6 6.1 B A: Γ d Q S(B) S(A) = S (6.1) T (e) Γ (6.2) : Γ B A R (reversible) 6-1

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6 6.1 B A: Γ d Q S(B) S(A) = S (6.1) (e) Γ (6.2) : Γ B A R (reversible) 6-1

(e) = Clausius 0 = B A: Γ B A: Γ d Q A + d Q (e) B: R d Q + S(A) S(B) (6.3) (e) // 6.2 B A: Γ d Q S(B) S(A) = S (6.4) (e) Γ (6.5) ( ) U + W = Q (e) S (6.6) 6-2

6.2.1 ( ) Q = 0 0 S 1 S =maximum 6.2.2 (V, ) = : Helmholtz W = 0 = 0 = (e) ( ) Q = U S = ( S) (6.7) (U S) 0 (6.8) Helmholtz F (V, ) F (V, ) U S = Helmholtz free energy (6.9) : F 0 : F =minimum F (V, ) F 1 homson Boltzmann H 6-3

2 df = du d( S) = ( ds P dv ) d S ds }{{} du = Sd P dv (6.10) ( ) F P = V ( ) F, S = V (6.11) F = dv (x)/dx Helmholtz Helmholtz V W W + (U S) = W + F 0 W F (6.12) X X 0 F = F [X 0 ] F [X] W F [X] F [X 0 ] F [X 0 ] F ( F F ) 6.1 U F U = 2 ( ) F V (6.13) 2 F (V, ) V, F = U S V, F 6-4

S = ( F/ ) V ( ) F U = F + S = F = 2 V ( ) F V (6.14) 6.2.3 (P, ) = : Gibbs P ( ) Q = U + P V = (U + P V ) S = ( S) (6.15) Gibbs G(P, ) G(P, ) U + P V S = F + P V = Gibbs free energy (6.16) : G 0 : G =minimum G (P, ) G dg = df + d(p V ) = Sd P dv + P dv + V dp = Sd + V dp (6.17) ( ) ( ) G G = S, = V (6.18) P P G 6-5

m h P = mg/a A G = F + P V = Helmholtz + 6.2 1 P 1 P 2 ( P 1 < P 2 ) Helmholotz Gibbs P V = R (P V ) = 0 F, G F = G = U S (6.19) U = C V + U 0 (6.20) S = C V ln + R ln V + S 0 = C V ln R ln P + ln R + S 0 (6.21) U F = G = S = R ln P = R (ln P 2 ln P 1 ) = R ln P 2 P 1 (6.22) 6.2.4 (S, V ) = : S = 0, W = 0 ( ) U 0 U S V U = U(S, V ) U =minimum 6-6

S, V U ( U ) S V du = ds P dv (6.23) =, ( ) U = P (6.24) V S 6.2.5 (S, P ) = : S = 0 U + P V 0 (6.25) P = P V = (P V ) (U + P V ) 0 (enthalpy) 3 H(S, P ) H(S, P ) U + P V = Enthalpy (6.26) : H 0 : H = minimum H dh = d(u + P V ) = du + V dp + P dv = ( ds P dv ) + V dp + P dv = ds + V dp dh = ds + V dp (6.27) 3 en+thalpy(= ) 6-7

( ) ( ) H H =, = V (6.28) S P P S : 1. P = dh = du + P dv = d Q H d 0 Q C P H H (, P ) ( ) H d Q = dh = d + P ( ) ( ) Q H C P = = P P ( ) H dp P (6.29) 2. F = P A h F = mg A = mgh = F h P V = P Ah = F h P V H = U + P V = + (6.30) 6-8

6.3 ( ) ( ) H V = + V (6.31) P P P U = H P V ) ds = du + P dv = dh V dp (6.32) (P, ) ds = 1 ( ) H d + 1 (( ) H P P V ds ( ( ) ) 1 H = ( (( ) )) 1 H V P P 0 = 1 (( ) H 2 P P ) V 1 ( ) V P // ) dp (6.33) (6.34) (6.35) 6.3 : [1] f {X, Y } df (V, ) = Sd P dv (6.36) dg(p, ) = Sd + V dp (6.37) du(s, V ) = ds P dv (6.38) dh(s, P ) = ds + V dp (6.39) 6-9

[2] f f(x, Y ) : U (S, V ) (, V ) (i) U = U(S, V ) : du = ds P dv ( ) U = (S, V ), S V ( ) U = P (S, V ) (6.40) V S S, V P S, V S, P, V U = U(, V ) = (S, V ) S = S(V, ) U = U(S(V, ), V ) (ii) U = U(V, ) : ( ) ( ) U U du = d + dv (6.41) V V P (V, ) ( ) du = ds P dv S P (6.41) dv ( U/ V ) P P ( U/ V ) S (iii) U = U(V, ) P (V, )( ) : : S(V, ), V S ds = 1 (du + P dv ) = 1 [( ) (( ) ) ] U U d + + P dv (6.42) V V 6-10

( U ), ( ) U V V P (, V ) ( ) S(V, ) P (V, ) : d Q = du + P dv (e) ds (6.43) : : P, V, S, U ) ( ) U, S, P, V, 5 U, S, V, V, U S V, du( ) = C V d 6-11

: (Legendre) : U, F, G, H ( ) (X, x) (X, x) = (P, V ), (S, ) (6.44) ( ) A(x, y, z,...) x X B(X, y, z,...) Legendre A da(x, y, z,...) = Xdx + Y dy + Zdz + (6.45) A (x, X) Legendre B A Xx (6.46) db = da d(xx) = Xdx + Y dy + Zdz + (Xdx + xdx) = xdx + Y dy + Zdz + (6.47) B (X, y, z,...) Legendre : U Legendre du(s, V ) = ds P dv (6.48) (S, ) Legendre Helmholtz free energy F F = U S (6.49) df (, V ) = du d( S) = Sd P dv (6.50) ( P, V ) Legendre Gibbs free energy G G = F ( P )V = F + P V (6.51) dg(, P ) = df + d(p V ) = Sd + V dp (6.52) 6-12

U ( P, V ) Legendre H = U ( P )V = U + P V (6.53) dh(s, P ) = du + d(p V ) = ds + V dp (6.54) 6.4 P, V, P, V, = ds = P dv : (1) E chem = i µ i dn i, i = (6.55) µ i = N i = (2) E mag = H d M, E ele = E d P, M =, H = (6.56) P =, E = (6.57) (3) E general = Xdx, X =, dx = (6.58) : X = x = 6-13

6.4.1 f = k( )x, k( ) = k 0 + k 1 (6.59) x V ( ) f P ( ) fdx P dv (6.59) x C x (C 0 ) ( : U ) = ( ) P P V V ( ) ( ) U f = f = k 1 x + (k 0 + k 1 )x = k 0 x (6.60) x x ( ) U C 0 = (6.61) du = ( ) U d + x x ( ) U dx = C 0 d + k 0 xdx (6.62) x U(x, ) = 1 2 k 0x 2 + C 0 + const. (6.63) 6-14

= 0 C 0 : ds = du + fdx = C 0 d + k 0 xdx + ( k 0 x k 1 x)dx = C 0 d k 1 xdx ds = C 0 d k 1xdx (6.64) S = C 0 ln 1 2 k 1x 2 + const. (6.65) ( ) F : U, S x, Helmholtz F (x, ) F = U S = 1 ( 2 k 0x 2 + C 0 + const. C 0 ln 1 ) 2 k 1x 2 + const. F = 1 2 k( )x2 C 0 ln + C 0 + a + b (6.66) : F = U S f = F/ x ( ) ( ) U S f = + = k 0 x k 1 x (6.67) x x 6-15

6.4 0 x x V S (6.65) C 0 ln = 1k 2 1x 2 + const. = 0 exp ( ) k1 x 2 2C 0 0 : x = 0 (6.68) 6.4.2 H M χ = M/ H Curie χ = C, C = const > 0 M = CH (6.69) ( ) : dm HdM HdM M V ( ) H P ( ) HdM P dv Curie χ = M H V P = V (6.70) : 6-16

d Q = ds = du HdM (6.71) M = CH (6.72) P = f(v ) U = U( ) d Q = ds = C M ( )d HdM, C M ( ) ( ) U M (6.73) 6.5 : 0 H 0 d = 0 Curie d Q = HdM = C HdH (6.74) Q = C H0 0 HdH = C 2 H2 0 (6.75) Q(> 0) 6.6 H, HdM M H, M S H, ds = 1 (C M ( )d C ) MdM = C M ( ) d 1 C MdM CM ( ) S = d 1 2C M 2 + const. CM ( ) = d C ( ) 2 H + const. (6.76) 2 6-17

6.7 : 0 H 0 0 U = a 4 (a = const > 0) 4 C M ( ) = ( U/ ) M = 4a 3 S = 4a 2 d C ( ) 2 H + const. 2 = 4a 3 3 C ( ) 2 H + const. (6.77) 2 4a 3 3 0 C 2 ( H0 0 ) 2 = 4a 3 3 3 3 0 = 3C 8a ( H0 0 ) 2 (6.78) 4 6-18

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