i 18 2H 2 + O 2 2H 2 + ( ) 3K
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- しほこ みのしま
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2 i 18 2H 2 + O 2 2H 2 + ( ) 3K
3 ii (2.11)
4 iii
5 iv
6 C 1 C 1 C 1 C 1 C = nr (1.1) n R = /nr (1.1) (1.1)
7 1 2 (1.1) = (1.2a) 1787 α = αt, : C : t C α 1 t ( ) = (αt + 1) = α (t + α 1 ) (1.2b) α α = (1.2a) (1.2b) R = R (t + α 1 ) (1.3) 182 (1.2b) t + α 1 = t (1.4) 1 (mol) C L R = J/mol K n (1.3) R = nr (1.1) (1.4)
8 A B A B A B A B A A 1 B B 1 (cal) 1 1g 14.5 C 15.5 C C
9 1 4 1g 1 C 1 25 C (cal/g) g cal=4.1855J J 1.6 (,, ) f (,, ) = (1.1) [ ( n ) 2 ] + a ( nb) nr = (a b ) (1.5) (1.1) (1.5)
10 N A = /mol 1.8 C (, ) 2 2 C (, ) C (, ) 1 1
11 W W W F S F = S (1.6) x S x x + x x F W = F x = S x = ( S x) (1.7) x ( x dx) d W = d (1.8) d d W C ( 1, 1 ) ( 2, 2 ) (1.8) W = 2 1 d = (, ) 1 1 C (, ) 2 2 (, ) W = = = C I I d d + d II II = d d (II ) C II I II (, ) 1 1 (, ) 2 2
12 y x y f (x, y) f (x, y) f x f (x, y) x lim x f (x + x, y) f (x, y) x (2.1) y x f (x, y)/ x (x, y) +x C 1 A 2.1 C (gradient) x y ( f (x, y) grad f (x, y), x f (x, y) x = ( ) f x y ) f (x, y) y 2 x (2.2) (2.3) f (x, y) = x 2 y f (x, y) x = 2xy, f (x, y) y = x 2
13 A C 1 C C 1 O C s r(s) = (x(s), y(s)) (s s s 1 ) (2.4) r(s) = (cos s, sin s) ( s 2π) (2.4) C s1 f ds f (x(s), y(s)) ds (2.5) C s s = x C f dx x1 x f (x, y(x)) dx 2.3 d f F x (x, y) dx + F y (x, y) dy = F (r) dr, { F (r) (Fx (x, y), F y (x, y)) dr (dx, dy). (2.6) F (r) r d f r r + dr F F (r) r d f dr F d f 2.1 A C 1 C 2 d C 1 C 2 h C 1 C 2 C F dr r+dr r O
14 2 9 d f A B(1, 1) C 1 C 2 1 y B 1. F (r) = (2y, x) (1) C 1 (, ) (1, ) r = (x, ), x 1 dr = (dx, ) F = (2, x) = (, x) d f = dx + x = C 2 A C 1 1 x (1, ) (1, 1) r = (1, y), y 1 dr = (, dy) F = (2y, 1) d f = 2y + 1dy = dy f d f = + C 1 1 dy = 1 (2.7a) (2) C 2 (, ) (, 1) r = (, y), y 1 dr = (, dy) F = (2y, ) d f = F dr = 2y + dy = (, 1) (1, 1) r = (x, 1), x 1 dr = (dx, ) F = (2 1, x) = (2, x) f d f = + C 2 1 (2.8) (2.7a) d f = F dr = 2dx + x = 2dx 2dx = 2 (2.8)
15 F (r) = (2xy, x 2 ) (1) C 1 (, ) (1, ) r = (x, ), x 1 dr = (dx, ) F = (, x 2 ) d f = (1, ) (1, 1) r = (1, y), y 1 dr = (, dy) F = (2y, 1) d f = dy f d f = + C 1 1 dy = 1 (2.9a) (2) C 2 (, ) (, 1) r = (, y), y 1 dr = (, dy) F = (, ) d f = (, 1) (1, 1) r = (x, 1), x 1 dr = (dx, ) F = (2x, x 2 ) d f = 2xdx f d f = + C 2 1 2x dx = 1 (2.1) (2.1) (2.9a) d f C 2.4 B A F (r) dr F x(r) y = F y(r) x (2.11) 1 2
16 F (r) = (2y, x) (2.8) (2.7a) (2.11) F x (r) y = 2, F y (r) x = 1, F x (r) y F y(r) x (2.12a) 2. F (r) = (2xy, x 2 ) (2.1) (2.9a) (2.11) F x (r) y = 2x, F y (r) x = 2x, F x (r) y = F y(r). (2.12b) x (2.11) 1 2 d f F x (r)dx + F y (r)dy F x(r) = F y(r) y x d f d f d f d f d f d f A f (B) = B A d f (2.13) B 2 F (r) = (2xy, x 2 ) (2.13) y d f = 2xydx + x 2 dy (2.14a) (2.12b) A(, ) B(x 1, y 1 ) f (x 1, y 1 ) C t r = (x 1 t, y 1 t), t 1 dr = (x 1 dt, y 1 dt) F = (2x 1 y 1 t 2, x 2 1 t2 ) y 1 C A x 1 d f = F dr = 3x 2 1 y 1t 2 dt B x
17 2 12 B f (x 1, y 1 ) = d f = 3x1 2 y 1 C 1 t 2 dt = x 2 1 y 1 (x, y) f = f (x, y) f (x, y) = x 2 y (2.14b) (2.14a) (2.14b) (1, 1) (2.14b) f (1, 1) = 1 (2.9a) (2.14b) f (x, y) x = 2xy, f (x, y) y = x 2 (2.14a) dx dy d f = F x (x, y)dx + F y (x, y)dy, F y (x, y) x = F x(x, y) y (2.15a) f = f (x, y) f (x, y) f (x, y) F x (x, y) =, F y (x, y) = (2.15b) x y ( ) f (x, y) f (x, y) F = (F x, F y ) f grad f (x, y), x y (2.15a) f 2 f (x, y) x y = 2 f (x, y) y x (2.15c) f (x, y) 2.5 (2.11) (2.11) (i) (ii) (2.11) 2.5.1
18 2 13 ( Fy (F x dx + F y dy) = dxdy x F ) x y C R (2.16) C R y R (region) C (contour) x (2.16) R x y x dxdy F x(x, y) y = = = R x2 x 1 dx Y2 (x) Y 1 (x) dy F x(x, y) y y x2 x 1 dx [ F x (x, Y 2 (x)) F x (x, Y 1 (x)) ] x1 x 2 dxf x (x, Y 2 (x)) + x2 x 1 dxf x (x, Y 1 (x)) C = F x (x, y)dx (2.17a) C (2.16) R y x y = R y2 dxdy F y(x, y) x = y2 y 1 dy X2 (y) X 1 (y) y 1 dy [ F y (X 2 (y), y) F y (X 1 (y), y) ] y2 = dyf y (X 2 (y), y) + y 1 = F y (x, y)dy C y1 y 2 dyf y (X 1 (y), y) dx F y(x, y) x (2.17b) y x=x 1 (y) (2.17a) (2.17b) y 2 y 1 y y=y 2 (x) y=y 1 (x) x 1 x 2 x x=x 2 (y) x
19 x y = Y 1 (x), Y 2 (x) x = x C 4 L C C 1 C 2 L C 1 C 2 C = C 1 + C (2.11) y x C C 1 C C 2 L x (2.16) C C 1 C 2 C 2 C (2.16) (F x dx + F y dy) C 1 (F x dx + F y dy) = C 2 R ( Fy x F ) x y (2.18) (2.18) C ( R F y x = F x y ) ( C 1 = A C 2 C C C 2 C 1 C 2 ) C 1 B 2.6 ( ) z x y ( ) x y z ( ) y = 1 (2.19) z x z = z(x, y) y = y(x, z) z = z(x, y) (x, y) x dz = ( ) z dx + x y ( ) z dy y x ( ) z y x y ( ) z y x
20 2 15 dy dz dy = ( ) z dx x y dy + ( ) z y x z = (x, y) (x + dx, y + dy) = ( ) z x y ( ) x + y z ) ( y z 2.7 ( ) z y x x ( ) z x y ( ) x = y z ( ) z y x (2.19) (, ) (, ) (, ) (,, ) = (, ) = (, ) = (, ) (2.19) ( ) ( ) ( ) = 1 (2.2) = nr = nr/ = /nr = nr/ ( ) ( ) ( ) = nr nr nr = nr 2 = 1 (i) (ii) (i) (ii)
21 16 3 (i) (ii) 3.1 du = d Q + d W (3.1) d Q d W du ( ) U ( ) d'q U U+dU d'w d d d W d W
22 U = U(, ) (gradient) gradu du = (( ) ( ) ) U U, ( ) U d + ( ) U d (3.2) d W = d (3.3) (3.1) d Q = du d W d Q = ( ) U d + [( ) ] U + d (3.4) (3.4) d = d d ( ) Q U d = n C 1 ( ) Q = 1 ( ) U n n (3.5) d Q/d d Q (, ) d Q d ( ) Q Q d
23 (3.5) (3.4) n d Q = nc d + [( ) ] U + d (3.6) F (3.6) d n C C 1 n ( ) [( ) Q U = C + ] 1 + n ( ) (3.7) (186 ) (1844 ) U(, 1 ) = U(, ) ( ) U = (3.8) U = U() (3.5) C = C () C ()
24 3 19 (3.5) U() = n C ( )d + U( ) (3.9) (3.8) = nr (3.8) ) = nr/ ( = nr/ (3.8) (3.7) C = C + R (3.1) (3.4) (3.9) (3.5) = nr d Q = du() d + d d = nc d + nrd = n(c + R)d d Q nc d nrd C C 1842 C C 3.58J/cal (3.6) d Q = (3.8) = nr/ = nc d + nr d nc R = C C d + ( C C 1 ) d = (3.11)
25 3 2 γ C C = 1 + R C (3.12) (3.11) d + (γ 1) d = ln + (γ 1) ln = ln = log e ln γ 1 = γ 1 = (3.13a) = /nr γ = (3.13b) He Ar γ C = 3 2 R, γ = 5 3 (3.14) (, ) (3.13) 3.4 (, ) ( 1, 1 ) (, ) 1 1 (, ) 4 4 ( 2, 2) (, ) 3 3
26 γ (3.3) d W = d 1 3 II I (3.13a) 3 γ 1 3 = 1 γ 1 2, 3 γ 1 4 = 1 γ 1 1 ( 3 / 4 ) γ 1 = ( 2 / 1 ) γ = 2 1 (3.15) I = 1 1 = nr 1 I W I 2 2 nr [ ] 2 1 W I = d = 1 1 d = nr 1 ln = nr 1 ln 2 = 1 1 (3.16a) III W III W III = nr 3 ln 4 3 = nr 3 ln 3 4 = nr 3 ln 2 1 (3.16b) (3.15) II (3.13b) γ = 3 γ 3 = 3 γ 3 /γ II W II 3 W II = 2 = 3 γ 3 γ 1 3 d = 1 γ γ 1 2 γ 3 3 γ d = 3 γ ] 3 3 [ 1 γ γ 1 = 2 = γ 1 = nr 3 nr 1 γ 1 (3.16c)
27 γ 3 = 2 γ 2 I W I W I = γ 1 = nr 1 nr 3 γ 1 = W II (3.16d) W (3.16a)-(3.16d) W = ( W I + W II + W III + W I ) = nr( 1 3 ) ln 2 1 (3.17) W Q 1 η W Q 1 (3.18) I Q 1 U 1 = Q 1 + W 1 U 1 = (3.16a) Q 1 = W 1 = nr 1 ln 2 1 > (3.19) (3.17) (3.19) (3.18) η Carnot = = (3.2) 1 1 Q 1 Q 1 W W Q 3 Q Q 1 W (3.17) (3.19) III Q 3 (3.19) Q 3 = W 3 = nr 3 ln 3 4 = nr 3 ln 2 1 > (3.21)
28 3 23 Q 1 = W Q 3 W
29 24 4 (i) ( ) (ii) 4.1 d Q ds (4.1) d'q ds S 1865 du = d Q + d W (4.2) du d W d Q d Q (4.1)
30 (4.1) (4.1) d Q ds = ds (4.2) = du = d Q + d W d Q = d W du W W d W = d Q W (4.1) 4.3 (4.1)
31 (4.2) B1 B3 = du = d Q + d W = d Q + d Q A 1 du A 3 1 A 1 B 1 B 3 Q Q 3 A 3 (4.1) A i B i ( j = 1, 2) = B1 B3 B1 B3 d Q + d Q 1 ds + 3 ds = 1 [S (B 1 ) S (A 1 )] + 3 [S (B 3 ) S (A 3 )] A 1 A 3 A 1 A 3 S (B 3 ) = S (A 1 ) S (A 3 ) = S (B 1 ) ( 1 3 )[S (B 1 ) S (A 1 )] (4.3a) 1 < Q = B1 B1 d Q 1 ds = 1 [S (B 1 ) S (A 1 )] A 1 A 1 S (B 1 ) S (A 1 ) > (4.3b) (4.3a) (4.3b) 1 3 (4.4) 4.4
32 Q 1 Q 1 W W Q 3 Q 3 3 Q 1 W Q 3 > W + Q 1 + Q 3 = η W Q 1 = Q 1 + Q 3 Q 1 = 1 Q 3 Q 1 (4.5a) W Q 3 Q 1 W Q W Q = Q 3 Q 3, W = W + W (4.5b) W = Q (4.5c) W (4.5d) (4.5d) (4.5c) (4.5b) W = Q = Q 3 Q 3 Q 3 Q 3 (4.5e) (4.5e) (4.5a) η 1 Q 3 Q 1 1 Q 3 Q 1 η (4.5f)
33 4 28 η η (4.5f) η = η 4.5 η Carnot = (4.6) (4.5f) η 1 Q 3 Q (4.7) K K = C K 1 3 Q 1 Q Q 3 Q (4.7) = Q Q < Q/ Q > K 4.6 (4.1)
34 4 29 (4.1) (4.7) Q 3 / Q / 1 Q 1 / 3 Q Q 3 3 (4.8) n (4.9) n j=1 Q j j (4.9) n n Q 1 Q 2 Q n 1 2 n Q 1 Q 2 Q n n n Q 1 Q 2 Q n E j ( j = 1, 2,, n) Q j j Q j Q j Q j (4.9) Q j + Q j j = Q j = Q j j j ( j = 1, 2,, n) Q j n E
35 4 3 Q Q = n Q j = j=1 W W = Q W n j=1 W = Q = n (4.9) (4.9) Q i n (4.9) lim n n j=1 Q j j = j=1 Q j j Q j j d Q d Q (4.1) (4.1) ds d Q (4.11) ds = (4.12) S (4.1) (reversible) (general) (R) B A(G) d Q B A(R) d Q A B (G)
36 4 31 (4.11) B A(G) d Q B A(R) ds (4.13) A B d Q ds 4.7 (a) Q 1 W Q 3 W 1 Q 1 Q Q 3 3 Q 3 Q 1 Q 3 1 Q 1 W 1 Q + Q 3 1 Q 3 3 (b) Q 1 W = Q 1 Q 3 Q 1 + Q 3 Q 3 (4.1)
37 (4.13) AB S (B) S (A) B A(G) d Q (4.14) (d Q = ) S (B) S (A) (4.15) S (B) S (A) (i) (ii) (4.1) d Q = ds d W = d (4.2) du = ds d ds = 1 du + d (4.16) 1/ / du d S
38 4 33 (i = 1, 2) U i i U = U 1 + U 2, = S = S 1 + S 2 U 1 1 U 2 2
39 d'q d'w du = d Q + d W (5.1a) U U+dU d Q ds (5.1b) du d W d Q d Q (5.1b) 5.2 (5.1b) (d Q = ds ) d W d d W = d (5.2) (5.1a) (5.1b) (5.2) du = ds d (5.3)
40 5 35 du ( ) ( ) = S S (5.4) (5.3) (, ) S = S (, ) ds = ( ) ( ) S S d + d ( S / ) ( S / ) (5.3) du = ( ) [ ( ) ] S S d + d du [ ] S (, ) = [ ] S (, ) (5.5a) (5.5b) 2 S (, ) = 2 S (, ) + S (, ) (, ) ( ) ( ) S = (5.6) (5.5b) d Q = ds C 1 n ( ) Q = ( ) S n n (5.5a) (5.5b) ds = n C d + [ du = nc d + ( ) d ( ) ] d (5.7) (5.8a) (5.8b)
41 5 36 C (, ) d = (, ) S (, ) U(, ) (5.8a) (5.8b) ( ) C = ( 2 ) n 2 = (, ) (5.9) C (, ) C = C (, ) = (, ) (5.9) (5.9) (5.5) ( S/ ) (5.6) = (, ) ( / ) 5.3 = nr (5.1) C C = (3/2)R C = (5/2)R (5.8a) (5.1) ds = n C d + nr d (5.11) C (, ) ( 1, 1 ) S ( 1, 1 ) = nc 1 d + nr 1 = nc ln 1 + nr ln 1 = nc ( ln 1 + R C ln 1 d + S (, ) + S (, ) = nc [ ln 1 + (γ 1) ln 1 = nc ln 1 γ S ( γ 1, ) ) + S (, ) ] + S (, ) ( R C = C C C = γ 1 )
42 5 37 ( 1, 1 ) (, ) S (, ) = nc ln γ 1 + S ( γ 1, ) = nc ln γ 1 + (5.12) γ 1 = = d Q = ds = nr (5.8b) ( ) U ( ) = nr = (5.13) U = U() (5.13) 19 - (5.13) (5.8b) du = nc d (5.14) 1 2 C U() = nc d + U( ) = nc ( ) + U( ) = nc + (5.15)
43 5 38 (5.13) S (5.11) (d = ) d S = nr 1 = nr ln > (5.16) (5.1b) d Q = ds (, ) (5.3) ds = 1 du + d (5.17) ( ) S = 1 ( ) S U, U = (5.18) U j = 1, 2 U j j U 1 + U 2 = U, = (5.19) S S j = S j (U j, j ) S = S 1 (U 1, 1 ) + S 2 (U 2, 2 ) (5.2)
44 5 39 (U j, j ) (U j, j ) (U j + δu j, j + δ j ) δ d (5.19) δu 1 + δu 2 =, δ 1 + δ 2 = (5.21) S S (δu j, δ j ) 2 [S j (U j + δu j, j + δ j ) S j (U j, j )] (5.22) j=1 2 ( ) ( ) S j S j = δu j + δ j U j=1 j j j U j 2 ( 1 = δu j + ) j δ j j=1 j j ( 1 = 1 ) ( 1 δu 1 + ) 2 δ 1 (5.23) (5.18) j j (5.19) (U, ) (5.21) (5.23) (δu 1, δ 1 ) 1 = 2, 1 = 2 (5.24) 1 2 S = S (U, ) (, )
45 4 6 du = d Q + d W (6.1a) d Q ds (6.1b) du d W d Q d Q (6.1b) (6.1b) d Q = (6.1b) ds (6.2)
46 (6.1a) (6.1b) d = d W = du + d Q du + ds = d(u S ) F U S (6.3) d W df (6.4) d W F F (6.3) U S d W = d = (6.4) df (6.5) F (6.4) (6.3) d W = d d = df + d W = df d = d(f + ) G F + (6.6) dg (6.7)
47 6 42 G G 6.2 (6.1a) (6.1b) d W = d du = ds d (6.8) U = U(S, ) = ( U/ S ) (6.3) S U S ( ) U, S F(, ) = U(S, ) S (6.9) F df = du d(s ) (6.8) d(s ) = ds + S d (, ) df = S d d (6.1) v L p L/ v H p v L 6.3 U = U(S, ) F = U S F U = U(S, ) du = ds d (6.11) ( ) ( ) U U =, = S S (6.12)
48 6 43 du ( ) ( ) = (6.13) S S (6.12) U 2 U(S, ) S = 2 U(S, ) S U = U(S, ) ( ) U, S H(S, ) = U(S, ) + (6.14) dh (6.11) d() = d + d dh = ds + d (6.15) ( ) ( ) H H =, = S S dh ( ) ( ) = S S (6.16) (6.17) U = U(S, ) ( ) U, F(, ) = U(S, ) S (6.18) S df (6.11) d(s ) = S d + ds df = S d d (6.19)
49 6 44 F = F(, ) S ( ) ( ) F F S =, = (6.2) df ( ) ( ) S = (6.21) du F = F(, ) ( ) F, G(, ) = F(, ) + (6.22) dg (6.19) d() = d + d dg = ds + d (6.23) G = G(, ) S G ( ) ( ) G G S =, = dg ( ) ( ) S = (6.24) (6.25)
50 W = 1 d 2 d = 1 ( 1 ) W U 2 U 1 U 2 U 1 = U = U H = U µ J ( ) H (6.26) H (6.15) dh = ds + d (6.27) ( ) ( ) S S ds = d + d = nc ( ) d d (6.28) C = ( S/ ) /n G (6.25) (6.27) [ dh = nc d + ( ) ] d d dh = (6.26) - ( ) ( ) µ J = (6.29) H nc
51 6 46 = (, ) - = nr/ µ J = nr nc = µ J -
52 47 7 H 2 O 7.1 du = ds d 1 1 N 1 N 2 2 N 2 du = ds d + µdn (7.1) dn ( ) U µ N S, (7.2) (7.1) ds = 1 du + d µ dn (7.3)
53 U N U 1 + U 2 = U, =, N 1 + N 2 = N (7.4a) (7.3) S (,, N) = S 1 ( 1, 1, N 1 ) + S 2 ( 2, 2, N 2 ) (7.4b) (U j, j, N j ) (U j, j, N j ) (U j + δu j, j + δ j, N j + δn j ) δ d (7.4a) δu 1 + δu 2 =, δ 1 + δ 2 =, δn 1 + δn 2 =, (7.5) S S 2 [S j (U j + δu j, j + δ j, N j + δn j ) S j (U j, j, N j )] (7.6) j=1 (δu j, δ j, δn j ) 2 ( ) ( ) ( ) S j S j S j = δu j + δ j + δn j U j=1 j j,n j j U j,n j N (7.3) j U j, j 2 ( 1 = δu j + j δ j µ ) j δn j (7.5) j=1 j j j ( 1 = 1 ) ( 1 δu 1 + ) ( 2 µ1 δ 1 µ ) 2 δn 1 (7.7) (δu 1, δ 1, δn 1 ) 1 = 2, 1 = 2 µ 1 = µ 2 (7.8) 1 2 S = S (U,, N) (,, µ)
54 G F + = U S + dg = S d + d + µdn (7.9) G (,, N) N (7.9) G (, ) x G x G(,, xn) = xg(,, N) (7.1) x N x xn G(,, N x ) N x dn x dx = G(,, N) x = 1 ( ) G N, N = G(,, N) (7.9) ( G/ N), = µ G = µn µ = G N (7.11) (7.11) dg = µdn + Ndµ (7.9) - S d + d Ndµ = dµ = S N d + d (7.12) N µ = µ(, ) ( ) µ = S ( ) µ N, = N (7.13)
55 (7.8) µ - (7.12) µ = µ(, ) µ 1 (, ) µ 2 (, ) = (7.14) + d + d µ 1 ( + d, + d) µ 2 ( + d, + d) = (7.14) d d [( ) ( ) ] [( ) ( ) ] µ1 µ2 µ1 µ2 d + d = (7.15) N A (7.13) ( S 1 S 2 )d + ( 1 2 )d = (7.16) S j j L 12 ( S 1 S 2 ) (7.17) - d d = L 12 ( 1 2 ) (7.18) dg = S d + d + µdn F G
56 7 51 df = S d d + µdn (7.19) F F N Ω ( ) F µ, Ω(,, µ) = F(,, N) µn (7.2) N, dω (7.19) d(µn) = Ndµ+µdN dω = S d d Ndµ (7.21) 7.6 j = 1,, 2, n α = 1, 2,, ν ν S = S α (U α, α, N α1,, N αn ) (7.22) α=1 (U α, α, N α j ) α j (U α, α, N α j ) (U α +δu α, α +δ α, N α j +δn α j ) S δs = ν S α = δu α + S n α S α δ α + δn α j U α=1 α α N j=1 α j ν 1 = δu α + n α µ α j δ α δn α j α α α (7.23) α=1 j=1
57 7 52 δu δ δn j ν δu α = α=1 ν δ α = α=1 (7.24a) (7.24b) ν δn α j = ( j = 1, 2,, n) (7.24c) α=1 (7.24a) (7.24b) (7.24c) λ U λ λ N j (7.23) ν ( ) ( ) 1 α + λ U ) δu α + + λ δ α α α α=1 n j=1 ( ) µα j λ N j δn α j α = (7.25) (δu α, δ α, δn α j ) 1 α = λ U α 1 α = λ α 1 µ α j = λ N j (7.23) (7.24) (7.25) 7.7 n ν f f = n ν + 2 (7.26) n 1 (number of variables) N v = ν(n 1) + 2 (7.27a)
58 7 53 j µ 1 j = µ 2 j = µ ν j (number of equations) N e = (ν 1)n (7.27b) (7.27a) (7.27b) f f = N v N e = ν n n = 1 (H 2 O) ν = 2 f = = 1 - n = 1 (H 2 O) ν = 3 f = =
59 54 8 d Q ds (d Q = d W = ) ds µ C n ( ) S >, κ 1 ( ) >, C > C, κ > κ S (8.1) κ κ S 8.1 S S f (x) x = x x x δx f (x + δx) f (x ) >
60 8 55 f (x + δx) x = x f (x )δx + 1 2! f (x )(δx) 2 + > δx f (x ) =, f (x ) > f (x, y) (x, y) = (x, y ) δx δy f (x + δx, y + δy) f (x, y ) > f (x + δx, y + δy) (x, y ) (x, y ) ( ) ( ) f f δx + δy + 1 {( 2 ) [( f (δx) 2 2 ) ( f 2 ) ] ( f 2 ) } f + + δxδy + (δy) 2 + > x y 2! x 2 x y y x y 2 ( 2 ) ( f 2 ) f ( ) ( ) f f δx + δy + 1 [ ] x 2 [ ] x y δx δx δy ( x y 2! 2 ) ( f 2 ) + > (8.2) f δy y x y 2 δx δy ( 2 ) f x 2 H ( 2 ) f y x ( ) ( ) f f = = x y ( 2 ) f x y ( 2 ) (8.3) f y 2 f H (Hessian matrix) (8.3) H ( 2 ) ( f 2 ) ( f 2 ) [( f 2 ) ] 2 f >, deth = > (8.4) x 2 x 2 y 2 x y
61 f (x, y) g(x, y) f ( f, g) (x, y) det x f y g x g y = f g x y f g y x (8.5a) ( f, g) (x, y) (u, v) ( f, g) (x, y) = ( f, g) (u, v) (x, y) (u, v) = f g u v f g v u x y u v x y v u (8.5b) 8.2 du = ds d ds = 1 du + d (8.6) ( ) S = 1 ( ) S U, U = (8.7) U j = 1, 2 U j j U 1 + U 2 = U, = (8.8) S S j = S j (U j, j ) S = S 1 (U 1, 1 ) + S 2 (U 2, 2 ) (8.9)
62 8 57 (U j, j ) (U j, j ) (U j + δu j, j + δ j ) δ d (8.8) δu 1 + δu 2 =, δ 1 + δ 2 = (8.1) S (U j, j ) 2 [ S j (U j + δu j, j + δ j ) + S j (U j, j )] > (8.11) j=1 δu j δ j (8.2) 2 ( ) ( ) S j S j 2 δu j + δ j U j=1 j j + 1 [ ] δu j δ j j U j 2! j=1 2 S j U 2 j 2 S j j U j 2 S j U j j 2 S j 2 j [ ] δu j + > δ j (8.12) (δu j, δ j ) (8.7) (8.1) δu 2 = δu 1 δ 2 = δ 1 2 ( ) ( ) S j S j 2 ( 1 = δu j + δ j U j j = δu j + ) ( j 1 δ j = 1 ) ( 1 δu 1 + ) 2 δ 1 j U j j j j=1 j=1 (δu 1, δ 1 ) 1 = 2 1 = 2 (8.12) (8.4) 2 S 2 S 2 S U >, det U 2 U >. (8.13) 2 2 S 2 S U 2 j = 1, 2 j (8.7) (8.13) ( U/ ) = nc 2 S U = ( ) ( ) S (1/) = 2 U U U 2 S = ( ) ( ) S (/) = 2 U = 1 ( ) = 1 1 ( 2 U 2 U U ) = ( ) 1 2 U = 1 nc 2 ( ) U (8.14a) (8.14b)
63 S U = ( ) ( ) S (1/) = U U = 1 ( ) 2 U ( / ) U ( / ) U (8.5) ( ) ( ) (, U) U U ( ) (, U) = U (, U) = (, ) = ( ) = (, U) U nc (, ) ( ) (, U) = U (, U) = (, U) (, ) (, U) (, ) = ( ) ( ) U ( U ( ) ) ( ) U = ( ) ( ) nc (8.14c) (8.15a) ( ) U (8.15b) (8.14b) (8.14c) ( U/ ) = ( / ) ( ) ( ) ( ) U U 2 S = 1 ( ) 2 2 nc nc = 1 ( ) + 1 nc 2 ( ) [ U ( ) ] = 1 2 S U = 1 ( ) = 1 2 U nc 2 ( ) + ( ) U (8.12) 2 S 2 S U det 2 U = 2 S 2 [ S 2 ] 2 2 S U 2 S 2 U U 2 S 2 = 1 nc 2 = 1 nc 3 = 1 nc 3 1 ( ) ( ) 1 1 ( ) + 1 nc 2 1 nc 2 [( ) ] 2 [ U [( ) ] 2 U (8.16a) 1 nc 2 (8.16b) ( ) ] 2 U (8.17) (8.14a) (8.17) (8.12) > C >, κ 1 ( ) > (8.18) κ
64 (8.5) C > C C = C + R n ( ) ( ) n S S (C (S, ) (S, ) C ) = = (, ) (, ) = = = ( ) S [( ) ( ) ( ) ] 2 ( ( S ) ) ( ) ( ) S = (S, ) (, ) (, ) (, ) (S, ) (, ) ( ) ( ) S ( ) ( ) S df = S d d = ( ) [( ) = ( ) ] 2 [( ) ] 2 = κ > (8.19) κ (8.18) C > C ( ) > ( ) S ( / S ) > ( / S ) C > C κ > κ S
65 8 6 (κ κ S ) = = ( ) ( ) = nc ( ) S = ( ) ( ) S [( ) ( S / ) = nc / κ (, ) (, ) (, S ) (, ) = (, S ) (, ) ( ) ) ( S ( ) S dg = S d + d (, S ) (, ) (, S ) (, ) ( ) S ( ) = ( ) S ( ) S = ( ) ] 2 < (8.2) > κ S ( ) S > ( ) 1/( ) S < 1/( ) > < 1/( ) S < 1/( ) / > 1 ( / ) S < 1 ( / ) κ S < κ
66 ds = d Q (9.1) d Q 1 S (, α) = S (, α) + (9.2) 1 α, S (, α) F = U S = U d Q 1 1 S (, α) S (, α) 195 F G
67 C α = n ( ) S α α =, n/ S ( = ) = S (, α) = C α ( 1 ) 1 d 1 C α () ν ν > D 1K Nk B R 3R C/3Nk B / D α α 1 ( ) (, ) dg = S d + d α 1 ( ) = 1 ( ) S (9.3)
68 ( ) (9.4) (, ) df = S d d ( ) ( ) S =
69 (a) nb (1.1a) n b (b)
70 1 65 N/ N n = nr = nr a ( n a ( n ) 2 + a (1.1b) = nr (1.1) [ ( n ) 2 ] + a ( nb) = nr (1.2) ) = (, ) ( ) ( 2 ) = = c 2 ( c, c, c ) c (1.3) = nr/ (1.3) (1.2) c = nr c c nb a n2 2 c (1.4a) = = ( ) = nr c n2 + 2a c ( c nb) 2 c 3 ( 2 ) 2 c nr c n2 = 2 6a ( c nb) 3 c 4 (1.4b) (1.4c) (1.4) ( c, c, c ) (1.4b) 2/( c nb) (1.4c) ( 2a n2 2 c 3 c nb 3 ) = c
71 1 66 c = 3nb (1.5a) (1.4b) c c = 2an2 3 c ( c nb) 2 nr = 2an2 27(nb) 3 4(nb) 2 nr = 8a 27bR (1.4a) c c = nr 2nb 8a 27bR a n2 9(nb) = 2 a 27b 2 (1.5b) (1.5c) (a, b) n R ( c, c, c ) = ( 3nb, a 27b 2, ) 8a 27bR (1.6a) 1 nr/ nr c c c = 8 3 = 2.67 (1.6b) 1 nr c / c c nr c / c c c ( C) c (atm) N/ c (g/cm 3 ) nr c / c c H 2 O O He nr c / c c 188 (,, ) = ( c r, c r, c r ) (1.7) ( r, r, r ) (1.6a) (1.7) (1.2) ( a 27b n 2 2 r + a 9n 2 b 2 r 2 ) (3nb r nb) = nr 8a 27bR r ( r + 3 ) ( r 2 r 1 ) = r (1.8) ( c, c, c ) ( r, r, r )
72 1 67 r = r ( r,,r ) ( r,,r, r ) = (1, 1, 1) r < 1 r 1 r < 1 r = r ( r ) κ 1 ( ) r r (1.8) K 1.3 S U ds du ds = nc ( ) d + d (1.9a) [ du = nc d + ( ) ] d (1.9b) (1.2) = nr nb a n2 2 ( ) = nr nb (1.1a) (1.1b)
73 1 68 (1.1) (1.9) ds = nc d + nr nb d (1.11a) du = nc d + a n2 2 d (1.11b) (1.11a) (1.11b) ( ) C = (1.12) C = C () (1.11) C ( 1 ) S = n d 1 + nr ln( nb) + (1.13a) 1 U = n C ( 1 )d 1 a n2 + (1.13b) c = (, ) B D ( / ) > B C D κ 1 ( / ) A B C D E = A A E A B E A C D - A E A - (, ) ABC CDE A A B C D E A S U d Q = U, = S = ds = = 1 d Q = Q
74 1 69 W = W = d = (ABC ) (CDE ) ( 1 > 2 ) µ J ( ) = H ( ) ( = ) nc nc ( ) (1.14) µ J > µ J = (boundary) = () r = 3 8 ( r + 3 ) ( r 2 r 1 ) 3 (1.15) r ( ) = = 3 {( + 3 ) ( 1 ) [ 6 ( 1 ) + ( + 3 )]} = 3 [ 1 ( + 3 ) + 6 ( 1 )] = = () () = 9 ± 9(9 ) (1.16)
75 1 7 (1.15) = () µ J > (, ) r µ > r
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
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現代物理化学 2-1(9)16.ppt
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Chap11.dvi
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mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
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1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
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変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
2.4 ( ) 2.4.1 ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec. 2. 4 p. 1/30 (2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i
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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)
73
73 74 ( u w + bw) d = Ɣ t tw dɣ u = N u + N u + N 3 u 3 + N 4 u 4 + [K ] {u = {F 75 u δu L σ (L) σ dx σ + dσ x δu b δu + d(δu) ALW W = L b δu dv + Aσ (L)δu(L) δu = (= ) W = A L b δu dx + Aσ (L)δu(L) Aσ
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5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
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