j A κ T (9.14-4 ( : L AA A L Au A α = L AA /T, β = (L Au µl AA /T 2 (2 µ c(= n A /a 3 T (8.14-4 j A = D c [ c + k T T T ] ( µ j u = [k T + µ]j A κ T (9.14-5 c T D c k T D c (8.4-7 k T ( : D c = α( µ/ c T, k T = T [α( µ/ T c + β] 101

9.14. 9. (3 T = G ( : j A = 0 c = k T G/T 102