0 (Preliminary) T S pv
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1 0 (Preliminary) S p p S (two variables function) (total differential) : ( ) (1) (2) Legendre F (3) (4) Maxwell 47 i c 2015 etsuya Kanagawa
2 Maxwell [ ] Joule Joule [ ] Mayer Mayer Joule homson [ ] Joule homson [ ] J Joule homson [ ] : ds [ ] ii c 2015 etsuya Kanagawa
3 : df : dg iii c 2015 etsuya Kanagawa
4 II ( ) : 3F305, 5254 kanagawa kz.tsukuba.ac.jp : 1. ( 6 ), ( 5 ),., , 60 II ( ).,,. 2. : 100. (a) (manaba ): 4 (b) 10 9 [1]: 5 (c) [2]: 7 (d) [3]: 7 (e) [4]: 7 (f) : 70, (a) (e) 30 2, (f) 70., (10:10 ), 3 4.,,. II ( ). 2 (manaba ) I.. 3 [ ],,, ( ). 4 [ ]. 5 II,.,, (4 1). 1 c 2015 etsuya Kanagawa
5 , 27 I., I,, 8.,, I,. 4. :.,,,. I,, 9. 10, ,.,,,,,., manaba, 13. 6,,.,,,,.,,,,.,, ( )., ( ),,. 7 [ ],,,,,, (, ). 8 I,, 27 I,,. 9 I,,,,., , ( I ),,, ,,.,,. 12 [ ],,,,., ( ).,,,., I,,.,, 6. 13,, (, 2 c 2015 etsuya Kanagawa
6 5. :, 14. : ( 0), ( 1), Maxwell ( 2), ( Joule, Mayer, Joule homson )( 3) 15, ( 4) ( ) A301 ( ), ( ) :, 18, 19. manaba ). ( ),,.. 14,.,,,.,,. 15 4,.,. II.,,,.. 16,. 17 ( ).,,,.. 18,,.,, (Legendre transform). 19 [ ],.,., (mechanics). (mass point) ( I). ( ) (particle mechanics),. ( ), (rigid body). (deformation), (continuum) ( ).,,,., Newton [, (Newton ), ]. (mechanical engineering) [ ( ) ], 2, 2,,.,.,,. 3 c 2015 etsuya Kanagawa
7 0 (Preliminary) 0.1 II 20, F F U S (0.1), G G H S (0.2) ,, U H S, F G, S p H H U + p (0.3) 20 [ ], (state quantity), (state variable),. [ ]. 21,. G. F 1 U, H. 22 [ ] (definition),. (formula) (law),, [ (0.1)(0.2) 1 ].,,,. 24 [ ] (free energy), Helmholtz ( )., (free enthalpy), Gibbs ( ). (Gibbs ), Gibbs G, (Helmholtz ), H F ( ). 4 c 2015 etsuya Kanagawa
8 G (0.2), G H S = U + p S = F + p (0.4), F G. (0.4), (0.1)(0.2)., S 25. (0.4), F G. U p 26, (0.3) 27., F G,. (0.1)(0.2),,,, (0.1) (0.3)., 29. : 25 [ ],.,, 100., (,,, ).,.,., G = H S S, G = F + p p.. 26 p, pd,.,, G = F + pd F, F pd ( ),,,.,,. 27, S p ( 1 )., I ( ),, 1. 28,, F G.,,.,, F G (, ). 29 [ ] : (i) (pressure) p (volume), (mechanics). (ii) 0 (the zeroth law of thermodynamics), (temperature) ( I 2 ). (iii) 1, (internal energy) U. (iv) (enthalpy) H, (isobaric process). (v) 2, (entropy) S. 5 c 2015 etsuya Kanagawa
9 (i), ( ) 30 31, p (ii),, U, H ( U + p ) 33, S 34., (0.1)(0.2) F G 2 35.,, 6,. 1. U, H, F, G, 4, 36., U, H, F, G, [ ] ( ) ( ),,, ( ).. 31 [ ] (extensive variable),, (intensive variable),. 32 [ ] 3, II (chemical potential).,, (concentration).,,, (specific enthalpy) h = H/m, (m )., II,. [ ] ( I ). 33 [ ] Gibbs., Gibbs ( ). 34 [ ] (reversible process),, S 2.,. (finite value). 35, p, (, U, H, S, F, G) ,, 4, ( ),,. 37,.,, ( ).,. 1 d Q 6 c 2015 etsuya Kanagawa
10 , p S J (0.4) ( ) F d = U d p d (0.5) 2 ( ) G d = H d + dp (0.6) 2 [ ] (0.1)(0.2), (0.7) 41, ( ) ( ) F U S ( ) = d = d = Ud du ds = U ds pd d + ds = ( ) (0.8) 2 38,. 39 1,, (0.1) (0.4), S p ( p) (S ) (, S p )., S, (heat quantity) ds (= d Q). 40 (0.4) 10 9 ( 1 ).. 41 ( 4), : d(fg) = fdg + gdf (0.7) (, : derivative, differential quotient), (differential),,. 7 c 2015 etsuya Kanagawa
11 . (0.6) , ( ) = ( ) ( ) 45., 46 : (0.10) du = d Q d W (0.11), U, ,, du., d Q d W, d 42, (0.5),,, dh = ds + dp (0.9).,, (1.15)., [ 26 II ( ) 27 I [ A]( ) ]. 43, d 1 d = 2 = d 1 = 1 2 d, 3 (, ).,,, df ( ).,,. 44 (0.19). 45 (system). (surroundings), (boundary). 46, [ ( : infinitesimal), ] (heat quantity) (work). (process), c 2015 etsuya Kanagawa
12 49., 2 : (i) 50, 51, ( ) d p, d W = pd (0.12)., I,,,,. 50,.,. 51 (displacement), (force). 52 I (0.12),. (0.12),. 9 c 2015 etsuya Kanagawa
13 (ii) 53 S ds d Q (0.17), d Q, d Q = ds (0.18). (0.17),, 57., (0.18) ( 0.2.1). 53 [ ],,. ( ), :,. 54 [ ] (irreversible process), S, ds d Q (0.13) (Q ).,.,, ( 4),. 55 [ ] : ds = d Q d. ( ), ( )..,,. 56 [ ] ds, S, (definite integral) S 2 S 1 S = 2 1 d Q (0.14). 2 : (i) = 0, S = d Q = Q 12 0 (0.15). 1 (integrand) (isothermal), 2., Q 12 1 (thermal equilibrium state) 2 12 (, I, 12, , Q 12 Q 1 2 ). (ii), d Q 0, ds = 0, S = 0. S 1 = S 2 (0.16). [ ],. 57 [ ], Clausius, d Q/, 10 c 2015 etsuya Kanagawa
14 , 58 59, 60, du = ds pd (0.19). (0.19) 61.,, p S,, S [(0.17) (0.18)] 62.,,,,, (0.18)., p,,, S 4. (i). p ,,,. 58,, (0.11) (conservation law of energy)., (0.19), S,., (0.19),, ( ) ( ).,,,., (0.19), 3 ( ),,. 59 [ 58 ] ds = d Q,, (0.19)., (i),, (ii) (0.19),, (0.19),. 60 [ ],., 3, (reversible) (quasi-static)..,.,,.,,.,, 1 1,., 2,,., (counterexample). 61,., (0.19), I ( 4, ). 62 (0.11) (empirical law), (0.12). 63 [ ]., 11 c 2015 etsuya Kanagawa
15 (ii)., , S. ds,. ds d.,, 2 66 : d W = pd (0.12) d Q = ds (0.18), p d., ds (0.12). 3. (0.19) 70.,,., ( ) ( ).,,, ( ),., (i) ( ), (ii) ( ), (iii), ( )., ( ) ( ),., ( ) ( ),. 64,., ( ), ( ) [,, (heat input),,, (heat output) ]. 65 [ ],, ( ), 20 C.. 3 : (i) ; (ii) ; (iii),. 66 [ ]., 99%,.,,. 67 [ ] (0.18),,., ( ),, ( ). d d, ( ). 68 d Q = d, S. S [J/K], 1 K., p S,,. 69, S.,,. d W = pd, p, d Q,, ds ( 2 ), (0.12) ( 12 c 2015 etsuya Kanagawa
16 0.3, ( 3) 71., ( 1),, Maxwell ( 2). I,,, II III,,.,,.,, (two variables function) f x y, x y, f(x, y) ( ) 74., x y,, f 75. f x y ,., ). 71, Boyle Charles. 72 d I, ( ). 73,,,,,.,,,,., (differentiation), :, ( ), pp. 1 6, ( ) (independent variable), ( ) (dependent variable)/ (unknown variable). 75, f, x y.,. (point). (curve).. 76, z = f(x, y)., f (x, y) z, z.,. 77 [ ] 1 y = f(x), x, y, 1, 2,., 13 c 2015 etsuya Kanagawa
17 , 78, 2,., 79 : p = mr = p(, ), = mr p = (p, ), = p mr = (p, ) (0.20) p,, 2 80., 81,, (inverse function), f, x = f 1 (y), x, y (x y )., 2,,., ( 0.4.6). 78, II. 1 ( Boyle, Charles, Poisson. )., 2 1,,,,. [ ],,,.. [ ] II, 3 n.,, 1 2,, 2 3 n., 2, m (mass), R (gas constant). 80,,, 2., p ( ). 81 [ ], t, v(t)., x, δ(x)., (t, x), u(t, x) p(t, x).,,,.,,,.. 82,,,., 1., c 2015 etsuya Kanagawa
18 0.4.2 (total differential) x y ( ), f(x, y) , df(x, y), df(x, y) = [df](x, y ) + [df](y, x ) = (x ) + (y ) = f f dx + dy (0.21) x y , (i) f/ x, y 87, ( ) f x lim f(x + x, y) f(x, y) f = lim x 0 x x 0 x (0.22) 88., f/ x (, ), x f ( ). x dx, ( f/ x)dx, x f, , x,, y.,. 84 [ ] f(x, y),, 3 : (i) x y, (ii) x, (iii) y.,, (i) = (ii) + (iii). (0.21).. 85 (0.21) 3., (0.21) 3,., (differentiability),., (proposition),..,. 87, x f. 88 (symbol), x, a., x,,.,,., aylor (higher-order terms), ( ). 89 f/ x., f(x, y) x 15 c 2015 etsuya Kanagawa
19 , y [ (0.21) 1 ] 90. (ii),. x dx f ( f/ x)dx, y dy f ( f/ y)dy, f ( ). (iii) df, dx, dy 3, (, ) 93, ( ) (iv), (0.21)., y 96, dy = 0, df(x, y), df(x, y) y=const. = f dx (0.23) x 97., x y,,, (verb). 90 (0.21) 3 1, y. x y, (i) dx y. x., (ii) f/ x y., (0.21) 3, d, (infinitesimal).,, 1 dy/dx (finite value), (fractional number). 92, 10 30, ,,,,. 93,, ( 1 ). 94 (product), (sum), ( )., (0.21) 3 ( ) = + = + = = ( ) ( ).. 95 (,, ),.,,, ( )., f/ x,.,,,, f/ x. 96, y,, x. 97, : df y=const. = f x dx, df y = f dx (0.24) x 16 c 2015 etsuya Kanagawa
20 . (0.23) ( 3) , 99. d(fg) = fdg + gdf (0.7) [ ] fg h 100, h, h(f, g)., z = f(x, y) y f/ y,, 1 y = f(x) f(x, y, z), y(x) dy/dx, x(y) 101 dx/dy 102., dy dx = 1 dx/dy (0.26),., 1 dy/dx, dx dy [!!!] (0.23), d, df(x, y) y = df dx (0.25) dx.,, dx, df., f 2 f(x, y)., y, y. 99, I, ( : mapping) f, y = f(x) x = f 1 (y). f. 102,,. 103 [ ] (,,, ).,,.,,. 17 c 2015 etsuya Kanagawa
21 , 2 f(x, y). II, f x 1 x/ f (0.27) 104., 2,,, x y, 105.,.. 1, 2,,, d dx f(x), x f(x, y), 2 f(x, y) (0.28) x2,, f , arcsin x, arccos x, arctan x. (0.26), I, C P, C P = dh d (0.29) 104. y. (chain rule). 105,, 2, [ ] x 2 (exponent) 2, x 2, ( x) 2. (1 ). : 2 ( ) 2 f x 2 = 2 x 2 f = 2 ( x) 2 f = f x 107 (inverse trigonometric function), sin 1 x, 1/ sin x = (sin x) 1.., (sine), (cosine), (tangent) : d dx arcsin x = 1, 1 x 2 d dx arccos x = 1, 1 x 2 d dx arctan x = x 2 18 c 2015 etsuya Kanagawa
22 . H 1 108,, (0.9), C P = ds + dp d (0.30). ( 3). I,, 3.2,, 2, 1.,, C P = H(, p) (0.31).,,.. H (0.3)., (i) dh H 109, (ii), / x, : 2 (2 ). (p,,, U, H, F, G, ), 2,.,. 2,, 108, 2,, H 1. Joule ( 3.2), U = U( ), H U + p = U( ) + mr = H( ) ( ). 109 [ ] ( ): C P = H(, p), : C P = H(, p) S + p ds + dp = U (U + p ) = + p ds + dp d 19 c 2015 etsuya Kanagawa
23 110, 111., (Boyle Charles ) p = mr = (κ 1)U (0.32) 112., p = mr/, 2 p = f(, ) (0.33), p = (κ 1)U/, 2 p = g(, U) (0.34) 113.,,, (0.33) (0.34) U., 2, (0.33)(0.34) 114., f g.,,.. p. f g, p(, ) p(, U) (0.35)., (d = 0) p/, (du = 0) p/ 115, ( )., p,., 110,, Boyle ( ) Charles ( ), 2 ( ), , [ ( ) ]., ( ),.,, p = (κ 1)U., Boyle Charles , ( ). 114, f g,.. 115,,. 20 c 2015 etsuya Kanagawa
24 (0.35) ( ),.,.,,, p(, ) ( ) p (0.36), 116.,, ( ) , : ( ) ( ) x y = 1 (0.37) y z x z ( ) ( ) ( ) x y z = 1 (0.38) y z z x x y ( ) ( ) ( ) ( ) y y y t = + (0.39) x z x t t x x z ( ) ( ) ( ) y y t = (0.40) z t z x x x 116, ( ) ( ),..,, z = f(x, y),,,., p(x, t), p t x,. 2,,.. 117, 2., ( A), ( B),,. ( ). 118,, : ( ) p(, ) p(, ),,,., ( ).,, ( 1) :, ( ), pp ( ). 21 c 2015 etsuya Kanagawa
25 , (x, y, z, t). (0.37). 2 z, 1 (0.26) 2,. (0.38), : 3 (x, y, z), 2, 1., 3., (x, y, z), (0.37)(0.38). (0.37)(0.38), (x, y, z),. x, y z., 2, 3, f(x, y, z) = 0 (0.41).,. (0.41),, (x, y, z), 122.,,,., (0.39)(0.40), 3,, 123., (0.39)(0.40), 4 (x, y, z, t),, 4 3. (0.37)(0.38),, 2,. 120 (denominator), (numerator), (subscript), (x, y, z) 1 (circulation),. 1 ( ) (0.26) , 2 [4 (0.37) (0.40),, ]. 123 (0.40), 2 (chain rule). 22 c 2015 etsuya Kanagawa
26 5. (0.37) (0.38) , 2, 127. ( 1) : (0.37), x = x(y, z) y = y(z, x) 128.,. dx(y, z) dy(z, x), ( ) x dx(y, z) = y ( ) y dy(z, x) = z z ( ) x dy + z ( ) y x dz + x y z dz (0.42) dx (0.43). (0.42) (0.43), dx(y, z) = ( ) x y z ( ) y dx + x z [ ( x ) y z ( ) y + z x ( ) ] x dz z y (0.44), dx dz 130,, 124,,,. 125,.,,,,.,,., (0.39)(0.40). 126 (0.38), II (Jacobi, Jacobian).. 127, 2, 3 : f(x, y, z) = 0..,., : (i), x, (y, z). (ii) x, 1, 2. (iii) (0.38), z = z(x, y),. 129, (0.43) (0.42). (0.42)(0.43). 130, dx dz (identity) : Adx + Bdz = Cdx + Ddz A = C B = D 23 c 2015 etsuya Kanagawa
27 dx, (0.37) : ( ) x y z ( ) y = 1 (0.37) x z, dz, : ( ) x y z ( ) y + z x ( ) x = 0 (0.45) z y, (0.38)., (0.45) ( z/ x) y : ( ) x y z ( ) y z x ( ) z + x y ( ) x z y, (0.37) ( ) x z y ( ) z = 0 (0.46) x y ( ) z = 1 (0.37) x y,, (0.38) : ( ) x y z ( ) y z x ( ) z = 1 (0.38) x y ( 2) : (0.37),, z ( )., (0.42), dz = 0 133, dx = ( ) x dy (0.47) y z 131 ( 1),,.,,. 132 : (0.37), x = x(y, z) y = y(x, z),, z, 1,, dx dy, 1 (0.26), 1. dy dx 133, z. 24 c 2015 etsuya Kanagawa
28 , dx ( 0) 134, : ( ) x dy y z dx = ( ) x y y ( ) y = 1 (0.48) x z 1, 135. (0.43), y,, (z, x) 2, (z, x) (, z) 136, ( )., (0.38) : (0.42) dx = 0 137, dz, 138 : ( ) x dy y z dz + ( ) x = z y ( ) x y z ( ) y + z x ( ) x = 0 (0.49) z y,, , (0.37), ( 1) : P (x, y)dx + Q(x, y)dy (0.50) 134 dx = 0 x, x, x, (0.37), ( ). 135,.,,, 2. ( 1) ( 2). 136 z, z. 137, x. (0.42), z y, x,. 138 [ ] d,,.,. 139,,, : y (x, z) 2,,,, dx = 0 x, x., y 2, 1, ( ) ( ). 25 c 2015 etsuya Kanagawa
29 , z(x, y), dz(x, y) = ( ) z dx + x y ( ) z dy = P (x, y)dx + Q(x, y)dy (0.51) y x., dx dy, : P (x, y) = ( ) z, Q(x, y) = x y ( ) z y x (0.52) 2, (0.52) , III, 143 : ( ) P = y x 6. (0.53). ( ) Q x y 7. (0.51) 144 (0.53) z(x, y) = C (0.54) (C ) (0.51) ( ). 142 (necessary condition), (sufficient condition), (necessary and sufficient condition).,. 143 (0.52) (0.53) (,, ). 26 c 2015 etsuya Kanagawa
30 1, U, H, F, G, (0.1)(0.2)(0.3), F, G, H,,, Legendre,,.,., : ( ) ,.,,.,,.,.,,., 2 ( ) 149, ( ). 2,,, 150. ( ).., ( ) ( ),, ,. 146,,. 147,, ,, ρ = m/, v = 1/ρ = /m, (m ) ( ),, 27 c 2015 etsuya Kanagawa
31 ,.,, ( ) (1), (0.19) 154 : du = ds pd (1.1), 2. (1.1), ds d., U 2, S, 155. :, (1.1), du(s, ) = ds pd (1.2)., du(s, ), du(s, ) = ( ) ( ) U U ds + d (1.3) S S (1.2) (1.3), [ du(s, )],.,,.,.,,,. 152,. 153,., Maxwell.,, ( ),,,. [ ],, ( 3). 154,.. 155,, U(S, ). ( ),.,, ,, ( ). 28 c 2015 etsuya Kanagawa
32 ds d ( ),., (1.2) (1.3), : ( ) U (S, ) = S ( ) U p(s, ) = S (1.4) (1.5) p, U , U U(, S), U(, S).,. (1.1) (thermodynamical identity), (1.1) U(, S) (thermodynamical potential) , (U,, S). 159,,.,., ( ),,. (1.4)(1.5).,,, (, I, ), (, ).,. 160 (thermodynamical characteristic function). 29 c 2015 etsuya Kanagawa
33 ,, [ (1.4)(1.5)].., 2, 2 ( ), 3, 4,. (1.4)(1.5),,, 163., , ( ), ( ).,, ( ).,,. 161,.., F Ω,, F = Ω/ x. ( ), ( ). 162 [ ], ( ) v ( ) Φ : v = gradφ.,,.,.,,, (,,,, ). 163,.,. 164,,.,.,,,,. 2,,.,,.,,.,,.,, ,, [ ( )].,.. 30 c 2015 etsuya Kanagawa
34 1.2 (2) [ (1.1)],, (4, 166 ) 167., 3, Legendre Legendre ,, ( ).., U(, S), (, S).,, S,,,,, 1.,. U(, S),., (1.4),. F (, ) 171, ( 1) U(, S) (1.4) : (, S) = ( ) U S = U = U F (1.6) S 0 ( ) U S + F = S + F (1.7) S 166, Maxwell, Joule, Mayer, Joule homson. 167,. 168, 2., 3,,,. 169 :, (, 2014). 170,, ( )., (analytical mechanics) Legendre. 171,. 31 c 2015 etsuya Kanagawa
35 (1.7) 2, F (0.1),, F U S (0.1). F, U S,. (1.7) 1 [ (1.6) ], 172, F S., S,, ( ) ( ) p, ( ), 1,.,, ds Sd, pd dp. ( ), Legendre. 8. (0.1) (0.4), Legendre (linear function)., (1.7) : ( ) U U = S y = ax + b = dy dx x + b S + = ( ) U S + F S (slope) a (= dy/dx) ( ), = ( U/ S). (intercept) b, F. 173 (1.6)(1.7),, F : ( ( )) =.. 175, U S U S U S 0 U S 0 ( ) ( ) ( ( )),, (U ), (S ). 32 c 2015 etsuya Kanagawa
36 1.2.2 F, F, (, )..,. F (0.1) ( 176 ) : df = du ds Sd (1.8) 1 2, (1.1), df = pd Sd (1.9)., 2 F (, ) 177. (1.9), (1.1),., (1.9), [ ].,.,,,, [, df, df(t, x) ].,,, du.,, du, du(, ), ,,..,,,. :... (1.9) : p, d, pd S, d, Sd pd Sd., df. 178 (1.9),,,,., (1.9), F, (1.1). ( ).,. 33 c 2015 etsuya Kanagawa
37 (1.9), F (p S ), (, ) 179., F (, ) df (, ) = ( ) ( ) F F d + d (1.10), (1.9) [ df (, ) ] 180, U, ( ) F p(, ) = ( ) F S(, ) =., F = F (, ). (1.11) (1.12) F (, ),, 181., S, F (, ) (1.12).,, ( )..,., (0.1), (1.11)(1.12),.,, F,, [ ] (1.9), F, p, S (, ).,, 2., (1.9), (, ), [ (1.10) ],, (1.11)(1.12).,,. 180, (1.9)(1.10) d d., ( ).,, (1.9) (1.10). 181 [ ] F (, ),., F (p, ) F (S, ) (. ) Legendre,,,..,., ( ),,. 34 c 2015 etsuya Kanagawa
38 (1.8) (1.9). (1.8).., 3, df = du ds Sd (1.8). F F (U, S, ), df (U, S, ), du, ds, d.,, 2, (1.8). F = U S. ( ), (1.10). F, H, G,,,. 1.3 (3) 3, 4, 185., 2. 2 F (, ), 1 U(, S), 3 U(, S)., F (, ),., ( ) 183,, 2,.,,., II 3 ( n ). 184 [, ] 3,, 1, 1 = ( F/ U),S., F = F (, S).. 185,, 2,.,,.,,.,,,,.,. 35 c 2015 etsuya Kanagawa
39 ,., ( ).,.,, ( 1) U(, S) p(, S) (1.5), H ( ) ( U U H p(, S) = S ) (1.13) H = U + p (0.3). H, (1.13), (0.3), H.. H, 3. H (0.3), 2 (1.1) 188, dh = du + pd + dp = ds + dp (1.14). 2., dh = ds + dp (1.15) 3., I,,, , ,, ( ). F,, (0.1) ( ). 188 ( 27 I [ A] 2 ). (1.15), (1.1).,.,,. 189,,, (1.15) 36 c 2015 etsuya Kanagawa
40 (1.15), H (S, p), dh(s, p)., H(S, p) dh(s, p) = ( ) H ds + S p ( ) H dp (1.16) p S. dh = dh(s, p) (1.15), (1.16).,,, H(S, p) : ( ) H (S, p) = S ( ) H (S, p) = p p S (1.17) (1.18), H, I , H(S, p)., (0.3),, (4) 3 U(, S), F (, ), H(S, p),., H(S, p), ( ) S.. ( ). 190 I,, ( ), ( )..,,,..,,,,.,,.,,. 191 [ 190 ], ( ).,.,,. 192, H(S, p),.,,. 37 c 2015 etsuya Kanagawa
41 , S. p,.,,., ( 3) H(S, p) (S, p) (1.17) : = ( ) H H G S p S 0 G, : (1.19) G = H S (0.2) G, (p, ). (0.2) G : dg = dh ds Sd = dp Sd (1.20) 1 1 2, 3 (1.15) 2., 4, : dg = dp Sd (1.21), G , (1.21), G = G(p, )., G(p, ) dg(p, ) = ( ) G dp + p ( ) G d (1.22) p,,, 193 (1.21),. 194,,,.,,,,,,. 38 c 2015 etsuya Kanagawa
42 S : ( ) G (p, ) = p ( ) G S(p, ) = p (1.23) (1.24),, p G(p, )., 4 G(p, ) Legendre, F, H, G ( ) 4 4, 4. : A. U, F U S, H U + p, G H S 4 197, : du = ds pd (1.1) df = pd Sd (1.9) dh = ds + dp (1.15) dg = dp Sd (1.21), G F [, G(p, ) F (, ) ].,, ( 4 ). 196.,,., ( ).,. 197,,. 198, ds pd,. [J]. 199, (1.25),, 39 c 2015 etsuya Kanagawa
43 B. U, F, H, G, ( ) 200., (1.1)(1.9)(1.15)(1.21) 2., U, F, H, G 201 : U(S, ), F (, ), H(S, p), G(, p) (1.25) (1.25). (0.1) (0.3)., (1.1)(1.9)(1.15)(1.21),,. C. (1.25), (1.25),, (1.4)(1.5)(1.11)(1.12)(1.17)(1.18)(1.23)(1.24)., 1, 2 (2 )., : ( ) ( ) U(S, ) H(S, p) = = S S p ( ) ( ) U(S, ) F (, ) p = = S ( ) ( ) H(S, p) G(p, ) = = p S p ( ) ( ) F (, ) G(p, ) S = = p (1.26) (1.27) (1.28) (1.29), (1.26), 1 (S, ), 2 (S, p). D. 2,., (1.26) (1.29),.,,,,,.,, ( ). 200,.. 201,., F (p, S),, F = U S.,. 40 c 2015 etsuya Kanagawa
44 p,,, S , 4 2,, ,, (S, ), (, ), (S, p), (, p), ,,. E. (1.26) (1.29), (1.26), 2 S,. F. (1.26) (1.29),, ( )., S,., (1.29) 1, , 1, 2.,, 207. G. ( ), ( ).,, ( ) , 0,, ( ),., p, S, , (combination), 4C 2 = 6, 4 ( ). 2 ( ) ,, (1.25), 1 1,. 206 (Otto ) (1.29) 1, (Diesel ) (1.29) 2,.,, 1., ,,,. 41 c 2015 etsuya Kanagawa
45 1.5.1, U, F, H, G 208.,,, : (i) ; (ii) (Legendre ), (i) ( ) ; (iii),, (ii) ; (iv) (i) G(p, ).,,.,..,,,, (1.26) (1.29), 1 ( ) 2 ( 2 )., [ (1.26)], ( ) U(S, ) (S, ) = S ( ) H(S, p) (S, p) = S p, 209.,.,,, (p, ) 210,, (1.26). 208, (Legendre ). F, H, G, ( ). p. 210, = p/(mr) = (p, ). 42 c 2015 etsuya Kanagawa
46 ,, 211,.,, ( 3 ) (1.26) (1.29), ( ).,, (1.26)(1.28), H(S, p).,, H(, ) H(p, ) 213.,. S p 2, H(S, p),. U, F, H, G, 214.,.,, U, F, H, G, 215.,,., ( ), 211.,.,,.. 212,.,, t x = (x, y, z), (kinematics),, (thermodynamics) ( ),. [ ] (mechanics) (kinematics) :,,., (statics) (dynamics) ( ),. 214,,. 215,,,., II,,,. 43 c 2015 etsuya Kanagawa
47 ( ).,, , (1.25) G(, p), p,, G 216.,, 217 ( )., ( ),..,, p = f(, ) (1.30) 218., 219, g(p,, ) = 0 (1.31). f g,., f g ,, f 216 G. 217,., Boyle, p = C/,,. 218, (1.30),. U S. 219 (1.31) :, 3, 3, 2,. (1.31), (implicit function representation). g = 0 p f.,,, f (explicit function). f, g, 3 ( ). [ ]. 220, p = p(, ),, p = f(, ) ( p f ).,.,,. 221 (ideal gas), f, I p = f(, ) mr/ 44 c 2015 etsuya Kanagawa
48 g, 222.,, 223,,., I 224,,.,,,,.., Maxwell. 6. F, H, G 225,, 4 (1.1)(1.9)(1.15)(1.21) , [U(S, ), F (, ), H(S, p), G(p, )] (1.26) (1.29) (m [kg], R [J/(kg K)]).,,,, mr. [ ] mr, Hooke Young, Newton ( ), Fourier ( ). 222,,,.,, 1.,. ( : real gas), van del Walls. 223, ( ). 224 Joule [U = U( )], ( ) Mayer, Joule homson. 225 U, II ( ). 227, I,,,, ( ) ( ).,. 228 (1.4)(1.5)(1.11)(1.12)(1.17)(1.18)(1.23)(1.24). 229 ( 6),.,,, U(S, ), F (, ), H(S, p), G(p, ) ( 199 ). 45 c 2015 etsuya Kanagawa
49 8. Gibbs Helmholtz 230. [ ( )] F (, ) U(, ) = 2 [ ( )] G(p, ) H(, p) = 2 p (1.32) (1.33) F (, ) G(, p), U(, ) H(, p) ( ) : x 2 ( y ) ( ) y = x y x x z x z 231 [ ( )], U(S, ) H(S, p), S ( ), S U(S, ) H(S, p)., F (, ) G(, p) (p,, ), F (, ) G(, p).,, F (, ) G(, p),, U(S, ) H(S, p)., U H,. 232 [ ],,,,.,,,., F G. 46 c 2015 etsuya Kanagawa
50 2 Maxwell (Maxwell s relation) , , P (x, y)dx + Q(x, y)dy (0.50), 239, 240 : ( ) P = y x ( ) Q x y (0.53), 4,, U(S, ), F (, ), H(S, p), 233 [ ] Maxwell,,. 234 [ ],,,,, ( ).,, ,. 236 [ ] ( ), ( 7,, 4 ),, ( 10, ), (,, ).,. Amazon., ( ). 237, (0.50). ( ). 238, P Q (x, y), P dx + Qdy, (, ). 239 [ ] z(x, y), : ( ) ( ) z z P (x, y)dx + Q(x, y)dy = dx + dy = dz(x, y) (0.51) x y y x, (0.53) ( III). 240 dx dy (differential)., (0.50) ( ). ( ),. 47 c 2015 etsuya Kanagawa
51 G(p, ) du(s, ) = (S, )ds p(s, )d (2.1) df (, ) = p(, )d S(, )d (2.2) dh(s, p) = (S, p)ds + (S, p)dp (2.3) dg(p, ) = (p, )dp S(p, )d (2.4) 242. (2.1) (2.4),, U(S, ), F (, ), H(S, p), G(p, ) 243,,.,, U(S, ), F (, ), H(S, p), G(p, ) 244.., (2.1) ds, U(S, ), = ( ) U S = (S, ) (1.4)., , 1. [ ],, (, ). 242, du, du(s, ). 243 [ ] (2.1) (2.4),.,. 244,.,,,. 245 (1.4), 7,. 48 c 2015 etsuya Kanagawa
52 2.2 Maxwell, (2.1) (2.4), (0.53) 246., (2.1),, du(s, ) = (S, )ds p(s, )d = ( ) ( ) U U ds + d (2.5) S S. (0.53), : ( ) ( ) p = S S (2.6), 3 (2.2) (2.4), (0.53), ( ) p ( ) p ( ) S p ( ) S = ( ) = S p ( ) S = p. (2.6) (2.9) Maxwell 247. (2.7) (2.8) (2.9), ( ) d, ( ) ( / ) p., ,,,.,,.,.,. 247 [ II ( )] Jacobian (Jacobi, ), (2.6) (2.9), ( ): (, S) J (p, ) / p / S/ p S/ = 1 [ ], J, : d ds = Jdpd 248 [ ] ( ),.,. : (i) 49 c 2015 etsuya Kanagawa
53 (2.7)(2.9), (2.6)(2.8) ( ). (2.6) (2.9), (i) [J] ( p S), (ii) 1 (p,,, S),, [ ]. (2.5),, ds d., ds d., (1.4)(1.5) : ( ) U = S ( ) U p = S (1.4) (1.5), (1.4), S, ( ) = S [( ) ] U = [( ) ] U S S S S = ( ) p S (2.10),, [ ( ), ( ) ]. (ii) d,,. (iii),, ( ). v = dx/dt. [ 1] 3, ( ),. [ 2],, (line integral). 50 c 2015 etsuya Kanagawa
54 ., , (1.5), Maxwell 1 (2.6) ,,. (1.26) (1.29), ( ) U, F, H, G 253, 4 ( p,,, S)., (2.6) (2.9), (, p,, ), F G,,, [ ( II)] 2 f(x, y), (i),, 2 f x y 2 f y x, (ii). (i) (ii), 2 : 2 f x y = 2 f y x,. 250, U, S,. 251 [ III],,. 252 [ ], ( ).,,.,.,,,,. 253 [ ], U, F, H, G,,. 254 [ ]. ( ), ( ). 255 p p, S S,., F U,,.,,. 51 c 2015 etsuya Kanagawa
55 (2.6) (2.9),, p,, S 4, 4.,, ( ) ( ) (closed set/system)., (2.6) (2.9) 258, (2.6) (2.9).,. (i) (2.6)(2.8), S,., S 260.,, : x + y + z = 0, x + y z = [ ( )],,.,, ( : computational mechanics).,, Newton Navier Stokes (2 ) (exact/analytical solution), ( )., (numerical/computational study), (approximate solution), (perturbation method) [ (weakly nonlinear phenomena) ].,. 258 [ ], Cauchy Riemann ( ), 2 ( ). 259 [ ] (partial differential equation)., ( ), ( ).,,., 1 ( ( III) ).,., ( 2 ) ( ),,, ( )., (diffusion equation), (wave equation), Laplace,, (parabolic), (hyperbolic), (elliptic) [2 (quadratic curve) (analogy)] ( ). 260 [ ],. d Q = ds., d Q < ds ( 4 ). 52 c 2015 etsuya Kanagawa
56 261.,,.,. (ii) (2.7) (2.9), p,,.,, 262., p,,, ( ),., (2.7), S(, ) = ( ) p d + S 0 (2.11) (S 0 )., p = f(, ) 263.,, (2.7) (2.9), (2.6) (2.8)., F H,, (0.1)(0.2). 3, Maxwell (2.7)(2.9),., Maxwell.,. 2. Maxwell, (2.8), : (2.7), : [ ] Boyle Charles,, p = f(, ), S = g(p, ), ( 3.2 ). 264 [ ], (speed of sound), (isothermal compressibility), (isothermal bulk modulus), (coefficient of thermal 53 c 2015 etsuya Kanagawa
57 9. Maxwell (2.6) (2.9) ,,, Mayer, Joule homson 3., (0.19),.,, 267.,, Maxwell (2.6) (2.9) 268,., expansion), (thermal pressure coefficient) : ( ) p : a (2.12) ρ S : κ 1 ( ) (2.13) p ( ) p : k = 1 (2.14) κ : α 1 ( ) (2.15) p ( ) p : β (2.16),,,,., Maxwell : β [ (2.7) ],, α [ (2.9) ]. [ ],,,.,.,,. 265,., ( )., (2.6) (2.9),, [ ],,. S. 268, (1.26) (1.29). 54 c 2015 etsuya Kanagawa
58 , , 1.5.4,.,, (i) Boyle Charles ( ) p = mr = f( ) (3.1). (ii), Boyle Charles, v ( /m = ρ), ρ, ( ): pv = R (3.2) p = Rρ (3.3),,. 11. ( ): p = f(, ) (1.30) g(p,, ) = 0 (1.31) (1.30), 2., 2, 1, f. 269,.,, I, Joule (Joule ) Mayer, Joule homson ( ),,. (general relation),. 270 ( 3.1), Mayer ( 3.2), Joule homson ( 3.3),,, 3.,, Maxwell,,,.,. 271 [ ( )] (i). (i), (ii) [ ]. 55 c 2015 etsuya Kanagawa
59 (1.31), , f g, Joule 3.1.1, (0.19) du = ds pd (0.19) 274. (0.19) U S,, U(, ) S(, ) d ( 0) 277 : ( ) U = ( ) S p (3.4), 278., 2 U S [ ] ( ).,., 2, 3 ( )., (implicit function). (, ). 273,. 274,,,.,,,,.,,,. 275 [ ].,,, [ ] du(, ) du(s, )!!. 277,, (,. ).,, d = 0,. 278 [ ] (3.4),, d du d = ds d p,, 1 2,. 56 c 2015 etsuya Kanagawa
60 279,., S.., Maxwell (2.7) ( ) U = ( ) p p (3.5), (, ). 1, d, Maxwell (2.7). 2,, (2.7) , Maxwell. (0.19),, (3.4),,., 284. (3.5) (,, ) 279 S/, , (2.7) ,,,.,, (, ). 281 [ ( )], 1 ( ). p, (0.19) H ( ).,,, (3.4)., U(, ξ),,, ξ. 282 [ 281 ] du = ds pd,. U, (S, ), (, )., U ( ),,. ( 3, 3 ),. 283 U = U(, U) ( ). 284, (0.19) (3.4), (, )., 2,,. 57 c 2015 etsuya Kanagawa
61 Joule,, (3.5).,. (3.1), (3.5), ( ) p ( ) mr/ = = mr = p ( ). (3.5), : ( ) U(, ) (3.6) = 0 (3.7), 1,, U = f( ) (3.8) 285 [ ] (0.19),, ( ) (3.5)., Maxwell (2.6) (2.9). 286, 285. I :,,,,.,,,,., ( ). 58 c 2015 etsuya Kanagawa
62 ., f( ) ( ) (3.8) Joule ( ) 291,, [ ( III )] (3.8),, (3.7).,. (3.7) (3.8),, [ 287 ] n, n., n, n ( ),.,,, ( ),,. 289 [ ],. du/d = 0, U = C (C ).,, [,,, ( )]., ( C),.,. 2,.,,.,,., ( ), F ( ),. 290 [ 1] (general solution), (particular solution), (singular solution),. (family of curves). [ 2]. [ 3]. [ 4],. ( ). 291, ( ),,. 292., ( I, ). 59 c 2015 etsuya Kanagawa
63 , Joule, (3.8) ,,, Joule., Maxwell ,, (3.5) , : (3.5) 1 ( ),, p = f(, ) f.,., (3.5) 293 [Joule ( ) ] A B (valve), A ( ), B (vacuum)., ( ), (insulation)., A B [ ],,, A + B,., ( ), : ( A B ),,,,., [ ],., Joule,,., U(, ), : U(, A ) = U(, A + B ) = U = f( ) (3.9) ( ), ( ),., Joule (3.8) ( ) U = 0 (3.8). [ ],, pp [ ] Joule 2., (1843) ( I). 297 Maxwell.. (3.4). 298., (3.4),.,,,,,,. 299 ( ( )), ( ). 60 c 2015 etsuya Kanagawa
64 , U ( )., (3.5) (p,, ) 300., U [ ] (3.5)[ (3.4)],, 301.,,. 1.1, U U(S, ), p 302., (1.3), S,, U 3., U(S, ) S,, S = S(, ξ) 303. S, 1 S U 300, Maxwell (2.7). 301 ( 0.4.1)., U, / ( 0.4.3) ,,. 302 [ ]. (0.19) ds d,., U = U(S, ), du(s, ), ) ) du(s, ) = ( U S ds + ( U S d (1.3)., (0.19) du = du(s, ), (1.3) (0.19)., ds d, : ( ) ( ) U U =, p = (1.4) S S 303 [ ] U(S, ) 2, S, S = S(, ξ),., = (S, ξ), (3.4)(3.5).,,, ( ) ( ) U = p (3.10) S ξ S ξ, ( S/ ) ξ, (0.37), (3.4) ( )., S,,. 61 c 2015 etsuya Kanagawa
65 , 2 ξ S(, ξ), ds(, ξ) = ( ) S d + ξ, (1.3) ds : [ ( U ) du(s(, ξ), ) = S + ( ) ( ) ] U S d + S ξ,, ( ) ( ) S dξ (3.11) ξ ( ) U S ( ) S dξ ξ (3.12). du(s, ) = du(s(, ξ), ) = du(, ξ) (3.13) (3.13), (3.12),, du(s, ), du(, ξ)., du(, ξ) du(, ξ) = ( ) U d + ξ ( ) U dξ (3.14) ξ (3.12) d dξ, ( ) U ( ) U ξ ( ) ( ) U U = + S S ( ) ( ) U S = S ( ) S ξ (3.15) (3.16) 2.,, 3 4 (0.39)(0.40),,. (1.4)(1.5), (3.15), ( ) ( ) U S = p (3.17) ξ ξ., ξ =, (3.4). 304, S 2, ξ = ξ = p (,, ). 305 [ ] ξ (, : xi), x. 62 c 2015 etsuya Kanagawa
66 (3.4)(3.5). (3.5), Joule (3.8) : f( )p = (3.18) (i), Joule (3.8). (ii) (3.18), (3.18).. [ (i)] (3.5) 1, (3.18), ( ).,, Joule (3.8). [ (ii)] f( ) =., f( ),, f ,. 12.,, (3.5).. (i) ( )(1.15) 308, : ( ) H p = + ( ) S p = ( ) p (3.19) [ ], 1,, Maxwell 309, (0.39) (3.15), U,, S, (quadratic function), (polynomial),,. 308 (2.3)., (2.3) (1.15) [ ] 4,., H(p, ), dp.,., H(p, S). 63 c 2015 etsuya Kanagawa
67 (ii), H [ ] (3.19),., : ( ) H p = 0 (3.21), H = g( ) [g( ) ]. 310 [ ] Joule (3.8), (3.19),., (0.3), Joule (3.8) (3.1) : H U + p = U( ) + mr = H( ) (3.20) 64 c 2015 etsuya Kanagawa
68 3.2 Mayer 3.2.1,,,., ( I)., C c C/m, (2 ),.,,, ( I).,,, 311. (2 ) C, C = C(, ) (3.22), 1. 1, 2 : (i) C P p 2, (ii) C 2 ( ): C P = C P (, p), C = C (, ) (3.23) C, d Q d : d Q = C(, )d (3.24) 311,,. 312 [ ],, C [, ( ) ]., d Q, C d,. 313, : C = d Q d,,, ( ).., ( 0.4.3), d /d.,, (,, 1 ).,, d, 1, 2 ( ).,,.,,., (,, ). 65 c 2015 etsuya Kanagawa
69 I,, C P C,, ( ) [ (Mayer s relation)] 314..,,. 315, Mayer : C P C = ( ) ( ) p p (3.25) 13., Mayer (3.25). [ ] (3.1) ( ) p = mr, ( ) = mr p p (3.26), (3.25), ( I) : C P C = mr (3.29) 314 [ ], Mayer, ( ) ( ).,,, Mayer (,, ). 315,,. 316 [ ] mr ( ). (3.25),,., (p,, ), (, ),., (3.1), p = f(, ),. 317 [ ] I, c P = C P /m c = C /m, (3.25).,, ( ) ( ) p v c P c = (3.27) (v )., pv = R, Mayer :, (3.25)(3.29) m,. v p c P c = R (3.28) 66 c 2015 etsuya Kanagawa
70 3.2.2, 318., 3. ( 1) 2 C : d Q = C(, )d (3.24) ( 2) 319 : d Q = du + pd (3.30) S.,. (3.24) d Q, d Q = ds 320. ( 3) U, (, ) 2. : (i) (3.24) d, ; (ii) (3.30) d 321, 322. d, 323., : du(, ) = ( ) U d + ( ) U d (3.32) 318,,,,.,. 319,. 320.,,.,,. d Q = Cd = ds (3.31) 321 [ ], U,,, U,. 322 [ ] : (i),, (ii). 323 [ ] d d., U(, ),. 67 c 2015 etsuya Kanagawa
71 . d d, (3.24)(3.30), ( )., (3.24)(3.30)(3.32), 324,. ( 1),, (3.24) : d Q = C d (3.33) ( 2) (3.30), 325 : d Q = du(, ) (3.34), (3.32) U(, ),. (d = 0), U [ ], ( ). : d 0 = U(, ) U( ),,. 325 I, U ( ). 68 c 2015 etsuya Kanagawa
72 ( 3) (3.32) : du(, ) = ( ) U d (3.35) (3.33) (3.35),, : ( ) U C = = C (, ) (3.36), C (, ). (3.36), 331., 326 [ ( )], d = 0, , : ( ) U du(, ) = du(, ) + du(, ),, du(, ) = d 327 [ ] U = U( ) 1, d : du = du d d /, U/,., U, U, U ( 0.4.2). [ ] ( 0.4.2),, ( U/ ),,,. 328 [ ] d Q...,. 329, ( ),,., (3.35),,. 330 (3.36),, : U = C, du = C d,, ( ). 331 [ ] (3.36),,,,. (3.36),.,,,. 69 c 2015 etsuya Kanagawa
73 C P : ( ) H C P (, p) = p (3.37) [ ]. ( 2) ( ) d Q = dh dp (3.38),, ( 3) H(p, ) 333 dh(p, ) = ( ) H dp + p ( ) H d (3.39) p 334.,,, (3.39) : C ( ) = du d, C P ( ) = dh d (3.40) (i) (3.36)(3.37),, (3.40), ,. 333 [ ]. (3.38), d dp.,,. 334, p.,,. 335 [ ], I,,,,., (3.40), 1., C [ ],, (1 ) (, )., U 1., (3.40),., (iii) (3.44) (,, I ).,, (3.40), U = U( ), 1,, C( ),. 70 c 2015 etsuya Kanagawa
74 (ii) (3.40), 337. U( ) = C ( )d + U( 0 ), H( ) = C P ( )d + H( 0 ) (3.42) 0 0 (iii) ( ), (1 ) : U( ) = C, H( ) = C P (3.44), 0 = 0 K. [ (i)] Joule (3.8), U = U( ) (3.36), 2 1, 341., (0.3), Joule, H U + p = U( ) + mr = H( ) (3.45),,. 337 [ ],, 0 C (θ)dθ (3.41)., 2, 0 1,. 338 [ ], ( ). 339 [ ],, c c P,, : u = c, h = c P (3.43) 340 [ ], U = f( ), U( ),., U f. 341 [Joule ], Joule (3.8), Joule, (, ). 71 c 2015 etsuya Kanagawa
75 [ (ii)] d, [ 0, ] 342 ( ) = 0 du d d = du = U( ) U( 0 ) = C ( )d = ( ) 0 0 (3.46),. H( ). [ (iii)] C, 0 = 0, U( 0 ) = 0 : U( ) = C d + 0 = C (3.47) 0 =0 (3.45), = 0 = 0, H( 0 ), H( 0 ) = U( 0 ) + mr 0 = 0 + mr 0 = 0 (3.48), U( 0 ).,. 12. (3.1),, U,, H 343 : p = κ 1 H = (κ 1)U (3.49) κ, κ c P /c = (C + R)/c = 1 + R/c [ ], 0,. 343 ( ).,,,,,. (3.1), (3.49). 344 [ ], S : p κ = A exp(s) S = Ã ln(p κ ) (3.50), A ( )., ( ). [ ], (p,, ) (p,, S)., (0.19), Boyle Charles (3.1).,, S.,., I. 345 [2 ] 2,,,., 72 c 2015 etsuya Kanagawa
76 [ ] (3.1), (3.44) 346 : U = C = mr κ 1, H = C P = mκr κ 1 (3.51) Mayer 347, Mayer (3.25). C P (, p) C (, ) (3.36)(3.37), (i) 4 (0.40) 348, (ii) [H(S, p) U(S, )] (S, p) (1.26) : ( ) H C P = ( ) U C = p ( ) H = S ( ) U = S p ( ) ( ) S S = p p ( ) ( ) S S = (3.52) (3.53), 349., : (i), p, ; = p mr = (p, ) (ii), (3.49) ; (iii), (3.50) ; (iv),.,,,,,,.,,,,,. 346,,,. ( I )., (ratio of specific heats), heat s. (2 ). 347 [ ],., d Q = ds,, Maxwell., Maxwell. 348 [ ],, 1 2. dy dx = dy dt dt dx 349 [ ] S [J], (1.26), H S,. 73 c 2015 etsuya Kanagawa
77 , S 350., : [ ( S ) C P C = p ( ) ] S (3.54). 1 S(, p), 2 S(, ), 351., ds(, p) ds(, ) 352 : ( ) S ds(, p) = ( ) S ds(, ) = d + p ( ) S dp (3.55) p ( ) S d + d (3.56) (3.54) Mayer (3.25)., S,, p, 4, d, dp, d 3 2. (3.55)(3.56), d 353, dp d., p = p(, ) 354 : dp(, ) = ( ) ( ) p p d + d (3.57) 350 [ ],,.., S., S. 351 [ ], S., S(, p) S(, ). 352 (3.55)(3.56),, S, (, ( ) ): ) ( S p ( ) S 353 [ ] (i) 3 ; (ii) 2, 1 ( ); (iii) d 2, ; (iv) dp d. dp, dp(, ),, d d,. 354 [ ] = (, p) (,. )., S, S. 74 c 2015 etsuya Kanagawa
78 (3.57) (3.55) dp, : ( ) S ds(, p) = [( ) S = p d + p ( p ( ) S p ) ] d + [( ) ( ) p p d + [ ( S ) ( ) S + p (3.58) 1,, ds(, p) = ds(, p(, )) = ds(, ) = p ] d ( p ) ] d (3.58) ( ) ( ) S S d + d (3.59), (, p) (, ) , (3.59). (3.58)(3.59),, ds(, ).,,, d d., d, 3 (0.39) 357 ( ) S = ( ) S + p ( ) S ( ) p p (3.60), d, 4 (0.40) ( ) S = ( ) S ( ) p p (3.61). ( 3.1.4), (0.39)(0.40), [ ] S p, p ( ).,, S [ ] S(,, p) (,,, ): ( ) y = x z ( ) y + x t ( ) y t x ( ) t x z (0.39) 358 [ ],, (0.39)(0.40).,,. (0.39)(0.40), (,,, ). 359, (3.60)(3.61) (0.39)(0.40).,. 75 c 2015 etsuya Kanagawa
79 (3.54). (3.60).., (3.54) : [ ( S ) C P C = p ( ) S p ( ) ( ) ] S p = p ( ) S ( ) p p (3.62) , Maxwell (2.9) , Mayer (3.25) : C P C = ( ) ( ) p p (3.25) (3.25) 2, mr,, , 366,, : C P > C (3.63), C [ (Maxwell )] ( S/ p)., ±( / ) p (, )., : (A) S (p, ), G(p, ). G (1.21). (B), ±Sd ± dp ( )., (p, ),, (, 4 ). 361 [ ],. 2, S. (2.9).,,. 362,. Maxwell ( ). 363 [ ] (2.9)., G = H S ( )., = (, p), (2.9) (2.7) ( )., F G,. 364,, mr > 0 ( ). 365 [ ],, (thermodynamical inequality).,,,. 366,, (c P > c ). 367 [ ].,. 76 c 2015 etsuya Kanagawa
80 , C P, (3.25), C., (3.25)., (i) C S (3.52)(3.53) 369. (ii) (3.60)(3.61),, (3.62). (iii) Maxwell, Mayer (3.25) (3.30), d Q = C d + [ ( ) ] ( ) U p p + d = C d + d (3.65) [ ] C (, ) (3.36), U(, )., : du(, ) = ( ) U d + ( ) ( ) U U d = d + C d (3.66) (0.39) ( ), : ( ) ( ) ( ) ] S S + p p [ ( ) S C P C = ( ) S = ( ) ] [ ( ) S S = ( ) ( ) ( p = p ) p (3.64) 4, Maxwell (2.7) [ (2.9),, ]. 2, 3 (0.39)., (0.39),.,,,. 369 [ ], Mayer, ,,.,,. C (. I ). 372,,,.,, 1 1,. 77 c 2015 etsuya Kanagawa
81 , (3.36)., (3.30), (3.65) 1., (3.5), d [ ], (3.65). [ ]Mayer, (3.65) ,. (i) C P C 374 : ( ) ( ) CP 2 = p 2 p ( ) ( ) C 2 p = 2 (3.70) (3.71) (ii) : ( C ) = 0 (3.72) 373 [ ],,,,, ( ). [ ], C (, )., C P (, p), C P, = (, p) [ (3.66) d, dp.,,, ].,, p ( ) : [ ( ) d (, p) p = d + p ( ) dp p ] p = ( ) d p (3.67) p, (3.66), (3.67), [ ( ) ( ) ] p d Q p = C + d p (3.68) p., C P, d Q p = C P (, p)d p (3.69) ( )., Mayer (3.25). 374 [ ],. 78 c 2015 etsuya Kanagawa
82 375 [ (3.72) ] p(, ) = f( ) + g( ) (3.73) (f g ) [ (i)] ( 1) (3.71) 378. (3.36), 379, (3.5) 380, : ( ) C ( ) = [ = ( p = [( ) ] U = [( ) ] U ) ] ( ) ( ) p 2 p p = + 2 ( ) p = ( ) (3.74) ( 2) 381 (3.53), 375 [ ],.,,.,,,.,. 376, ( ),. 377 [ ] n, n (arbitrary constant)., n n ( ).,.,,, ( ). 378 (3.70). ( 1) (3.19), ( 2) Maxwell (2.9) , 2, 2,,. 380 [ ] (3.5), Maxwell (2.7),,., F. [ ](3.70), (3.5), (2.9), G. 381,. 79 c 2015 etsuya Kanagawa
83 , Maxwell (2.7), : ( ) = = [ ( ) ] S ) ] [( S = = [( ) ] S [( ) ] p = ( ) (3.75) 2,, ( ) 382. [ (ii)], p, (3.71) (3.72), 0, ( ) 2 p 2 = 0 (3.76)..,, 1 : ( ) p = f( ) (3.77), ,, : p(, ) = f( )d + G( ) = f( ) d + G( ) = f( ) + g( ) (3.78), G g,, G g, G, g [ ],,,,,,.,,. 384,,. 2.,,.,. 385,,, C = K( ) (K ) [ ( III )] (general solution) 80 c 2015 etsuya Kanagawa
84 3.3 Joule homson [ ],,.,,,., Joule homson (Joule homson effect) 387,, Joule homson (, J ) 388 : µ ( ) p H., 389 : ( ) = 1 p H C P [ + ( ) ] S = 1 p C P [ ( ) ] p (3.80) (3.81) ( : particular solution).,, p =, p = C 1, p = C 2, (C ) (3.79), (3.76),., (3.79) (3.76) ( ), (3.76).,,. (3.79) (3.76),., (, ),,.,,,. 1, ,,,.,,, (, ),. 388 J ( ).,,., II,, µ,. 389 [ ] (i) ( S/ p), ±( / ) p ( ); (ii) (p, ), G(p, ) ; (iii) G. 81 c 2015 etsuya Kanagawa
85 ,, Maxwell (2.9) 390, 391. J (3.81). C P = ( H/ ) p [ (3.37)] J µ = ( / p) H 3 (H,, p). 392.,,, 2 (0.38) 393 : ( ) ( ) p p H H ( ) H = 1 (3.82) p 1, 3,, J,, : ( ) p µc P H = 1 (3.83) ( H/ p), 1 (0.37) 394, µ = 1 C P ( ) H p (3.85)., (3.19) , G (2.4). 391 [ ],,,. 392,,.,,, (0.26) 2 : ( ) ( ) dy dx y x 1 : = 1, 2 : = 1 (3.84) dx dy x z y z 395 [ 12 ] dh = ds + dp (1.15) dp ( ) H = p ( ) S + p (3.86) p. 2, 1 p, 2, ( ),,, =. [ ] 82 c 2015 etsuya Kanagawa
86 , J (3.81) [ ] J H (1.15) dh = ds + dp (1.15), S. S, (H,, p),, µ, H, ds(, p) : ds(, p) = ( ) S d + p (3.87) (1.15) 1 dh = ( ) [ S d + + p ( ) S dp (3.87) p ( ) ] [ S dp = C P d + + p ( ) ] S dp p (3.88) 398,, dh : 0 = { [ C P d + + ( ) ] } S dp p H (3.89), H dp, ( / p) H, (3.81) : ( ) = 1 p H C P [ + ( ) ] S p (3.90),, (,, ). 396,, (3.19),, ,,., C P µ, S. S. 398 (3.52). 83 c 2015 etsuya Kanagawa
87 19. Joule homson (3.81) Joule homson J (3.81),.,,,.,,. I, Joule, Joule homson, Joule homson , C P J µ. 3.,. 400 [ ], J [, ( ) ].,,, ( ). 401 [ ],,,,., J, (,, ).,,. 84 c 2015 etsuya Kanagawa
88 p., H., ( )., 402 [Joule homson (1) ] Joule ( 293),.., (heat bath),,,,.,,, ( )., [ 402 (2) (throttling) ( )],, ( : pore valve). A, B, B. A A.,,., A B,. A, A,, B,. A B,,. 404 [ 402 (3) ]., A < B.,, A ( ), B ( )., p A > p B.,, Joule.,., (, ) [ 402 (4) ],,. : (i),, A B ; (ii) U = U B U A ; (iii),, p A A, ( ) A( ), W A = A 0 p A d = p A A (3.91) (,, ). B, W B = B 0 p B d = p B B (3.92), W = W B W A,, U = W = W A W B = p A A p B B (3.93), : H B = U B + p B B = U A + p A A = H A (3.94), W, (U A > U B ). [ ],, p A A = p B B, U A = U B, Joule. 406 [ 402 (5) ],., 85 c 2015 etsuya Kanagawa
89 . J,,., J,, ( ) 407.,, J µ 408 : 1 ( ) > 1 p (3.95) J,,. J inv, : inv = ( ) p (3.96),. Joule homson. J, , µ = , (200 K) (100 K),,,.,., 1908 Kamerling Onnes., (superfluidity) (superconductivity). [ ], pp , ,,. 408,,. 409,. J,,. 410,,,. 86 c 2015 etsuya Kanagawa
90 4,.., 411.,,,.,.,.,...., , 1 3., ,. 2, 2, ,, ( I)., 2,..,, 416.,,.,,,, 411,,. 412 [ ], ( II ),,,. 413 [ ],.,, Le Charelier ( ). 414,,.,,. 415, 0. 0,,. 416, = 2 1 d. 87 c 2015 etsuya Kanagawa
91 417.,, ( 1 2 ), , C 419., C ( ),. C, A, B. A B. : (i), A B. (ii), A B. (iii), A B 420., A B,., A B C , A B ( ), C,.,., ,,,,,.,,. 418, [ ],,. 421 [ ],,. 422 A B, C.,, 2. : 1,.,., (heat bath).,,,,.,,,,.,,.,, [ ], C A B, A B. 424, 0 I. 88 c 2015 etsuya Kanagawa
92 4.1.2 C,,., A = B (4.1).,, p A = p B (4.2) ,, A B,, , (4.1)(4.2),.,, , 1 2,,, 430., (4.1)(4.2),. ( ), A B. ( ), C, 425 [ ], ( ). 426, 2,. 427 [ ] 2.,. 428 [ ],., ( 0 ).,,. 429 [ ] ( ).., ( ).,. (thermal equilibrium), (thermodynamic equilibrium).,,,, ( ). 89 c 2015 etsuya Kanagawa
93 (4.1)(4.2) C ( ),, 432 : du C = d Q d W = 0 (4.3), U C., U C 433 : du C = 0 = U C = const. (4.4) ,, U, S 434. C, A B, 435 : C = A + B = const. (4.5) U C = U A + U B = const. (4.6) S C = S A + S B (4.7) 431,,,., (intensive variable)., (extensive variable).,.,. 432, (isolated adiabatic sysytem), d W, d Q, : d Q = d W = ,,.,,. 434 [ ], U, S. (4.1)(4.2),,, p. [ ] (1.1),,, U, S A B, C, A B. A B,, A B.,, c 2015 etsuya Kanagawa
94 , A B, ( ) C 436., (4.4), U C. (4.5) (4.7),, 9, U C C, , ( ),,. (4.5) (4.7),, 2 438,, S A ( A, U A ), S B ( B, U B ) (4.8)., 4, (4.5)(4.6), S A ( A, U A ), S B ( C A, U C U A ) (4.9). C U C, ( A, U A ) 2., C S C, (4.7), S C = S A + S B = S A ( A, U A ) + S B ( C A, U C U A ) = S C ( A, U A ) (4.10) 439.,, [ ] : ds 0,,. 436 [ ].. 437, A, U A, S A, S B 4 ( A B, S A S C ). 438,, [ ],. 91 c 2015 etsuya Kanagawa
95 ( ) , : d Q = du + d W = du + pd ds (4.14) 1, 2, (4.13)., du ds pd (4.15) 442. ( ), ( ) (4.15), (1.1)., (4.5)(4.6) U, : ds 0 (4.16), ( ). ds > 0,.,,.,,. ds = 0, ( ) S = S max. 440 [ ]. 441 [ ] ds d Q = d Q = ds (4.11).,, ds > d Q = d Q < ds (4.12)., (,., ). (4.11), (4.12), : d Q ds (4.13) [ ] (4.13), Clausius,.,. I, ( ). 92 c 2015 etsuya Kanagawa
96 , [ ] 5, ds = 0.,, (4.15) 444 : ds = 1 du + p d (4.17), S = S(U, ) 445. ds(u, ) ( ) S U = 1 ( ) S, U = p (4.18) 446., S(U, ), ( 5 ), (U, ) p(u, ) 447. (4.17), S(U, ) ( 5 ) , C.,. C,, (4.16) 449. C 443. S, 2 : (i) (, ); (ii) [ ( )]., 1 ds = 0 S = S max, ( ) ds > 0. (ii), ( ),. 444 [ ]., C., C,.,,. 445 (4.10),. 446 [ ].,,, p(u, ) (U, ). 447 [ ] (U, ) ( ). 448 S(U, ) [J/K], 4 U(S, ), F (, ), H(S, p), G(p, ), ( )[J]. 449,,, c 2015 etsuya Kanagawa
97 S C ( A, U A ),, C., ds C = 0, : ( ) SC U A A = ( ) SC A U A = 0 (4.19), S C (U A, A ), U A, 452 : ( ) SC U A A = ( ) SA U A ( ) SA = U A A + A ( ) SB U A ( ) SB U B A B = 1 A 1 B (4.20), (4.10) 453., (4.19), 454. A = B (4.22) S C A, 450 S C ( A, U A ). S C / U C. 451 [ ] (4.18),.,, (,, ): (i) A B,, C ; (ii) C ; (iii),, ; (iv),, ; (v), ( S C / U A ) A = 1/ A. 452 [ ], (U A, A ),., 1.,., 4,,, (,, 3, ) ( ): ( ) SB U A ( S B (U C U B ) ) du B d(u C U B ) ( ) SB U B A = C B = B ( ) SB = (4.21) U B B 454,,. 94 c 2015 etsuya Kanagawa
98 ., ( ):, (4.22), ( ) SC = p A p A = 0 (4.23) A U A A B p A = p B (4.24) , C, A B. (4.22)(4.24),., [ (4.1)(4.2)].,, ( 1 2 ),,, ,, 459.,, 460 : d Q = du + d W ds (4.25) 455,. 456 [ ] S = S max ( ds = 0),,,., (4.22)(4.24),,. 457 [ ], 2,. 458 [ ] ( ), 2.,, ( 4.2.2).,,,,.,,,,. 459 [ ],, ( )., ( ),. 460,. 95 c 2015 etsuya Kanagawa
99 (4.13). : d W (du ds) (4.26), F, 461 : df = (du ds Sd ) = du ds (4.27) (4.26) (4.27), d W df (4.28)., 12 : W 12 (F 1 F 2 ) (4.29) W 12, ( )., F 1 F , 465, 461, du ds,,. 462 [ ],., ( ). 463 ( 4.2.2), F, F 1 F [ ], W 12 = , 60,, [ ],. : d W = (d Q du) = du = W 12 = U 1 U 2 (4.30),,.,. [ ],, (, ).,,. 96 c 2015 etsuya Kanagawa
100 : df 0 ( U) (4.15), F (1.9), 470., F. F U S df = d(u S) = du ds Sd (4.32)., 1 2, ( ) (4.15) 471, df + Sd = du ds pd (4.33), F : df pd Sd (4.34) 466 [ ] F, : U = F + S = ( ) + ( ) (4.31) U 2. U ( ) F.., (maximum theoretical work). 467 [ ] F, S,. S.. U = F + S,,.. S (bound energy),,., (0.1),. 468 [ ],,., d Q < ds. S, S.,,. 469 [ ] U = F + S, S, F ,,,, (,, ). 471,,. 97 c 2015 etsuya Kanagawa
101 ,,, : df 0 (4.35) (df = 0), F ( ), F., F,.,,.,.,, ,,, , C 473., C ,, C ,, A = B = const. C (4.36)., (p A = p B ). F, S. (4.34), F (, ), (4.36), 472 [ ]., ( ),,. 473 [ ] 4.1.1, A B.,, C. 474 C ( ). 475 [ ] C,. 98 c 2015 etsuya Kanagawa
102 F : F C = F A ( A ) + F B ( B ) = F A ( A ) + F B ( C A ) = F C ( A ) (4.37) (4.35), F ( df = 0 ),,., F C A, df C ( A ) d A = df A( A ) d A + df B( C A ) d A = p A + p B = 0 (4.38) 478, : p A = p B (4.39) ( ), A B, : dg 0 F, G (1.21). G F + p 479 dg = d(f + p ) = df + pd + dp (4.40) 476 [ ] (i) A B, C (F C = F A + F B ); (ii) C ; (ii) F A = F A ( A ) F B = F B ( B ); (iii) B = C A C. 477 [ ],,,.,,, 3,., 1, 2 ( ). 478 [ ] (4.34), F (1.9), p, ( ) F p = (1.11) ( ).,,. 479 G H S,,, F (4.34). 99 c 2015 etsuya Kanagawa
103 1 2, F (4.34) 480 dg dp Sd (4.41).,, dg 0 (4.42), G 481., G., G,, , 4.2.3,.,,. 482, 483..,,, A = B = const. C (4.45) 480, S, F, G.,. 481 [ ], (variational method) : δs = 0, δf = 0, δg = 0 (4.43) d ( ), δ ( ). (4.43),., 2.,, : δ 2 S < 0, δ 2 F > 0, δ 2 G > 0 (4.44), S,,,., 1 δ, 2 δ 2 S., ( ) c 2015 etsuya Kanagawa
104 p A = p B = const. p C (4.46).,, G ( ).,,.,, G (4.42) 484.,,., 485.,,,,,, ,, : 1., S. 2., 487, F. 3., G.,,, S ( F G ).,., 484,.,. 485,. 486 [ ] (chemical reaction),,. (combustion), (fuel cell), ( ).,, (Gibbs ) (chemical potential). 487,.,., 2, 3,. 101 c 2015 etsuya Kanagawa
105 ,,. F, G,,, F (, ), G(p, ) 488. F (, ), G(p, ), ( ).,., 489 :, (, 1980)., (, 1989)., (, 1989)., (, 1997). ( ( ), 2002)., (, 2014) ,. ( ) c 2015 etsuya Kanagawa
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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)
3.2 [ ]< 86, 87 > ( ) T = U V,N,, du = TdS PdV + µdn +, (3) P = U V S,N,, µ = U N. (4) S,V,, ( ) ds = 1 T du + P T dv µ dn +, (5) T 1 T = P U V,N,, T
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3 3.1 [ ]< 85, 86 > ( ) ds > 0. (1) dt ds dt =0, S = S max. (2) ( δq 1 = TdS 1 =0) (δw 1 < 0) (du 1 < 0) (δq 2 > 0) (ds = ds 2 = TδQ 2 > 0) 39 3.2 [ ]< 86, 87 > ( ) T = U V,N,, du = TdS PdV + µdn +, (3)
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