5 0 1 2 3 (Carnot) (Clausius) 2 5. 1 ( ) ( ) ( ) ( ) 5. 1. 1 (system) 1)
70 5. (isolated system) ( ) E N (closed system) N T (open system) (homogeneous) (heterogeneous) (phase) (phase boundary) (grain) (grain boundary) 5. 1. 2 (state quantity) x = (x, y) z = f(x, y) dz = df = f(x, y) x dx + y f(x, y) y dy (5.1) x
5. 1 71 f f dx df(x) = f(x) dx (5.2) 5.1 f(x) dx = C 1 f(x) dx = f(x 1 ) f(x 0 ) C 2 (5.3) C 2 f(x) dx + C 1 f(x) dx = C 2 f(x) dx = 0 (5.4) 0 x 1 x 0 5.1 (state variable) (state function)
72 5. 3 T, N 3 ( N ) (extensive(additive) variable) E S (intensive variable) P T 5. 1. 3 (irreversible) (reversible)
5. 2 0 73 ( ) 5. 2 0 1 0 θ (empirical temperature) 2 ds = δq/t (thermodynamical absolute temperature) (ideal-gas temperature) 5. 3 1 1 de δq δw de = δq δw (5.5) 1 ( de)
74 5. ( δq) (δw ) δ δw rev = P d δq rev = C dt 1 de = 0 (5.6) de 5. 4 2 5. 4. 1 3 3 2 D D P = NkT = 2 D E (5.7) C v = E T = D Nk (5.8) v 2
5. 4 2 75 1 δq = 0 δw rev = P d de = δw rev = P d (5.9) d (d < 0) de = C v dt C v dt = P d (5.10) P (, T ) C v dt = NkT d (5.11) T (T 0, 0 ) (T, ) T T 0 C v dt Nk T = 0 d C v Nk ln T T 0 = ln 0 (5.12) C v = D/2 Nk ( ) D/2 T = 0 T 0 (5.13) T P P T ( ) (D+2)/2 T = P T 0 P 0 (5.14) ( ) (D+2)/D = P P 0 (5.15) 0 1) 1)
76 5. 5. 4. 2 (Carnot) 2 1, 2) 2 Carnot 5.2 (1 2, 3 4) (2 3, 4 1) 1 P=2/ P 2 P=2/ 2 4 3 5.2 T H T L D P = NkT = (2/D)E ( Q) ( W ) ( E) 5.1 5.1 step E/Nk Q/Nk W/Nk 1 2 0 +T H ln( 2/ 1) +T H ln( 2/ 1) 2 3 D 2 (TH TL) 0 + D 2 (TH TL) 3 4 0 T L ln( 3/ 4) T L ln( 3/ 4) 4 1 + D 2 (TH TL) 0 D 2 (TH TL) total 0 (T H T L) ln( 2/ 1) (T H T L) ln( 2/ 1) E = D/2 NkT de = D/2 NkdT E = D/2 P de = D/2 (P d + dp ) de = P d
5. 4 2 77 5. 4. 3 (isothermal) f NkT E = 0 = Q W = Q i d = Q NkT ln f (5.16) i (adiabatic) Q E = D Nk T = Q W = W (5.17) 2 T (5.13) 1 4 3 2 = 1 4 = ( ) D/2 TH T L ( ) D/2 TL T H 1 3 = 2 4 (5.18) 5. 4. 4 2 W 12 + W 23 + W 34 + W 41 Q 12 = T H T L T H (5.19) ( ) 2
78 5. 5.1 Q 12 T H + Q 34 T L = 0 (5.20) δq/t δq T = 0 (5.21) P δq/t T δq δq/t (entropy)s ds = δq rev T 1 δq rev S 1 S 0 = T 0 (5.22) δq irr δq rev δq irr < δq rev = T ds (5.23)! δq rev ) 2 5. 4. 5 P T S 1
5. 5 : 79 de = δq rev + δw rev = T ds P d (5.24) (T, P ) (S, ) 1 5.3 P T S 1 W = P d = T ds (5.25) P T S Q = Q H Q L = T H S T L S (5.26) 1 1 T H (d) (b) P 2 T 4 3 T L (c) S (a) 5.3 P T S 5. 5 : 5. 5. 1 1 de = T ds P d
80 5. ( ) E S ( ) E S = T = P E(S, ) T, de = E T dt + E d T dt, d S P a. d(t S) = T (ds) + S(dT ) d(e T S) = SdT P d S, T, E T S F (Helmholts) P d(e + P ) = T ds + dp. E + P H d(e T S + P ) = SdT + dp. E T S + P G (Gibbs) F, H, G E 1
5. 5 : 81 b. µdn µ 1 µdn E de = T ds P d + µdn (5.27) E N = µ. (5.28) S, 5. 5. 2 - T, P, µ E, S,, N d(e T S + P µn) = SdT + dp Ndµ (5.29) E(S,, N) n n ne(s,, N) = E(nS, n, nn) E (S,, N) = T (ns) n P (n ) + µ (nn) n n E T = E (ns, n, nn) (ns) n,nn E = T S P + µn (5.30) (5.29) SdT + dp Ndµ = 0
82 5. T, P, µ - (Gibbs Duhem) 1, p.59) (5.30) T S P G = E T S + P = µn 5. 5. 3 Maxwell E(S,, N) de (S,, N) = 0 (5.31) de (S,, N) E 2 E S = 2 E S 2 ( ) E = ( ) E S S ( ) ( ) S ( P ) = (T ) S (5.32) F, G, H ( ) ( ) ( ) ( ) ( ) ( ) S P S T =, =, = T T P T T P S P P S (5.33) S (Maxwell) dsdt = dp d 1 N S N
5. 5 : 83 5. 5. 4 4) 1 de = T ds P d T d Maxwell ( ) ( ) ( ) E S P = T P = T P (5.34) T T T N T θ ( E/ ) θ,n ( ) ( ) P P dθ(t ) = T,N θ,n dt (5.35) ( ) ( E P = T θ,n θ ) d ln T dθ = ( P θ ),N dθ(t ) dt P /{ ( E ),N θ,n + P }. (5.36) T 0, θ 0 { } θ ( P / θ),n T = T 0 exp ( E/ ) θ,n + P (θ,, N) θ 0 (5.37) T = f(θ) P 0 θ P lim ( = const.) (5.38) θ 0 P 0 0 P 0 θ
84 5. P = Nkθ (5.39) k (Boltzmann constant) k = 1.3803 10 23 J/K θ E/N = f(θ) (5.40) (5.37) (5.40) ( E/ ) θ,n = 0 (5.39) ( P / θ),n = (N/ ) k = P /θ T = T 0 exp ( θ θ 0 ) ) dθ = T 0 exp (ln θθ0 = T 0 θ. θ θ 0 θ θ 0 = 273.16K T = θ 1) W. L. H. ( 1999) 2) Wm. G. Hoover ( 1999) 3) ( 1993) 4) ( 1981)