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1 (A2) , 0, (system) (surroundings) (boundary) (wall) (closed system) (isolated system) , (thermal equilibrium) (thermometer) , 0, (state) (quantity of state) (intensive variable) (extensive variable) (specific state quantity) (equation of state) , i c 2018 Tetsuya Kanagawa
2 1.5.1 SI kg, m, s ,, [ ] ( ) [ ] (quasi-static process) p dv p V p V (ideal gas) Boyle Charles ii c 2018 Tetsuya Kanagawa
3 3.2.1 Boyle ( ) Charles ( ) Boyle Charles [ ] C P C V c V c P κ (isothermal process) (adiabatic process) iii c 2018 Tetsuya Kanagawa
4 5.2.2 (Poisson ) S (specific entropy) (T, v) (p, v) (T, p) p, V, T, S T S ( : C ) (thermal efficiency) Carnot p V T S T S iv c 2018 Tetsuya Kanagawa
5 7.3.3 Clausius p V Carnot Clausius Carnot Carnot Clausius Carnot Clausius ( ) Clausius Clausius (1) Clausius (2) [ ] Carnot Kelvin Clausius Kelvin Clausius Kelvin Clausius Clausius, Kelvin v c 2018 Tetsuya Kanagawa
6 10.5.3Kelvin, Clausius Carnot, A (i) η A > η C (ii) η A = η C (iii) η A < η C [ ] (thermodynamic temperature) 173 vi c 2018 Tetsuya Kanagawa
7 2018 I : 1 2 3F305, 5254 kanagawa kz.tsukuba.ac.jp. 105, 60, ( ) 3. A B, 4 5 : ( A) : 1) [5 ](, ) 2) [ 40 ] = 8 ( ) 3) [60 ] ( B) : 1) [5 ](, ) 2) [100 ] 1. : 1 1) (5 8 = 40 ): 5/1, 8, 15, 22, 29, 6/5, 12, ,, 2017,.,, ABC 1, 2. 2,,,.,. 3 [ ], A B,.,,.,, B, A B,. 5, :,. 6, 2 (4 24 ). 7 I,,.., 11:25, 8:40. 8,,. 1 c 2018 Tetsuya Kanagawa
8 1 2) : 6 26 ( ) 8:40 11:25 ( AB ) 1 3) 8:40 = ( ) 9 1 4), 1 5) 20, ),, ) ( 12 ) ) , D, A+., ( ).,,. 9 [ ], 8:40, ( ). 10,,, (, ).,,. 11 [ ],,.,,. 12,.., 40 %, ( ). [ ],. 13,. 14 [ ] , ( ), ( ). 2017,. AB,,. 15 [ ],,.,,,.,,,,,.,, ( ).,,,,. 2 c 2018 Tetsuya Kanagawa
9 2. (5 ) 16 = manaba.. 3. (. ):,,,,.,, , (,, ), (3F305), 3F , 20 21, ( ): 4 1) (, II),, 23.,., 24,,. 4 2), %,. 4 5, manaba ( ) ,,.,.,. 19,.. 20.,.,, A+. 21,,. 22,.. 23 [ ],,,.,,, c 2018 Tetsuya Kanagawa
10 ,, 27,,,, ),. 29.,., 2 : i),,, ,,,,.,,, 50 %.,, 2 ( ),,. 26,,,. 27 [ ],,,. (i). (ii),, (,, ). (iii),,. (iv),. (v). (vi) (, ).,,,. 28.,,. 29 [ ( )] (i), (ii) ( ) 2. (i),. (ii),,.,,.,,.,.,,,.,, ( ). 30,., ( ),.. 4 c 2018 Tetsuya Kanagawa
11 ii) i), ), ( ) 32.,, ,,.,, ), ,, 38., ( ). 31,.,.,,.,,,,,,.,,,. 32,, (unknown) (known). 33, ( ), ( ),. 34,,.,,,.,,,.,. 35,,. 36, II, 2. I.,. 37 [ ],., (1 ). 38 [ ] ( ),.,,., ( ) ( ),.,,,. 5 c 2018 Tetsuya Kanagawa
12 , 41.,, , manaba 44.,, manaba PDF 46,,, , I II ( ). 39 ( ).,,. 40,. 41,,. 42 [ ],.,,.,,. ( ),,,.,.,,. 43 [ ],. 3,..,. [ ],,,. 44,. 45,,,. 46 PDF,. 47,, ( ) ( ). 48 [ ] I, ( ),,., II,, II, III. 6 c 2018 Tetsuya Kanagawa
13 8. 49.,,,., : 9 1) ( 5 ) :,., 53,.,, 49,,.,,,,,. 50 [ ],, 2.,, (,, ), 1,, 1.,, A B, A B.,,.,, (,, ). 51 2,, 6. [ ] II.,,.,. 52, 6.,, II,.,,,,., ( ). 53 [ ] 1, ( ), ( ).,,.,,.,,.,,. [ ],,.,,,,,, ( ). 7 c 2018 Tetsuya Kanagawa
14 54.,,, 55.,,. 9 2),, 56.,, 57.,,, 58.,,,,,.,, (entropy),,,.,,,,. 55 [ ],.,,.,,. 56 [ ] 21,,, (, 2000), (, 2007). [ ],. II.,. 57,, (,,, ). 56 (2000) (2007),. 58 [ ], ( ). [ ],.,,,.,. 59 [ ],..,,..,.,,. 8 c 2018 Tetsuya Kanagawa
15 , 60,,. 9 3) ( ), 61.,, ( )., 62,, ),.,, 64, , ( ). 61,,. 62,..,. [ ],,. 63 [ ],, ( ).,.,,,. 64 [ : ] (heat engine ( ) ),,,,. 65 [ ] (engineering thermodynamics), (mechanical engineering),., (strength of materials), (engineering mechanics), (fluid mechanics).,,. 66 [ ], 64 65,.,., I,., II,,. II, I,.,, I II,, I. 9 c 2018 Tetsuya Kanagawa
16 0,.,,.,., Newton (Newton s second law of motion), (the first law of thermodynamics) 67, (mechanics) 68,. ( ), ( ) 69.,, (equation of motion) 70.,, ( ).. ( ),.,,,.,. 67.,,,.,,,.,. 68 [ ],. (science/technical english). 2.,,,.,,,,. 69.,, ( ),.,,. 70,. 10 c 2018 Tetsuya Kanagawa
17 : (Q1). (Q2),,. (Q3),.,, 1 : (A1) Newton 2 ( ). (A2). (A3). (Q1).,, (conservation law of momentum) 72, (A1).,., (A3) (A2).. 75., [ ],,.,,..,,,.,,,. 72 [ ],. 73 [ ] (law),,,., ( ),..,, (definition).,. 74 [Hooke ] x F,,. [ ( Duffing )], F = kx + βx 3 (k, β ). 75 [ ],. 76,,,.,. 11 c 2018 Tetsuya Kanagawa
18 , Newton,, m d2 x dt 2 }{{} = F (t) }{{} (0.1) , v, m dv dt = F (t), dx, v dt lim x(t + t) x(t) t 0 t (0.2) 80., x(t) (position) 81, t (time), m (mass), F (t) (external force) t 82.,, (i) m 83. (ii) x(t), ( ) ( ). (iii) F (t) t,. m,., m, d(mv) dt = F (t) (0.3) 77 [ ], 1 (one-dimensional problem). ( ) 2 ( ) ( ) 78 [ ( )] d2 dt 2 = d2 d d d (dt) 2 = dd =,, dt dt dt ddt 2. dt 2, (dt) 2,.,,. 79 [ ], F (x, ẋ, t) ( ). 80 (definition). [:= =: ( ), def = ]. 81 [ ], (position vector).,,. 82 [ ] x (unknown variable) (dependent variable), t (independent variable)., F (inhomogeneous term). F = 0, (homogeneous equation). [ ], ( ). 83 [ ( )],, ( ). 12 c 2018 Tetsuya Kanagawa
19 84. dt, [t 0, t] 85 86, t (mv) = F (t)dt }{{} t 0 }{{} (0.4) ( ) 87.,, 88, (mv) = mv(t) mv(t 0 ) (0.5), (t 0, t ). (0.4) (momentum) (impluse),,.., : (A),.,,.,, (B), ,,. 85 [ ] (domain) (range), a t b, t ( ), [a, b] ( )., a < t < b (a, b). 86 [ ] dt,. (integration by substitution).. 87 [ ( )] (0.4),,, t,.,. t t 0 F (ξ)dξ 88 [ ] f = f 2 f 1 (difference).,.. [ ] (Greek letter) (delta) δ,. 89 [ ], ( )., ( ). 90 [ ] 100,, (i) 100,. (ii) 90, 10, 100. (iii) 90, 60, 150. (iv) 130,. 91 (0.4).,,. (mv) = c 2018 Tetsuya Kanagawa
20 ,.,, (A), (B) ( 2) , 94, ,,,..,,.,., (mass), (momentum), (energy) 3 96, 97.,, 98.,,, 92 [ ], ( ). 93 [ ] (mechanical energy), (kinetic energy) (potential energy). 94 [ ] m, k (Hooke ). x = 0.,,, x(t) 1 (2 ) : ( ) 2 1 dx 2 m + 1 dt 2 kx2 = const. 95,,, ( ( ) ).. 96 [ ],,.,,. 97 [ ],. (classical mechanics), 1,,, (, ), ( ). (quantum mechanics) I,,, c 2018 Tetsuya Kanagawa
21 ., ( ), (internal energy) , (A2), 102.,.., 103.,. (i), (ii), (iii),, (iv), 104. (iii) (iv),,., Newton ( ) ,,,,,. [ ] (elastic body) (plastic body),, ( ).,, ( II ). 100 [ ] 2 (i),.,,. (ii),,., ,,.,,. 102,, ( ). 103,,. 104 [ ],,,. 105 [ ],.,, ( )., ,. [ ],,. 15 c 2018 Tetsuya Kanagawa
22 , 106,,.,., ( 2).,, ( ).,.,,,.,,.,,,.,,,, = Newton ( ) = 106 [ ],. 107., (6 ).,. 108 [ ] 1 : ( ) 2 :.,.,. [ ], 1 : (Newton ) 2 : 3 :., Newton,,.,. [ ] 1,. 16 c 2018 Tetsuya Kanagawa
23 1, 0,,.,, ( )., ( ),,.., 110., 111.,,. 1.1, 112.,, (system) 113,, (matter). 109 [ ],,, 100 %. 1 1,. 110 A,. 111 [ ],,,,.,,. 112 [ ],,,,. [ ],.,,.,,..,,. 113 [ ], (thermodynamic system). 17 c 2018 Tetsuya Kanagawa
24 ,,.,., (degree of freedom),,. I, (surroundings) 115. (infinite).,.,., (heat source) (boundary),, (wall) 117, (box).,,, :.,, ( 5) [ ] II,,,. 115 [ ], (environment). 116 [ ] (heat bath). 117 [ ] II. I, [ ] (rigid body). (mass point), (elastic body). 18 c 2018 Tetsuya Kanagawa
25 1.1.5 (closed system) , ( ),.,,, ( ) (isolated system), ,,., (expansion)., ( ), ( ),.,,,,, 122.,, (i), (ii) ,, ( ) [ ] I, II (open system).,, ( ),. 120 [ ].. [ ], (turbine) (compressor) (throttling valve).,,., ( II), I,. 121,, ( II) ,,,,. 123 [ ],,. 19 c 2018 Tetsuya Kanagawa
26 1.2 0, 124.,, (thermal equilibrium) 20 C ( A), 100 C ( B).,,,., ( ) 1, ( A+B), 30 C,.,,.,.,,., ( ) 126 ( ),, [ ( )] (thermal equilibrium) (thermodynamic equilibrium).,,. I, II. 125 [ ],,.,,.,,,,, ( ). 126, ( 1.3). 127 [ ],. 128 [ 1],, (fluid mechanics, 2 ABC) (heat transfer, 3 AB). [ 2], T, T (p, V ) p V ( )., ( ) ( ),, T (x, t), x t.,,.,,. II. 129 [ 128 ] (2 ABC), ( ) T t = T κ 2 x 2,, T (x, t)., κ [m 2 /s], c 2018 Tetsuya Kanagawa
27 2, , 133, ( ) , 1 (law).,,,., : 0 (the zeroth law of thermodynamics) A B., B C., A C.,., ,,.,, Twitter (diffusion), ( ).,. [ ], T (x, t), T (p, V ) (p, V )., (,, ),.,.,, ( ). 130 [ ] ( 1.3 ).,. 131 [ ], (working fluid),,,. 132 [?],., 2, 2,, 2,.,. 133 [ ].,,, (,, ).,,. 134 [ ], 2,.. 2.,. 21 c 2018 Tetsuya Kanagawa
28 1.2.3 (thermometer),., : (i) A B. (ii) B C. (iii) C ( A),,. (i)(ii)(iii),,,,, ,,, ( ). 135 [ (1/2)],.,. (,,., ) A ( A) B ( B),,, AB AB ( AB A B ),., (i) (ii),, (ii) (iii).,,, A AB., AB C,, ABC ( ABC AB C )., 0, A C,. 136 [ (2/2)] 135 (a) A, B, (, ). (b) B, C,. (a), A B, AB, A., A B, B A.,, B, A.,, A C. (b), A C,, C A ( A)., C, A, A. [ ]..,,.,,.,, (,, ) , (a),, (b) ( ), (c),. (a)(b)(c),.,,,. (, ),,. 22 c 2018 Tetsuya Kanagawa
29 (i)(i)(iii),, 0..,,., 1, 1. 0., A B, 138. (thermometer), 0.,, (temperature),., , 0, 0, A, B, C, C 140.,,, ( 0), [ ]..,,. 139 [ (Twitter)] A C, B (A C ), A., ,..,,,.,,,,. 141 [ ],,. 2,,,.,, A B, ( ).,, 0,.,,,.,,,. 142 [ ], 2,,., 0, 1, 2, ,,,. II. 23 c 2018 Tetsuya Kanagawa
30 1.3,, 144., (state) ( ) 146, 147.,, ,.,,.,, 1 150,,.,,,. 144, ( ).,,,,,. ( ),,,. 145 [ ]. ( )., ( ),, ( ). 146 [ ],,, ( ). 147 [p V ] V, p, (starting point) (terminal point)., ( 2). 148 p V ( 2.5) T S ( ) 1., t = 3 x = (0, 2, 1) 1.,,. 149 [ ], 2 (plane), 3 (space). ( ), ( ) 2 ( II)., 3, 3 ( ) [z ], 2 [x y ]., I p V ( ),. 150 [ ], 25 C,.,. 24 c 2018 Tetsuya Kanagawa
31 1.3.2 (quantity of state),, 151.,.,.,,.,,,, ( 1.3.3) ,, 155.,,.,., ,,. (i) 1.2,,, (, ) t [ C], 151 [ ] I,.,,,. 152 [ ] 1. (state variable) ( II ). 153 [ ( )],, p V ( ).,,,,,.,. 154 [ ( )],,., ( ),.,,,. 155 [ ],,.,,.,,. 156 [ ( )],,. [ ],,., II,. [ ] 2 ( ) ( 2 ). 25 c 2018 Tetsuya Kanagawa
32 (absolute temperature) T [K] ( ) 157 : T [K] = t [ C] (1.1), 0 K 158.,, (ii) (pressure), p 161. Pascal ( ), Pa ,, (volume) 164, 3, 165., t x,,.,,., p T [ ], (thermodynamic temperature)., 6,, [ ] II (triple point),, ( ) ( ) ( ), 0.01 C, [ ],,. 160,. [ ] (rational number),. ( ). 161 [ ] p, p ressure. P. 162 [ ] Pa = N/m 2 = (kg m/s 2 )/m 2 = kg/(m s 2 ),,, ( ), atm. 1 atm = Pa.,.,,,. 164 [ ],,. 165 [ ], ( ), ( ), ( ),,.,. [ ] (kinematics), (velocity) ( (displacement) (acceleration) ), [ ], x t, p, T, V. 26 c 2018 Tetsuya Kanagawa
33 1.3.4 (intensive variable) (extensive variable) 167 2, ( ).,, ,,.,, ( 2.2), ( 2.6),.,,. ( ), ( : atmospheric pressure).,, ( ) (specific state quantity),, ( ) 171,. ( ) ( ),., ( ),., 172.,.,,. (specific 167. I 4 II,. 168 [ ],,.,. 169 [ ] 3, II. 170 [ ],.,. 171 [ ].,. [ ] ( ),,. 172 ( ),. 27 c 2018 Tetsuya Kanagawa
34 volume) v 173 v V m [m3 /kg] (1.2)., V [m 3 ], m [kg].,, v, ρ 175 : ρ m V = 1 v [kg/m3 ] (1.3) v [m 3 /kg],, ρ [kg/m 3 ],. I,, ( ) 176..,,,,. 2. V, v, ρ, m [ ],,.,,. [ ( )] ( ),,, ( ). 174 [ ],,.,,,,,,. 175 [ ], (density). 176 [ ],,..,,. 28 c 2018 Tetsuya Kanagawa
35 1.3.6 (equation of state) ,., p, T, V 3, : f(p, T, V ) = 0 (1.4), p (p = ) 179 : p = g(v, T ) (1.5) f g.,, f g 180. (1.5), 2. (1.4) 3,, 2., ,, 3. 1,, [ ] (Equation Of State) EOS. 179 [ ] (1.5) (explicit function), (1.4) (implicit function). 180 f g (, 1 ),. 181 [ ],, c 2018 Tetsuya Kanagawa
36 1.4, , , 184., ( 1.2)., ( ) , , 1.,. p V ( 2.5)., 1 1, , , 1 2, , 189,,. 182 [ ],, [ ], (change) (process) (, ).,.,,,.,,, (p 1, V 1, T 1 ) (p 2, V 2, T 2 ). 1 2,,., (path). 185 [ ], p V..,,. 186 [ ],, ( ). 187, ( ),. 188 (curve) (straight line). 189 p V. [ ]. 30 c 2018 Tetsuya Kanagawa
37 ,, ( ), ( ) ,,,., (explosion)., 193,.,. 194,.,.,, 195.,., 1 2.,. 190,. 191 [ ], 1 20, , 2 ( A) 30, ( B) , A B,.,. 192 [ ],.,.,. 193 [ ] (non-equilibrium) ( ). 194 [ ],. 195, ( 2.4). 31 c 2018 Tetsuya Kanagawa
38 1.5 SI (The International System of Units). SI (fundamental unit) SI (derived unit) 196 SI (prefix) SI,, (length): m (, meter) (time): s (, second) 198 (mass): kg (, kilogram) 199 ( ; mole number): mol (, mole) (absolute temperature): K ( ) 200 SI 19,, 4 : (pressure): Pa = N/m 2 (force): N = kg m/s 2 (work): J = N m (power): W = J/s 3. 4, SI. 4. Pa m 3 J, SI, kg m 2 /s SI. 197 [ ] (luminous intensity) cd (current) A, [ ] [J/s].,,. 199 [ ] g ( ). 200 Kelvin,. 32 c 2018 Tetsuya Kanagawa
39 1.5.2 kg, m, s 1 N... N J W,,.,.., Newton ( ) F g = m = 1 N 100 g (1.6) 9.8 m/s2,, kg, m, s., 201., (i), (ii), (iii), , N ( ).,,., (1.6). 203,,,,,. 201 [ ],.,.,,. 202 [ ],,,, [ ] K ( ), K.,,. 33 c 2018 Tetsuya Kanagawa
40 2, ,, ( 2.2 ).. 1 ( )., : 2,,., : 3,,.., : (the first law of thermodynamics), ( ),., [ ],, ( 0). 205 [ ] ( ), ( ).,,, ( ). 206 [ ], ( ),, ( 0). 207 [ ] 100, 80.,, 20.,., 100, 120,.,,. 34 c 2018 Tetsuya Kanagawa
41 2.1.1,,,,,, : 1). ( ), ).,, )., 210., : ( ) }{{} ( ) = ( ) }{{} (!!) ( ) }{{} ( ), 211.,, 212, ( )..,,.,.,,, [ ],,. 209 [ ], ( ). 6.,,. [ ],. 210 [ ],..,,,,,,.,,,. 211,. 212., ( ). 213 [ ],.,.,,, (,, ). 35 c 2018 Tetsuya Kanagawa
42 , [ ] 216,,,., Joule 217., (cal) 218 : 1 cal = J (2.1) , 1000.,, 219. ( ), ( ), ,,. 214,., ( ),. 215 [ ].,.,,,.,,. 216., ( ). 217 [ ( II )], (mixing)., (, ( 5)). 218 [ ] (mechanical equivalent of heat).. 219, (molecule), (atom), (quantum),,. 221,,. 36 c 2018 Tetsuya Kanagawa
43 2.2.2 (internal energy),, ,,,.,., ( ) ,, 227., [ ],. 223 [ ], 0 (, 0 ).. (analogy),,, ( ). 224 [ ],, (statistical mechanics)(!!). (kinetic theory of gases),.,,., ( ), (2 ). [ ] ( ).,. 225 [ ],, ( ).,,,, 3 (i),,,. (ii),. (iii),,, (,,., ). 226 [ ] ( ).,,. 227 [ ],., ( 0.3 ). 228 [ ],., (kinetic energy) ( (rotation), (translation), (vibration) ), (potential energy). 229 [ ] Joule ( II),, (i), (ii).,,.,. 37 c 2018 Tetsuya Kanagawa
44 2.3 1 ( 1) ( ) U 1 U , Q 1 2, W ,, U 2 U 1 = Q 1 2 W 1 2 (2.2)., 233 : U 2 = U 1 + (Q 1 2 W 1 2 ) (2.3) (2.3), : 2 ( ) = 1 ( ) + ( 1 2) (2.4) 2, 1, 1 2. (2.2), U U 2 U U = Q 1 2 W 1 2 (2.5) 235.,,, 230 [ ]..,.,.,, x p. 231 [ ] 1 2, W [ ] 1 2. U ( ) 1,, W Q, ( ), [ ],. 234 [ ], 1 2 ( 1 2)., [ ( )],. U = Q W.,,. ( ), ( ), ( 38 c 2018 Tetsuya Kanagawa
45 ,, (2.2) ( ), 2. W Q. 1 2 ( )., A B,.,, ( ), 237 : W ( 1 2), }{{} Q ( 1 2) }{{} (2.6) ( ) 238. ( ( )), ( ( ))., 239. ).,, (1 ),, ( ).,,, (i), (ii), (iii) (ii) 1 ( ). 236 [ ] ( 1) 50. ( ) 30, ( ) 20, ( 2), (2.3), U 2 = U 1 + Q 1 2 W 1 2 = = ( ).,. [ ],.,, ( ) 238 [ ] (2.6). (2.6), ( ). 239 [ ] ( ).,, ( ) ( ), 1.,,.., ( ) ( ), ( )., 39 c 2018 Tetsuya Kanagawa
46 , A, B., A B , 2 U 2. (2.2),.,,,., (2.2) U = U 2 U 1 ( ), Q 1 2 W 1 2 (, )., Q 1 2 W 1 2 U ,., ( )., 244,., 1/,,, ( ( )), ( ) ( ). 240 [ ] 1 2, A, B, C., : W 1 2A W 1 2B W 1 2C, Q 1 2A Q 1 2B Q 1 2C (2.7) 241 [ ] A, B,., W 1 2, (2.2), U 2. [ ] Q [ ] (2.2), U = 40, 40, U = Q 1 2 W 1 2 = }{{} A = }{{} B = 40 }{{} Q 1 2 W 1 2.,, [ ] 242, ( ),,.. U = 40 ( ). 244 [ ] (infinitesimal). 40 c 2018 Tetsuya Kanagawa
47 1., 245.,,. U 2 U 1 ( U 0) 246., , ( ), du = d Q d W (2.8) , d, 250. (2.2)(2.3) (2.8),., (2.8) (2.3),,. (2.8), (definite integral), 251 : 2 1 du = [U] 2 1 = U 2 U 1 (2.9) 245 [ ] ( ),.,,.,,. 246 [ ],., ( ).,,,. 247 [ ] U 0, 2 U 2, U 1 1 ( 1)., W 1 2 d W. Q 1 2 d Q. 248 [ ],,.. ( ). 249 [ ] du., d Q d W, ( ),.,.,, ( ). 250 [ ]. d.., 100.,, d δ., d. 251 [ ], U2 U 1 (1 2). du = [U] U2 U 1,, 41 c 2018 Tetsuya Kanagawa
48 ,. d W, 252., ( ) 1 2,, W 1 2 ( )., 253 : 2 1 d Q Q 1 2, 2 1 d W W 1 2 (2.10) (2.8) (2.9)(2.10), (2.2) ( ) 254., I, ( ) [ ],.,. 253 [ ], d,. 254 [ ] 2 1 d W = W 2 W 1.,., W 1, 1 ( 1).,. [ ] 1 2, ( ),., ( ), ( ). 255 [ 1/3 ( )] I,. ( ) : dy dx lim y(x + x) y(x) x 0 x (2.11),, ( ). ( )..,, df, y x,,.,. ( ),. 256 [ 2/3 ( )], df. df,.,, (, )., 2 (,, ), ( )., df f f. [ ],. 257 [ 3/3 ( )], df, II (total differential) df(x, Y ) (f, X, Y ). df(x, Y ) = f f dx + dy (2.12) X Y, ( II )., I,,., ( III). 42 c 2018 Tetsuya Kanagawa
49 1. (2.8), (2.2)(2.5) , (2.2)(2.5), (2.8) 258.,, (2.10) ( 1)., 259,,,.,.,,, ( ) [ ] V, dv., dv, ( ), 260.,,. (2.8),, ( ).., 261, U du,., (2.8),., (2.2), U. ( )., [ ],,,., ( ) [ ],, [ ] ( ), ( )., ( ). 261 [ ],,.. 43 c 2018 Tetsuya Kanagawa
50 , ( ) du (2 ) U 262. U, (Q 1 2 W 1 2 ). Q 1 2 W 1 2, (Q 1 2 W 1 2 ).. 2. ( 1.1.6) 263., 264. [ ],,, du = 0 (2.13)., 265 : U 1 = U 2 (2.14), [ ], du U.,. 263 [ ],,,..,. 264, ( ). 265 [ ],. 266 [ ] [ ].,,.,. 268 [ ] 267, (proposition)., ( I, 1 ),,,. 44 c 2018 Tetsuya Kanagawa
51 2.4 (quasi-static process),, ( 1.4).,, : (i),. (ii) p dv 270,.,.,,., , , ( ) W 1 2, W 1 2 F in x (2.15) 273., x x., F in,, F out ( ) 274., ( )., 269 [ ],,.,, ( ) ( ). 270 [ ],,.,,. [ ] (engine) (cylinder). 271 [ ] (deformation) (displacement). (continuum mechanics) ( (strength of materials) (fluid mechanics)). 272 [ ],,.,,. 273, ( ),, F in x ( ).,, [ ] F in, out. 45 c 2018 Tetsuya Kanagawa
52 x 0., (2.15), x dx, W 1 2 d W ( ) 275., d W = F in dx (2.16) 276.., F out = F in (2.17) 277. (2.17),,., F in F out,, 278.,,,., p, F in F in = pa (2.18) ( 1.3.3)., A (cross-sectional area). (2.18) (2.15), d W = padx = p dv (2.19)., V, (2.19) dv Adx (2.20) 275 [ ] ( )., d,,. 276 [ ] (2.15), (2.16).,. 277 [ ] (2.17), (,, )., Newton ( ). [ ],,,,. 278 [ ],,... F in F out.,,,. 46 c 2018 Tetsuya Kanagawa
53 . dv ( ) : (i), F p.,,., ( ) , , (ii) d W, p V,. d W. p V, dv ( )., p dv.,. 1 1,,.,, 279 [ ], dv = Adx. 280 [( ) ] (2.20)., Adx = ( ) ( ) A 1/ 1/ = ( ),.,, : }{{} A }{{} dx = dv }{{} 281 [ ],. 282 [ ],, [ ],,..,,.,. 47 c 2018 Tetsuya Kanagawa
54 284. (iii).,, ( )., ( II).,,.,,., ( ). (iv) (2.17),, ( 4).,, ,,,, , (2.19). 4., d W = p dv., (2.19), (2.16)( (2.15)) 287. [ ] (2.17), p. 2.5 p V, p V. 284 [ ], ( ( )),, [ ],. 286 [ ] d W = p dv,,.,,,.,.,,.,,.,,,,, ,. 48 c 2018 Tetsuya Kanagawa
55 , (2.19) 288. W 1 2 (2.10) (2.19), W 1 2 = 2 1 p dv (2.21) ,,. p V,, p,., ( ), (2.21)., p.,, ,, p V (2.21) W, p V, p = f(v ) V 292 : W = 2 1 p(v )dv (2.23) 288 [ ], (2.19). 289,, [V 1, V 2 ], V [1, 2]., [ 1, 2],, V,. 290,. 291 [ ],,,,,. 292 [ ] ( ):, S = x2 x 1 y(x)dx (2.22), x y x 1 x x 2, y = f(x) x (area). [ ] [x 1, x 2 ].,, [y 1, y 2 ] [y 2, y 1 ] (,, y 1 y 2 ). 49 c 2018 Tetsuya Kanagawa
56 p, V, p V ( ),,. 5. p V, p 0 ( ) 295. V 1 V 2,, p 0 (V 2 V 1 ). (i). (ii) 296. (iii) p V. [(i) 297 ] (2.21)., p ( ) p 0 298, (2.21) : W 1 2 = 2 pdv = 2 2 p 0 dv = p 0 dv = p 0 (V 2 V 1 ) (2.24) , [ ], x y, x, y., p V, p, V,., 5,. 294 [ ] p, (indicator diagram). 295 [ ], (isobaric process).,. 296 p V. 297 [ ],,,,. 298 [ ],,. 299 [ ],, p [ ],,. 301,,,,.,, ( ). 50 c 2018 Tetsuya Kanagawa
57 U pv 303, H : H U + pv (2.25) ( ).,. 7. pv U , 305. (enthalpy),, ( 7) ,. ( ), U., U, 302, [ ] p dv pv., U p dv,, U + p dv. pv p dv.,., pdv = p 1/ 1/ 0,. 304 [ ],.,,.,. 305 [ ] (2.25),,.,,.,,,.,,, I,., II,.,,. 306 [ ]., (entropy) [ ] (open system), (jet engine), (throttling valve), (turbine), (compressor), (pipe) (duct), (heat exchanger) ( ).. 51 c 2018 Tetsuya Kanagawa
58 , pv 308. pv 309., U. H 310 : ( ) }{{} H = ( ) }{{} U + ( ) }{{} pv,,, ,. I, df 312., f g fg, ( ) : d(fg) f dg + g df (2.27) 308 [ ] p dv pv. ( ) dv, ( ) V ( p ).,.,,, V ( ). ( ), p V., F x = pax = pv (2.26). F x ( ), pv ( ). 309 [ ] ( ),, (flow work) (eliminate work),., pv,., pv. 310 [ ] U ( ), pv ( ), H. [ ] p dv,. 311 [ ],,., ( : supersonic flow) ( ), (3 ). 312 [ ] (,, ), ( ). 52 c 2018 Tetsuya Kanagawa
59 , , (2.27), H ( (2.25)) ( ): dh = du + d(pv ) = du + p dv + V dp (2.29), H, du = d Q p dv (2.30). (2.30) du 315. (2.29), (2.30) du du }{{} = dh p dv V dp (2.31) 313 [ ] ( 314). [ ] (product), d(fg) dt = f dg dt + g df dt (2.28)., I, (2.27) (2.28). 314,. 314 [ ] pv, p V., pv (p, V ) (, pv f = f(p, V ) )., II (total differential) : d(pv ) = ( pv p ) dp + V =const. ( pv V ) dv = V dp + p dv p=const., 2 (. II ). 315 [ ], U ( ), U, H., H, U : }{{} U = H pv }{{} 1,. 53 c 2018 Tetsuya Kanagawa
60 316. (2.30), p dv, ( ): dh }{{} = d Q +V dp }{{} (2.32).,, 317., (2.32) 318.., (2.30) du = d Q p dv }{{} (2.30) : du = d Q p dv (2.33) dh = d Q + V dp 6. (2.30), (2.32). 7. : U = H 2 p dv V dp (2.34) [ ] (2.31), 1 2. (i), (ii) 316 [ ].,.,,,. 317 [ ] (2.31) (2.32),. ( ), ( ). ( 0). 318.,. 319 [ ].,,,. 54 c 2018 Tetsuya Kanagawa
61 320., , d Q = du + p dv = dh V dp }{{} (2.35), 2 : (i) [ ],, dp = 0., (2.35) : d Q }{{} p = dh (2.36) p=const.., : Q 1 2 = H(= H 2 H 1 ) (2.37) (ii) [ ] 323,, dv = 0, d Q V = du (2.38). 1 2, : Q 1 2 = U(= U 2 U 1 ) (2.39) 320. (2.31),.,,. 321 [ ],,. 322 [ ],,.,. 323 [ ],,,, ( ). 55 c 2018 Tetsuya Kanagawa
62 ., (i),. (ii),.,, H U. ( ),, 2.6 ( 1 )..,.. (2.37)(2.39), 324. (2.36)(2.38), , (2.37)(2.39). 324 [ ], (2.37)(2.39),, ( 4 ). 325 [ ] x, : f x=const. f x (2.40),,., = const.,. 326 [ ( II )] y ( ) f(x, y) f f(x, y), (x, y) x x y,,., y ( )., ( II ).,,.,. II,. 327 [ ]. y y = const..,. 328 [ ] ,,.,,,. 329 [ ],,., 0, (,, ). 56 c 2018 Tetsuya Kanagawa
63 (heat engine) (cycle) 331,,., 1 ( ) 2 ( ). p V, (closed curve) ,,., ( ), ( ) ( 333 ),., (perpetual motion machine of the first kind) 334.,., 335. (2.8). 9., ( ). 330,. 331 [ ]. Carnot ( ),, (. ), Otto (. ), Diesel (. ), Sabathé (, 2 ). (. ),, (Stirling)., Brayton (.,,,, ). Rankine,. ( ). 332 [ ],, (,,, ). 333 [ ],,, (power). 334 [ ]. 335 [ ],,,.,.,,. 57 c 2018 Tetsuya Kanagawa
64 [ ] (i). (ii),. (iii) (2.8),, W = 0 d W = 0 (2.41). (i) (ii). [ ] 336.,,,, (2.41). 2.9,,.,,, 337.,,.,.,,.,,..,,. J K, kg C.,.,.,,,. 336 [ ],, (2.8) (2.2).,. 337,,.,.,,,, ( ). 58 c 2018 Tetsuya Kanagawa
65 , p V, pv = const. (2.42),, U, U = const. (2.43) , p 1, V 1, p 2, 2. 1) 2.. 2) p 1 p 2.. 3) (2.42), p 1 V 1. 4) 1 2,. 5),.,,,.. 6) p 1 = 1 atm 340, V 1 = 1 l 341, p 2 = 3 atm, 4) 5). J., 1 atm = Pa, 1 l = 10 3 m 3, ln 3 = [ ] (isothermal process) ( 5). 339 [ ], (2.42) Boyle ( 3 ), (2.43) Joule ( II ) [ ] 1 1 atm, Pa.,.,. 341 [ ] 1 l = 1000 cm 3, [ ], ln Napier e, log 10.,, ln log,.,,. 59 c 2018 Tetsuya Kanagawa
66 p 0 [Pa], m [kg] 344, v 1 [m 3 /kg], Q 1 2 [J]., ( ) 1, ( ) ),. 2),, U, du = d Q mp dv (2.44) (m ).. 3), U,. U = Q 1 2 mp v (2.45) 4) U, H., H : H = U + mp v (2.46) 5) Q 1 2 = 2257 kj, m = 1 kg, p 0 = MPa, v 1 = m 3 /kg, v 2 = m 3 /kg, U H ) 1 H 1., 343 [ ]., ( ).,,. 344 [ ] SI, g kg.,,,. 345 [ ],. 346 [ ], ( 3). 347 [ ],, ( ).,. 348 [ ], ( ). 349 (2.44)(2.45) (2.34) [ ],.,,. 60 c 2018 Tetsuya Kanagawa
67 351., U 1 = 30 kj 352, 5). 7) 2 U 2 H (2.45) (2.46) 2, [ ],.., H = U + pv. 352,. 353 [ ]. 354 [ ], 1 2 ( ). 355 ( ). 61 c 2018 Tetsuya Kanagawa
68 , (ideal gas) I, , 359,., 2, 360 : (i) Boyle Charles (ii) Boyle Charles, m. 356 [ ],.,.,,. = =.,,. 357 [ ] (perfect gas). [ ] (fluid mechanics/dynamics/engineering),,,., (gas) (liquid) (,, ). 358 [ ] engineer ( ), engine ( ). 359 [ ]., (physics) (engineering),, (definition)., a.,. 360 [ ]..,, I,., (virial expansion). (real gas), (interaction), van del Walls ( ). ( ). 361, ( 4 ).,, (specific heat). 62 c 2018 Tetsuya Kanagawa
69 3.2.1 Boyle ( ), (isothermal process), Boyle : pv = const. (3.1) 9. (2.42). p V Charles ( ), (isobaric process), Charles : V T = const. (3.2), T [K]. 10. (3.2). p V Boyle Charles,,., Boyle (3.1) Charles (3.2) 364 : pv T = const. (3.3) 362 [ ],., (3.1).., (differential equation) (general solution) (arbitrary constant) ( III )., 1,, III. 363 [ ] 362, 1 ( ), 2., 2,., I, ( ),,. 364 [ ] ( ), Boyle Charles ( ),, ( ) Boyle Charles. 63 c 2018 Tetsuya Kanagawa
70 , Boyle Charles , 2 3,, i (i 1) i, p 1 V 1 T }{{ 1 } 1 = }{{} 1 2 p 2 V 2 T }{{ 2 } 2 = }{{} 2 3 = p iv i T i }{{} i (3.4)., 1 1, (3.3). : pv T (3.5) (3.4), p 2 2,, (3.3).,,., pv T = const. mr }{{} (3.6) 365 [ ] Charles, Boyle Charles,, (numerator) (denominator), T, K. ( ). 366, [ ], (experimental error), (, ). 368 [ ],,.,. 64 c 2018 Tetsuya Kanagawa
71 ,. pv/t,,.,, m, R, ,,, : pv = mrt (3.7),, 1.3.6, ( ) (1.4)(1.5) f g : p = mrt V = g(v, T ) f(p, V, T ) = 0 (3.8) 2, p, V, T ( ) 3, 2 (1 ). 11., (3.8), (3.6)(3.7), mr. m 369, ( ), R R [J/(kg K)], 372, [ ].,,. 370 [ ]...,. 371 [ ],.,, = m.,,,., [ /kg]., m, R,. 372 [ ], [ ]. 65 c 2018 Tetsuya Kanagawa
72 3.3.2 R [J/(kg K)] 374.,,,, 375, (3.7) R = pv mt (3.9),, : Pa m 3 kg K = N/m2 m 3 kg K = N m kg K = J/(kg K) (3.10), (3.10),,,,,, R ( ). 374 [ ],,. 375 [ ] 4.,,,,,. 376 [ ], ( ).,,,,. 377., ( ). 66 c 2018 Tetsuya Kanagawa
73 3.3.3 (3.7).. V , p }{{} V = mrt (3.7) m, v = V/m ( (1.2)), pv = RT (3.11)., ρ = v 1 = m/v, p = RρT (3.12) 380. (3.11)(3.12), (p, T, v, ρ). (3.7)(3.11)(3.12) ( ). 381, (3.7)(3.11)(3.12)., (3.7) (3.11)(3.12),, [ ],,., ( ),, ( ).,, ( 1.3.5). 379 [ ],..., ( ).,. 380 [ ], p = ρrt,,,., R,.,,. 381 [ ],. 382 [ ],, ( II). 383 [ ],,, 4., ( ),. 67 c 2018 Tetsuya Kanagawa
74 3.4,, ( ) , Boyle Charles (3.3), m pv T = const. nr 0 }{{} (3.13), n., pv = nr 0 T (3.14)., n [mol] ( ), R 0 [J/(mol K)] R, R 0, : R 0 = J/(mol K) (3.15), (3.3) R R 0., 384 [ ( )], pv = nrt, R. R, R 0,, ( ). R, ( ).,,. 385 [ ] (universal gas constant).,. 386 [ ], R 0.,,. 387 [ ],, (subscript) R [ ], R R 0,,.,,. 68 c 2018 Tetsuya Kanagawa
75 ,,.,.,. : Boyle Charles (3.3),.,. ( ),.,, 389., 390., ( ).,, , 394.,,,., 395, 389 [ ], [ ],,... ( ),. 391 [ ] ( ),.,, ( )., [ ], (3.14),, (3.15),. 393 [ ],, R 0,, R.,.,,,,,. 394 [ ], II. [ ],,,,. I,,, (3.7). 395 [ ],, pv = NkT 69 c 2018 Tetsuya Kanagawa
76 .,.,,,,. 12. (3.14), Boyle Charles m [kg], n [mol], M [g/mol] m = nm (3.16)., (3.16) k (kg g ). (3.16), (3.7) pv = mrt = nmrt nr 0 T }{{} (3.17)., R 0 R : R = R 0 M (3.18), R [J/(kg K)], M [g/mol] ( ) R 0 [J/(mol K)],., (i) R 0 ( ), (ii) M ( )., k Boltzmann, N ( ).,,, pv T. 396 [ ],,. 397 [ ( )], (molecular weight) (nondimensional number),,., ( ), g/mol,., k. 398 [ ],, kg, g/mol, kg g. 10 3,. 70 c 2018 Tetsuya Kanagawa
77 399.,. (3.18) (3.16), (3.17)(3.18)., (kg g) , 403.,,, , ) R [J/(kg K)]. M = 29 g/mol, ( ) (3.15) ) p = MPa, T = 25 C, V = 250 m 3 m [ ].,, (air), (oxygen), (nitrogen),. 400 [ ],,, (3.18).,. 401 [ ], (3.18). (physical property),,,,. 402 [ ]. 403.,,, (, ) ( ). 405 [ ],,,..,. 406 [ ],,. 407 [ ], (5 ) ( ) 3A304.,,,,. 71 c 2018 Tetsuya Kanagawa
78 17. V 1 = 47 l 408, p 1 = 14.7 MPa, T 1 = 20 C, 1., M = 32 g/mol, (3.15). 1) R. 2) m. 3) 1 2, p 2 = kpa, T 2 = 35 C, 2., V 2,. 4) 3),, Boyle Charles ,, 100 J, 70 J ( ). [ ], Q., ( A). ( B) p 0, V. 2, ) A B, ) A B, p V. 408 [ ] ( ),.,,.,,, ,,.,,,. [ ] 170 J A B. 413 [ ] ( A) Q. ( B) Q p 0 V. 72 c 2018 Tetsuya Kanagawa
79 4 1 K, (heat capacity).,,,, (specific heat)..,,, 414. I 415.,,, [ ], U H, ( 2.6): d Q = du + p dv }{{} = dh V dp }{{} (2.35),,, ,. d Q p=const. = dh (2.36) 414 [ ],, II. 415 [ ] II,, 9.,, ( II).,,. ( ). 416 (5/1/2018),. 417,. 418 [ ] [J],. 73 c 2018 Tetsuya Kanagawa
80 , 419 : d Q V =const. = du (2.38), 420., m ,, , 424.,, d Q, dt , 427 d Q dt (4.1) 428., (4.1)., C 419 [ ], V = const. V. 420,,,. 421 [ ] I ( ),. 422 [ ],.,,,. 423 [ ] ( 3.1),.,.,,. 424 [ ] (physical property). ( ),. [ ] google. 425 [ ] ( ), ( ). 426 [ ],,,. 427 [ ],.,,. 428 [ ] A B, A B. 74 c 2018 Tetsuya Kanagawa
81 : d Q }{{} CdT (4.2) C C [J/K] (4.2) 431. C, 432., d Q dt. C, C P (4.2) (2.36) : d Q P }{{} = dh }{{} C P dt (4.3) (2.36) (C P ) 1 (2.36) , C, P, C P. C P. 1, [ ] (4.2) C d Q/dT, C., ( ).,,,, ( )., d Q dt,,,. 430 [ ] (restoring force) Hooke ( ) F = kx k F/x ( ) k., (elastic body) (strain) (stress) Hooke ( ). 431 [ ],,.,, ( II)..,,. 432,,. 433 Q T, Q = C T [ ].,,. 75 c 2018 Tetsuya Kanagawa
82 ( ) 435.,., (4.3), Q 1 2 = H = C P T }{{} = (4.4)., H = H 2 H 1, T = T 2 T , C P 438.., (, ), 439., T, H, Q 1 2.,, C P 440, T., (4.4),,.,,., [ ], (4.3), d Q P = dh dh C P dt., ( ). 436.,. 437 [ ] H = H 1 H 2., 2 1.,, [ ] ( II).,,,. 439 [ ] II..,,.,,.,,. 440 [ ],, google.,,,,. 441 [ ].,, 76 c 2018 Tetsuya Kanagawa
83 ,. : dh = C P dt, H = C P T, H 2 = H 1 + C P (T 2 T 1 ) (4.5) C V, H U. (4.2), (2.38), d Q V = du C V dt (4.6) 1 (2.38), 2 C V ( )., 1 2 : Q 1 2 = U = C V T (4.7),, , (4.3) (4.7).,,, ,,, ( : ) 444.,, ( ),,. 442 [ ],,.,. 443 [ ],. 444 [ ].,. 77 c 2018 Tetsuya Kanagawa
84 ( 1.3.5)., c [J/(kg K)],, d q c dt (4.8)., q [J/kg], Q [J], C [J/K], m [kg] 445 : q = Q m, c = C m (4.9), ,, du = d q p dv (4.10) 446., ( ) u [J/kg], ( ) v [m 3 /kg], q [J/kg] u = U m, v = V m, q = Q m (4.11) 447., u, v,. 445 [ ],.,,,,. 446 [ ] pdv m, (p/m)dv, (p/m). pd(v/m) = pdv, v, m. 447 [ ] m,,., m,. 78 c 2018 Tetsuya Kanagawa
85 4.2.3 c V c P (4.10),, ( ): d q V = du (4.12), ( ( )) c V d q V = c V dt (4.13)., : 2 d q V = 2 du = , 449 : c V dt = q 1 2 = u = c V T (4.14) u 2 = u 1 + q 1 2 = u 1 + c V (T 2 T 1 ) (4.15) 21. (4.14) 2 (i) (4.10) ( ). (ii) (4.7). [(ii) 450 ] (4.7) m 451., q = Q m, u = U m, c V = C V m (4.16) 452, , (i) c P. 448 [ ],.,,. 449 [ ], u 1 T 1 ( ), u 2 ( ).. [ ].,. 450 (i),. 451 [ ].,,,. 452 [ ] c V T, T m., ( ).,,. 453 [ ], du = c V dt dh = c P dt. 79 c 2018 Tetsuya Kanagawa
86 (ii) c P h : dh = c P dt (4.17) (iii) T, h,, ( ) q : q 1 2 = h = c P T (4.18) 4.3 (4.3)(4.6), dt ( 0) 455,, 456 : C P = dh dt, C V = du dt (4.20) (4.20), C P C V : dh C P C V }{{} = dt du dt = d (H U) dt = }{{} H d dt pv = }{{} d mrt = mr (4.21) dt 1, (2 ) (2.25) ,, ( ). 455 [ ], 1.,. II,,. 456 [ ( II)],, : C V = ( ) U, C P = T V =const. ( ) H T p=const. (4.19) 457 [ ],.. 1.,, [ ] pv mrt., pv 80 c 2018 Tetsuya Kanagawa
87 (3.7) , R m. (4.21),, I, , (4.21) (4.21) 463,, c P c V = R (4.22)., 464., c P, c V, R, 465. [ ],, (. ) 466. T.,, mrt, T. 459 [ ],. 1, ( ). 460, II,.,. 461 [ ] (4.21),.,,,.,.,,. 462 [ ], d W = p dv,,.,,,. 463 (4.21) m,,. 464 [ ] Mayer ( ) ( II),.,, ( ).,,. 465 [ ] m.,. 466 [ ] ( ),. 81 c 2018 Tetsuya Kanagawa
88 4.4 κ (ratio of specific heats 467 ) κ 468, c P c V, ( ): κ c P c V (4.23), C [J/K] c [J/(kg K)], C = mc, κ = C P /m m/c V = C P /C V.,, (4.22). R c P c V,, c P = c V + R = c P > c V (4.24).,, 1 : κ c P c V = c V + R c V = 1 + R c V > 1 (4.25), (4.22)., (4.22) (4.25), ( ) κ > 1, (4.25) (4.22), c P = κc V, κc V c V = c V (κ 1) = R (4.26) 467 [ ] ( ), heat s. 468 [ ], (kappa) (gamma) γ.,,. 469 [ ].,,.,.,,. 82 c 2018 Tetsuya Kanagawa
89 .,, : c V = R κ 1, c P = κc V = κr κ 1 (4.28), ( ). Wikipedia,, κ R 472. (4.28), c P c V., m, 473, 474.,, ( ) ( (4.27) (4.28) ) (4.28). (4.22) ,, pv = mrt 470 [ ].,.,,. 471 [ ] (4.28) m, : C V = mr κ 1, C P = mκr κ 1 (4.27) 472 [ ], R, M R 0, (3.18) R = R 0 /M, : c V = R 0 M(κ 1), c κr 0 P = M(κ 1) (4.29) 473 [ ]., google, ( ). 474 [ ], (4.27) m,. 475 [ ], ( ) [ ], (4.22), ( ).,.,,.,,. 83 c 2018 Tetsuya Kanagawa
90 .,,.,, (real gas).,. 4.5 c P c V = R.,, 477.,,. 27. p, T, c V, m., Q..,. [ ] C 3 kg ( ), MPa, J/(kg K), 287 J/(kg K). 1). 2). 3). 477 [ ],,,. 478, ( ), [ ( )],., ( ),,,..,,.,. 480 [ ] k = 10 3, M = c 2018 Tetsuya Kanagawa
91 4)., ). 29., A, B, C, D, A. A B C D A ( 2.8). A,, (p A, V A, T A ),.,, p B = p A, p C = p D = p A /2, V B = V C = 3V A, V D = V A., A. A ) p V. 2) A B, B C, C D, D A,. 3),, (i), (ii) (i). [ ] ), [ ] m, R, 1 ( p 1, V 1 ), 2 ( p 2, V 2 ), 485 T , Q 1 2, 481,,,,,. 482,, ( ),.,,.,. 483 [ ], m, c V, 4,.,,,.,, p A V A., Boyle Charles,,. 484 ( ),.,,,,.,,,. 485, ( ) (isothermal process),. 85 c 2018 Tetsuya Kanagawa
92 W 1 2 Q 1 2 = W 1 2 = mrt 0 ln ( V2 V 1 ) = mrt 0 ln ( p1 p 2 ) (4.30).. [ ] Boyle (3.1). 31. [ ]., c P c V d Q = p c P R dv + V c V dp (4.31) R, R, p V, Q. [ ] , pv = mrt, c P c V = R, du = mc V dt. 86 c 2018 Tetsuya Kanagawa
93 5 4, ( 2.7),, (isothermal process),,.,,, 488., : du = d Q p dv (2.30),,.,, (2.30),,, dt = 0 T = const. (5.1) , U, C V c V ) ( du = C V dt = mc V dt (5.2) 487 [ ], 1. p, V, T, ( ) S,. 4., S. p T, V S,. 488 [ ],,,, [ ],,.,,.,. 87 c 2018 Tetsuya Kanagawa
94 ( (4.6))., (5.1) (dt = 0), du = 0 (U = const.) (5.3) 490.,, 491. (2.30), (5.3), d Q = p dv = d W (5.4)., 1 2, : Q 1 2 = 2 1 p dv = W 1 2 (5.5),, ( )., (5.2) : (i) (5.2), C V,,., (5.2), C V (d Q V =const. C V dt ),, (d Q V =const. = du). (5.2),,, d Q V =const (ii) (5.2) ( U = C V T ),., U = f(t ) (5.6) ,, 490 [ ] U = 0. [ ],, (absolute zero),.,. 491 [ ].,,,,. 492 [ ( )], dh = C P dt ( ). 493 [ ] (5.6) Joule, ( II ). 494 (5.6), C V = du/dt., du/dt = du(t )/dt = C V (T ) 88 c 2018 Tetsuya Kanagawa
95 ( )., 495. (iii). 496., (5.2),. 32.., (5.2)(5.5) W 1 2.,, p ( ) (p V ).,,, (3.7) : p = mrt V = f(v, T ) = f(v ) }{{} (5.7), 2 (V, T ),, T., 1 ( V ). (5.5) (5.7),,, : W 1 2 = 2 1 p dv = 2 1 mrt V 2 dv dv = mrt }{{} 1 V = mrt [ln V ]V 2 V 1 = mrt ln ( V2 V 1 ) (5.8)., U(T ), C V (T ).,,., I,. 495 U T,.,,,, (phenomenological). 496 [ ] ( ),.,. 89 c 2018 Tetsuya Kanagawa
96 , m R, T ( ) 497. V, p 498.,, ( ) : W 1 2 = mrt ln ( V2 V 1 ) = mrt ln ( p1 p 2 ) = p 1 V 1 ln ( p1 p 2 ) = (5.9) 33.,., ( ). (5.8), p V,.. [ ] Boyle : p = C V (5.10), C, 500., 1 501, C = p 1 V p = p 1V 1 V (5.11) 503., (5.8)(5.9). 497 [ ] ln (base), 10, Napier ( ) e = , log, (,,, ).,, log ln,,,. 498 Boyle (3.1).,, Boyle Boyle Charles, Boyle Charles,. 499 [ ] (logarithmic function) (antilogarithmic), 1., V 2 /V [ III]. (ordinary differential equation: ODE) (general solution) (arbitraty constant), (family of curves) ( ) ). 503 [ III] (5.11), ( ), c 2018 Tetsuya Kanagawa
97 36.,, p 1 < p 2. 1) ( ) 1 V 1 2 V [ ] Boyle 505,, p 1 < p 2 V 1 > V 2. ( ). 2) p V, 1, 2, 1 2 ( ). ( ) ) ( ),,.,.. [ ],,., U = 0.., (5.8)., ) ( ), p V ( ) [ ] Boyle Charles,,., Boyle, Boyle Charles,. 506 [ ( )], ( ).,, ( )., 1,,. 507 [ ( )].. 1 2, [ ], p 1 V 1 ln(p 2 /p 1 ) p 1 V 1 ln(v 1 /V 2 ). mrt ln(p 2 /p 1 ). m R T,. 91 c 2018 Tetsuya Kanagawa
98 5.2 (adiabatic process) 509, ,, : d Q = 0 ( Q 1 2 = 0) }{{} (5.12), ( )., 513., ( ) 514 : du = d Q p dv (2.30) (2.30) (5.12),,, : du = p dv (5.13), 515., (5.13)., 509,. ( ) (isentropic process) [ ]. 511 [ ],. 512 [ ] ( ), Diesel ( )., 30 atm, C., (heavy oil) (light oil) (fuel), (ignition), (combustion).,,, ( ). [ ], (, 1997);,, (, 2013)., (2 ABC). 513 [ ],, [ ],.,. 515 [ ]..,. 92 c 2018 Tetsuya Kanagawa
99 ., p = 1,,. 37. p 0, 1 2 (5.13).. [ ] U 2 + p 0 V 2 = U 1 + p 0 V 1 = C (C )., U + p 0 V = 0 (, 1 2 )., 1 1 ( ), (5.13), 516., p, 3 (U, p, V ) , (5.13) 519, 516 [ III],, ( )., ( II). 517 [ ( )], dy = z dx (5.14). ( ). z, (x, y).,,, dy = z(x, y)dx (5.15)., z(x, y)., z(x, y) = xy, dy = xy dx (5.16), (variable separable). [ ] ,,.,,,,.,. 519 [ III],.., ( ) ( ). [ ], (general solution), ( ; particular solution), (singular solution) 3., (initial value problem), (boundary value problem),,, ( ). [ ], ( ),,. 93 c 2018 Tetsuya Kanagawa
100 ., 520,, du = mc V dt (5.2) , U, T 523., p, p = mrt V = p(v, T ) (5.18)., (5.13), c V dt = RT V dv (5.19)., V T 2,, [ ],,.,,. 521 [ ] c V,.., d Q V, ( ). 522 [ ], mc V dt = p dv (5.17), 3 (p, V, T ),.,, p p(v, T ), [ ],.,., U = f(t ). 524 [ ], III. 94 c 2018 Tetsuya Kanagawa
101 5.2.2 (Poisson ) 1 2, : 2 1 dt T 2 = R dv 1 c V V (5.20) c V R,. ln T 2 T 1 + R c V ln V 2 V 1 = 0 (5.21) 527, : ln ( T2 ) ( ) R/cV V2 = 0 (5.22) T 1 V 1, c P c V = R 528, R/c V 529 : R c V = c P c V c V = κ 1 (5.23). R c V 2, κ (= c P /c V ) 1 ( ) [ ] (definite integral), (indefinite integral).,,.,, 2, 1.,, [ ] ( : arbitrary constant),. 1, 1. (V 2, T 2 ), (V 1, T 1 ). ( 525 ). 527 [ ] ln A + ln B = ln AB, ln A ln B = ln(a/b), a ln C = ln C a.,, ln e. 528 [ ] R = c P c V, ( ).,,.,. 529 [ ],. 530 [ ],,, κ.. 95 c 2018 Tetsuya Kanagawa
102 (5.23) (5.22), 531 : ( V2 V 1 ) κ 1 ( ) T2 = e 0 = 1 (5.24) T ( ), 532 : T 1 V1 κ 1 }{{} 1 = }{{} 1 2 T 2 V2 κ 1 }{{} 2 = = T V κ 1 = const. (5.25) ( 1) ( ), ( 2, 3,...) ( ) 533., (V, T )., 534., pv = mrt (= g(t )) 535, T V, : (i) (T, V ). T V κ 1 = const. (5.26) (ii) (p, T ) (5.26) V = mrt/p, V p : T κ p 1 κ = const. (5.27) (iii) (p, V ), T = pv/(mr) 531 [ ] Poisson ( ). 532, (5.24) [ ] 1 2, V 1 V , 2,.,,,. ( ),. 534,.,,,. [ ],.,. 535 [ ], Boyle Charles pv/t = const.. 96 c 2018 Tetsuya Kanagawa
103 : pv κ = const. (5.30) (5.26) (5.30), m R.,,., 3 (3 ), (5.26)(5.27)(5.30) 2,, :, pv T = const., pv κ = const.. 38., Boyle Charles ( ),, 3 (5.26)(5.27)(5.30). 39. : (i) T 1 V 1,, V 2,. (ii) p 1 V 1,, V 2,. [ ] (5.26)(5.30). p 1 V κ 1 = p 2 V κ ,., Boyle pv = const. = p 1 V 1 = p 2 V 2 = (5.28),, κ : pv κ = const. = p 1 V κ 1 = p 2 V κ 2 = (5.29) 537 mr ( ), T = pv ( ) Boyle Charles.,, mr., (5.27). 538 [ I, II], 2 (2 ), (5.26) (5.30) 1,, 1 1 (1 ).. 97 c 2018 Tetsuya Kanagawa
104 5.2.3,., (5.13) 541, W 1 2 = U = U 1 U 2 (5.31) 542,, W 1 2 W 1 2 = 2 1 p dv (5.33)., (5.31), , ( ) U = U 1 U 2, W W 1 2, (5.31) U 546,,.., 539,, ( )., (. ) [ ]..,. 541 [ ],.,,. 542 [ ( )], : U = (U 2 U 1 ) = 2 1 du = 1 2 du (5.32) [ ].,. 543,, [ ] [ ] (, ). 546, ( ). 98 c 2018 Tetsuya Kanagawa
105 , : W 1 2 = U 1 U 2 = mc V (T 1 T 2 ) = c V R (p 1V 1 p 2 V 2 ) = 1 κ 1 (p 1V 1 p 2 V 2 ) (5.34). 2, (5.2) , T = pv/(mr) 1 2,. m ( ) , R = c P c V κ, κ.. p 1 V 1, Poisson (5.30), : [ W 1 2 = p ( ) ] κ 1 1V 1 V1 1 = p ( 1V 1 1 T ) [ 2 = p ( ) ] (κ 1)/κ 1V 1 p2 1 κ 1 V 2 κ 1 T 1 κ 1 p }{{}}{{} 1 }{{} (5.35),., 2, 1/3, 550,. V 1 /V 2, T 1 /T 2, p 1 /p 2 3,, 551. (5.35) 3, 552,. 547 [ ] : U = mc V T. [ ],. 548 [ ].., ( ).,., ( ), m,.,,. 549,., 1 [ ( ) ] 1 κ p 1 V 1 V2 (κ > 1, ): 1 κ [ ],, (ratio), (nondimensional number).., ( ) [ ] ( ),. 552,, p 1 /p 2 = (V 2 /V 1 ) κ. V 1 99 c 2018 Tetsuya Kanagawa
106 (5.35) 1, ( ) V , 1 p 1 V (5.34)(5.35). 41. (5.35), (5.2), (5.30) 554. [!!],, W 1 2 = mrt p dv = 1 V dv = p(v, T )dv = 1 }{{}!! (5.36).,,, 555. [ ], (5.30)., : pv κ = p 1 V κ 1 (= const.) (5.37), p V 1 p(v ),., : W 1 2 = p dv = p(v )dv = p 1 V1 κ V κ dv = p 1 V1 κ [ = p 1V1 κ 1 κ [V 1 κ ] V 2 V 1 = p (V2 ) 1 κ 1V 1 1] 1 κ V V κ dv (5.35) (, ). (5.38) 553 κ ( ), p 1 V 1 1, 2,, V 2 ( 1 ). 554,,., (Boyle Charles (5.26)(5.27)(5.30) ), (5.35),. 555, 1 V., T V,. 556 (Boyle Charles ) pv/t = p 1 V 1 /T 1, pv κ = p 1 V1 κ., p V, p 1 V c 2018 Tetsuya Kanagawa
107 [ ], 557.,., U 2 U 1. (5.35). [ ], U = U 2 U 1 = W p V 558,, ,, Boyle. 559 : p(v ) = C V = CV 1 (5.39), C 560,, C = pv = p 1 V 1 }{{} = p 2 V 2 = (5.40) [ ],,.,. 558 [ ], V p.,, p V,., V p, p V,,. V p, ( ), p, V., p, V.,, p, V, V p, p V. 559 [ ( )]., C, p V ( ). 101 c 2018 Tetsuya Kanagawa
108 ,, : dp dv = d dv CV 1 = CV }{{ 2 = } p V C = pv (5.41),, , p(v ) = D V κ = DV κ (5.42), D C 564, D = pv κ = p 1 V κ 1 = p 2 V κ 2 = (5.43), : dp dv = DκV κ 1 = pv κ κv κ 1 = κp V (5.44), ( )., p V,, dp/dv. 561 Boyle Charles, pv/t, T., T C., Boyle Boyle Charles. 562 [ ],. ( ).,,., p V ( ) (p V ),. 563, [ ( )].,.,, pv = C pv κ = C. C D.,,.,,,.,,,, c 2018 Tetsuya Kanagawa
109 5.3.3 (5.41)(5.44) κ.,,,., p V,., p V 1,, 0 p 0 V 0. (5.41)(5.44), 565, p 0 V 0 }{{} < κp 0 V 0 }{{} (5.45).. (4.25) κ > , ( p 1, V 1 ), 2 ( V 2 ),, 2., 2, : (i) 2 p 2, (ii) W 1 2, (iii) U 2 U 1, (iv) Q 1 2. [ ] p V,., ( ), 1 1. [ ] (i) 568. (ii) (iii). 565,,.,. 566 [ ] κ > 1,., κ = c P /c V c P = c V + R, ( ). 567,, Carnot ( 7),. 568 p V,,. κ > ,. 570 [ ],, ( ),., Carnot ( 7),. 571 [ ]., Newton, c 2018 Tetsuya Kanagawa
110 (U 1 > U 2 ) 572. (iv)., d Q = CdT (5.46), dt = 0,, d Q = 0 (5.47).,..,,,. m/s,. Laplace,,, 340 m/s [ ], du = C V dt ( ). [ ]. 573 [ ].,., ( ), ( ),. 104 c 2018 Tetsuya Kanagawa
111 6, (the second law of thermodynamics).,,.,,., (entropy),,, 574., 575,,.,,,.,,. 6.1 (reversible process) 576, (irreversible process),,. 1) (friction),.,.,,,. 2), ( ) ,.,, ( ),,.,. 575.,. 576 [ ] (reversible). 577 [ ], (i).,.,,., (ii).,, 12,. 105 c 2018 Tetsuya Kanagawa
112 3).,, ( ) ) (i), 1,., (ii).,, , ( ).,,. ( ), , (i), (ii) 585., ( ),, 586., 578, (i) mẍ + kx = 0, (ii) mẍ + cẋ + kx = 0., x, m, k, c,. 581 [ ],, [ ( )].,,,.,,., 1 1,. [ (counterexample)] 2,, [ ],., (reversible) (quasi-static),,. 584,, (6.12),., I. 585,,. 586.,. 106 c 2018 Tetsuya Kanagawa
113 ,,,.,.,,,.,, 589.,,.,,.,,,.,, d Q = du + pdv }{{}}{{}!! (6.1),., d Q.,,..,,.,. [ ],.,. 587,,. 588 [ ( )].,. 589 [ ],,,. 107 c 2018 Tetsuya Kanagawa
114 6.3.1,. du,, du = mc V dt (6.2) dt., (3.7) p = mrt V (6.3). (6.2)(6.3) (6.1), : d Q = mc V dt + mrt dv V (6.4),.. T,., 590. T 0,, (6.4) T 591, d Q T = mc V dt T + mrdv V (6.7) : mc V ln T 2 T 1 + mr V 2 V 1 (6.8) 590 [ ] (6.4) 2,.. T dv (6.5) V. T V., dv dt 2,.,, T = pv/(mr), T V dv = pv dv mr V = 1 mr pdv =? (6.6)., p = mrt/v,,. [ ].,, [ T > 0] T 0.,,. 108 c 2018 Tetsuya Kanagawa
115 6.3.2, (6.7).. (6.7),,,.,., d, d. d Q T (6.9), 592., S., (6.9) ds. S, [J/K]., S,.,, : S ds d Q T (6.12).,, d Q/T ds 595., 592,,. 593 [ ] (6.12).. [ ] (i),, ( ).,. (ii), S,, : ds > d Q T (6.10) 594 [ ] : ds = d Q dt (6.11). ( ), ( ).. [ ] ( ). 595 S,, d,.,,,. 109 c 2018 Tetsuya Kanagawa
116 ,. : (i) d,. (ii) d,. (iii) T,,,. (iv),, S,. (6.12) 2 1 d Q T = 2. S, ds = [S] 2 1 = S 2 S 1 = S (6.13), S 1 S 2, (specific entropy) [J/K],.,, s 598 : s S m [J/(kg K)] (6.14) 596.,.,.,, [ ( )] S, (finite value)., d Q/T. [ ] dx, 1 x 3, 3 1 dx = ,. 110 c 2018 Tetsuya Kanagawa
117 6.3.5 (6.13), (i), T 0, T 0 S = 2 1 d Q T 0 = 1 T d Q = Q 1 2 T 0 (6.15)., Q 1 2 T 0, S 600. (ii) 601, d Q = 0, ds = 0., 602 : S = 0 S 1 = S 2 = const. (6.16). p, V, T 3, S , 3.,,,, 604.,,., s = S/m v = V/m, ,,. 600 [ ],,,.,,. 601 [ ], ( ). 602 [ ], (isentropic process) ( ). 603.,,,. 604,, ( ). 605, c 2018 Tetsuya Kanagawa
118 6.4.1 (T, v), (6.14),, ( ) (3.11) 606 : d q ds }{{} T du }{{} = T + pdv T dt }{{} = = c V T + Rdv v (6.17), dt ds = c V T + Rdv v, s = s 2 s 1 = c V ln ( T2 T 1 ) + R ln ( v2 v 1 ) (6.18) (6.19), s = s 2 s ,., (6.18),,., (p, v) 609 (6.18)(6.19) T, p. (3.11), T = pv R (6.20). ( ),, dt = d(pv) R pdv + vdp = R (6.21) 606 ds d q/t, du = d q pdv, du = c V dt, pv = RT. 607 [ I].,.,., (fundamental theorem of calculus). 608,, s 2 s 1., s = s 1 s ,,. 112 c 2018 Tetsuya Kanagawa
119 610. (6.20)(6.21) (6.18) : pdv + vdp R ds = c V R pv + Rdv v ( dp = c V p + dv ) + R dv v v = c V ( dp p ) + (c V + R) }{{} c P (Mayer) ( ) ( ) dp dv = c V + c P p v ( dv v ) (6.22), 3 4, (Mayer ) 611., : dp ds = c V p + c dv P v,, : s = c V ln ( p2 p 1 ) + c P ln ( v2 v 1 ) (6.23) (6.24) (T, p) 44.. dt ds = c P T Rdp p ( ) T2 s = c P ln + R ln T 1 ( p2 p 1 ) (6.25) (6.26) [ ]., (6.23) v, T [ ],., T = T (p, v), dt (p, v), (6.21). 611.,. c V R 2, c P 1...,,. 113 c 2018 Tetsuya Kanagawa
120 ., [J/K]. dt ds = C V T + mrdv V dp ds = C V p + C dv P V dt ds = C P T mrdp p (6.27) (6.28) (6.29) 6.5 pv = RT (3.11), (p, v, T ) ,,.,,.,, (3.11) T, s., 3 (p, v, s).,. (6.23) c V : s = ln c }{{} V ( p2 p 1 ) + c P ln c }{{} V κ ( v2 v 1 ) = ln ( p2 p 1 ) + κ ln ( v2 v 1 ) = ln ( p2 p 1 ) ( v2 v 1 ) κ (6.30) 612 v [m 3 /kg], V [m 3 ] ρ [kg/m 3 ]. 613 [ ],,, II,,, 3., ( ). 614,. 615 [ ] ln A + ln B = ln AB, ln A ln B = ln A/B, a ln A = ln A a, ln A = C A = exp(c). 114 c 2018 Tetsuya Kanagawa
121 , 616 : (v 1 /v 2 ) κ ( p2 p 1 p 2 p 1 = ) ( ) κ ( ) v2 s = exp v 1 c }{{ V } ( v1 v 2 ) κ ( ) s2 s 1 exp c V (6.31) (6.32)., s = s 2 s (6.32) v, ρ V : p 2 = p 1 p 2 = p 1 ( ρ2 ρ 1 ( V1 V 2 ) κ ( ) s2 s 1 exp c V ) κ ( ) S2 S 1 exp C V (6.34) (6.35) S (6.12) T, d Q, d Q = T ds (6.36) 616 [ ], exp(x) e x.,. 617 (3.11), (6.32),, ( ) s pv κ s0 = C exp c V (6.33)., 1 0 ( ), 2. [ ] C 1, [ ] (6.34), (3 ). ( ). 619 [ ], v V, s S, c V C V.,,. 115 c 2018 Tetsuya Kanagawa
122 620.,, du = T ds pdv (6.37)., d, d p, V, T, S,,.,,, (6.36)., p, V, T, S 4 : (i). p.,. V 621. (ii), T [ 1] (6.36). [ 2],, s S.,. 621 [ ( )].,,.,., ( ) ( ).,,, ( ).,. (i) ( ), (ii) ( ), (iii),., ( ) ( ),. [ ],. 622,., ( ), ( ). 623 [ ],, 20 C ( C ). 3 (i), (ii), (iii). 624 [ ],,.,.,,,.,.,,, 116 c 2018 Tetsuya Kanagawa
123 , S 625. ds,.,, ds dv.,, : d W = pdv (6.38) d Q = T ds (6.39), ( ) p ( ) dv., ( ) T ( ) ds 627., , 630.,.,,,.,.,, ( ) ( ).,. 625.,. 626 [ ]. 99 % ( ),,.,,,,. 627 (6.36),,., ( ) d Q,, ( ) T ds ( ). d d,,. [ ] d Q ds,,, S. 628 d Q = T d, S. S [J/K], 1 K. 629 V, S.,. d W = pdv, p V d Q T ds. 630 [ ], ( ), ( )..,, ( )., ( ).,,, ( ).,. 117 c 2018 Tetsuya Kanagawa
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国際物理オリンピック 研修用テキスト Ⅲ 熱物理 相対論 量子力学 特定非営利活動法人物理オリンピック日本委員会 1 1.1 1 1. 1.3 3 1.4 4 1.5 6 1.6 7 1.7 9 11.1 11. 0.3 1 6 3.1 6 3. -9 3.3 - -- 31 3.4 --33 39 4.1 39 4. 40 4.3 4 4.4 44 4.5 47 5 5.1 5 5. 5 5.3
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現代物理化学 2-1(9)16.ppt
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現代物理化学 1-1(4)16.ppt
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IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
0 = m 2p 1 p = 1/2 p y = 1 m = 1 2 d ( + 1)2 d ( + 1) 2 = d d ( + 1)2 = = 2( + 1) 2 g() 2 f() f() = [g()] 2 = g()g() f f () = [g()g()]
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Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math))
Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math)) 2001 1 e-mail:s00x0427@ip.media.kyoto-u.ac.jp 1 1 Van der Pol 1 1 2 2 Bergers 2 KdV 2 1 5 1.1........................................