m d2 x = kx αẋ α > 0 (3.5 dt2 ( de dt = d dt ( 1 2 mẋ kx2 = mẍẋ + kxẋ = (mẍ + kxẋ = αẋẋ = αẋ 2 < 0 (3.6 Joule Joule 1843 Joule ( A B (> A ( 3-2

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3 3.1 ( 1 m d2 x(t dt 2 = kx(t k = (3.1 d 2 x dt 2 = ω2 x, ω = x(t = 0, ẋ(0 = v 0 k m (3.2 x = v 0 ω sin ωt (ẋ = v 0 cos ωt (3.3 E = 1 2 mẋ2 + 1 2 kx2 = 1 2 mv2 0 cos 2 ωt + 1 2 k v2 0 ω 2 sin2 ωt = 1 2 mv2 0(cos 2 ωt + sin 2 ωt = 1 2 mv2 0 = (3.4 3-1

m d2 x = kx αẋ α > 0 (3.5 dt2 ( de dt = d dt ( 1 2 mẋ2 + 1 2 kx2 = mẍẋ + kxẋ = (mẍ + kxẋ = αẋẋ = αẋ 2 < 0 (3.6 Joule Joule 1843 Joule ( A B (> A ( 3-2

3.2 3.2.1 U O U O O (adiabatic ( W A A A U A U O + ( W A (3.7 U B U O + ( W B (3.8 W A = = ad A B A ad B U = U B U A = W }{{} = (W B W A (3.9 3-3

3.2.2 (, U + W 0 U + W = Q = (3.10 or U = W + Q = (3.11 U = W = ( W > 0 U < 0 U + W = 0 3.2.3 Calorimetric 1 14.5 C 1g 15.5 C 1 calorie 1 Joule 1 calorie = 4.185 10 7 erg = 4.185 Joule (3.12 3-4

4.185 J 1 Kg 40cm ( mgh ( Joule 3.2.4 ( e e e = dw = e d = d, d = (3.13 du + }{{ d} = d Q = (3.14 dw d 3-5

3.2.5 (partial derivative or partial differentiation 2 x, y f(x, y f x f (x, y lim y y 0 f(x + x, y f(x, y (x, y lim x 0 x f(x, y + y f(x, y y (3.15 (3.16 3 ( ( f f, (3.17 x y y x f(x, y = x m y n { f x = mxm 1 y n f y = nxm y n 1 (3.18 (total differential f(x, y x, y f(x + dx, y + dy f(x, y 2 f(x + dx, y + dy f(x, y = [f(x + dx, y + dy f(x, y + dy] + [f(x, y + dy f(x, y] (3.19 1 f(x + dx, y + dy f(x, y + dy = f (x, y + dydx x f (x, ydx (3.20 x f(x, y + dy f(x, y = f (x, ydy (3.21 y 3-6

df(x, y f f (x, ydx + (x, ydy (3.22 x y f(x, y du U,, ( f(,, = 0 du 2, U(, ( ( U U du = d + d (3.23 /N 1 = R 1 d Q 2 2 x, y ( u(x, ydx + v(x, ydy (3.24 f(x, y f x = u, f y = v (3.25 u y = f y x = 2 f y x = 2 f x y = f x y = v x 3-7 (3.26

u y = v x (3.27 ( ( d 0 Q U (, du + d = U U U d + d + d = d + ( U + d (3.28 x =, y = (3.27 = U (3.27 = 2 U = ( U + = 2 U + (3.29 (3.30 d 0 Q Q = Q(, 1 dq d Q 3.1 (, (, d Q 1 3-8

du + d = U ( U d d + d + d d + d ( U = + ( U d + + d d d (3.31 LHS = RHS = ( U + ( U + = 2 U + + 2 = 2 U + 2 (3.32 (3.33 LHS RHS // df(x, y r 0 = (x 0, y 0 r 1 = (x 1, y 1 1 s r = r(s r 0 = r(s 0 r 1 = r(s 1 Γ r(s r(s 1 r(s 0 f( r = f(x, y Γ ( f f df( r(s = dx + Γ Γ x y dy s1 ( f dx = s 0 x ds + f dy s1 df( r(s ds = ds y ds s 0 ds = f( r(s 1 f( r(s 0 = f( r 1 f( r 0 = r 0, r 1 (3.34 Γ 3-9

(integrable ( 3.2.6 (heat capacity = ( 1 (specific heat = 1g = 1 ( d Q C = d ( d Q C = d ( (3.35 ( (3.36 3-10

(expansion coefficient α = 1 ( (compressibility κ = 1 κ ad = 1 ( (3.37 ( ( (3.38 (3.39 3.3 3.3.1 (, du du = ( [( U U ( d + ( ( U U d + d (3.40 3-11 ] + d = d 0 Q (3.41

(, du d ( U du = ( d = ( U d + ( d + [( ( ] U ( + d [( ( U + + d (3.42 d (3.43 ] d = d 0 Q (3.44 3.3.2 ( = d = 0 ( d 0 ( Q U C = = (3.45 d ( = d = 0 ( d 0 ( ( Q U C = = + (3.46 d ( 3-12

3.3.3 C C (1 U (2 ( (U (2 1 = R (1 U = U( U( Joule A B A B ( A = B Joule Q = 0. 3-13

A + B W = 0 U = 0 ( U/ = U = U( // C C (1, (2 (1 ( U = ( U = du d (3.47 (2 ( = ( R = R (3.48 (3.45 (3.46 Mayer( C ( = C ( + R (3.49 U( C ( C, C C 1 1 R 2 C C γ = C /C 3 1 3 R 5 R 5 2 2 3 5 2 5 = 3 + 2 R 7 R 7 2 2 5 4 6 = 3 + 3 3R 4R 3-14 3

2 (3+ (2 (2 2 (3+ (3 3 C U( C = du d U = C + U 0 (3.50 γ 3.2 an der Waals C C an der Waals 1 = R b a (3.51 2 a 1/r 6 1/ 2 b a, b litter/mol a (atm (litter 2 /(mol 2 b (litter/mol He 0.034 0.024 H 2 0.24 0.027 N 2 1.35 0.039 O 2 1.36 0.033 CO 2 3.96 0.043 H 2 O 5.47 0.031 an der Waals 3-15

C = C ( = ( U ( ( U = a (3.52 2 C C ( ( U U du = d + d = C ( d + a d (3.53 2 ( du + d = C d + + a d = d Q (3.54 2 C (, (, d = Rd R d b ( b + 2ad (3.55 2 2 R/( b = + (a/ 2 d ( d = a + 2ab 1 (Rd ( bd (3.56 2 3 d = 0 ( d 0 Q C = = C ( + d a 2 + a 2 + 2ab 3 R (3.57 C C 1 a 0 (b 0 3.3.4 3-16

,, = R = constant Boyle (3.58 d Q = 0 U = U( C d + d = d Q = 0 (3.59 (, C d + R d = 0 (3.60 C = d = R C d = C C C d = (γ 1d (3.61 ln = (γ 1 ln + const (3.62 `1 = const. (3.63 γ 1 > 0 γ C d = d C,, = const. (3.64 = const. (3.65 `1 3-17

2 ( (, (3.65 log d = γ 1 γ d (3.66 h dh h h + dh h d = gρdh (3.67 ρ ρ = nr (3.68 m =, M = 1 (3.69 = n = m M (3.70 2 3-18

ρ = m = M R (3.71 ρ, (3.67 d = Mg R dh (3.72 (3.66 d d dh = γ 1 γ Mg R = const (3.73 γ = 7 5 (2, g = 9.8m/s2, M = 0.029Kg, R = 8.3J/deg (3.74 d dh = 9.8 deg/km (3.75 3.3 n i f (< i d = 0 d = d Q ( = nr d = nr d (3.76 2 d Q = d = nr d (3.77 Q = nr f i d = nr ln i f > 0 (3.78 3-19

3.4 κ κ ad κ ad = 1 γ κ (3.79 = a = const. = a/ κ = 1 ( = a = 1 2 (3.80 κ ad = 1 γ = b = const. = b 1/γ (3.81 ( ( = 1 ( 1 b (1/γ 1 = 1 ad γ γ (3.82 κ ad = 1κ γ 3.3.5 X, Y, f(x, Y, = 0 (3.83 1. ( ( = 1 (3.84 3-20

X (Y, Y (X, ( d = 0 ( ( dx = dy, dy = dx (3.85 (3.84 // 2. X, Y, W ( W ( W ( = W (3.86 X (Y, W Y (, W W (dw = 0 ( ( ( dx = dy + dw = dy (3.87 dy = ( W W d + W ( W W Y dw = ( ( dx = d // ( W W W d (3.88 d (3.89 3. Chain relation ( ( ( X Y = 1 (3.90 3.5 X = X(Y, Y = Y (X, ( ( dx = dy + d (3.91 Y ( ( dy = dx + d (3.92 X (3.93 3-21

( [( ( ] ( dx = dx + d + d X Y ( ( [( ( ( ] = dx + + d Y X }{{}}{{} 1 0 (3.94 ( 2 (3.84 (3.90 Y // 3.6 1 κ ad = C C κ C, C (3.45 (3.46 κ ( = C ( (3.95 C ad (1 U (, d Q = 0 (, ( U, ( U ad ( (2 U (, U C ( ( (3 U (, U, C, ( (4 (2, (3 (1 (3.95 3-22

(1 0 = du + d = ( [( ] U U d + + d (3.96 d d ( = + ( U ad ( U (3.97 (2 U (, 2 C ( ( ( ( U U = = C (3.98 (3 U (, C ( ( ( ( ( ( U U = = C ( ( ( ( = C = C }{{} 1 (3.99 (4 (2 (3 (1 ( = C ( ( ad C (3.100 chain relation ( ( ( = 1 ( ( ( = (3.101 (3.100 (3.95 // 3-23

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