September 25, 2017 1 1.1 1.2 p = nr = 273.15 + t : : K : 1.3 1.3.1 : e.g. 1.3.2 : 1
intensive variable e.g. extensive variable e.g. 1.3.3 Equation of State e.g. p = nr X = A 2 2.1 2.1.1 Quantity of Heat Q 1 2 3 1 > 3 > 2 Q Q = C 1 1 3 = C 2 3 2 C 1 : 1 heat of capacity C 2 : 2 C Q = C 2
2.1.2 : Calorie 1cal=1g 14.5 15.5 =1cal/gK 2.1.3 Lateut heat e.g. e.g. L=80cal/g e.g. L=540cal/g 2.1.4 calorimeter thermometer 2.1.5 calorique: 2.1.6 work: W = W = ps h 2.1.7 1N m = 1J 1N = 1kg m/s 2 1J = 1kg m 2 /s 2 MKS 3
2.1.8 2 ln = log e natural logarithm e 3 = 20.08 20 ln 20 = 2.9957 3 W = 2 = nr pd = nr ln 2 1 1 d 2 e = lim 1 + 1 n = 2.71828 n n { y = ln x x = e y d ln x dx = 1 x 2.1.9 1cal=4.186J ˆ Rumford ˆ Mayor ˆ Jouls 2.1.10 : 2000kcal 1 = 24 60 60 = 8.64 10 4 Q = 2000kcal = 2000 10 3 cal = 8.4 10 6 J 4
2.2 1 2 Q W 1 : P 1, 1, +Q,+W 2 : P 2, 2, 2 Q + W = U = Q + W = U 2 U 1 U U Q W 2.2.1 p- 1 2 A B = Q A Q B 2.2.2 A B W A = pd A W B = pd B pd A B pd = W A + W B Q A + W A = Q B + W B W A + W B = Q A Q B 2.2.3 A: + 5
B: 1 1mol 15 2 = 2 Q = P 1 2 pd 2 = 2, p = nr B 2 1 Q = p 1 R 1 d 2 = p 1 R 1 ln = R 1 1 ln 2 = 8.3 273 + 15 1 0.693 J = 1.66 10 3 J 2.2.4 = 2.3 ˆ dt, dp, d, du ˆ d W, d Q d W = pd du = d Q + d W = d Q pd d Q = du + pd 6
2.3.1 du d 2.3.2 d, d du U, du = U U d + d U U U U 2.3.3 2 f x, y f x = lim f x + x, y fx, y x 0 x f y = lim f x, y + y fx, y y 0 y f x, y x, y 2.3.4 a, b, c f x, y = ax 3 + bxy 2 + cy 3 f x = 3ax2 + by 2 7
f y 2 3 2 f x = 2 x 2 f y 2 = y 2 f x y = x = 2bxy + 3cy2 f = 6ax x f = 2bx + 6cy y f = 2by y 2 f y x = y f = 2by x f x, y x, y 2 f x y = 2 f y x 2.3.5 1 f x y = f x x, y df dx = f x 2 z=fx,y f x x 0, y 0, f x y 0, y 0 x 0, y 0, z 0 = f x 0, y 0 x y U U : : 8
2.4 2.5 2.6 adiabetic: quasi-static: : : dq = du + pd = 0 2.6.1 du = C v d 2.6.1 p = nr p = nr d d C v d + nr d = 0 1 2 C v d = nr d 2 1 2 1 C v d = nr 1 d 9
C v ln 2 1 = nr ln 2 ln 2 1 = nr ln C v = ln 2 2 nr Cv 2 1 = = 2 2 nr Cv nr 1 Cv nr C v 1 1 = 2 2 nr C v nr C v = C r C v C v = C r C v 1 = γ 1 2.6.2 γ = Cr C v 1 γ 1 1 = 2 γ 1 2 γ 1 = cond 2.6.3 p = nr p p γ 1 = const 10
p γ = const 2.6.4 p = nr = const isotherm p γ = const adiabat ˆ ˆ U = C v 2 1 < 0 2.6.1 n 1 2 W W = 2 2 1 pd = c d p γ = c = p γ 1 γ 1 = p 2 γ 2 p = c γ x α dx = 1 α + 1 xα+1 x, α γ 11
1 [ W = c ] 1 γ 2 1 γ c [ ] = 1 γ 2 γ 1 1 γ 1 [ ] = p2 γ 2 γ 2 p 1 γ 1 γ 1 1 γ 1 = 1 γ [p 2 2 p 1 ] 1 = 1 γ [nr 2 nr 1 ] = nr 1 γ 2 1 C p C v = nr γ = C p C v nr 1 γ = C p C v 1 C p C v = C p C v C v C p C v = C v W = C v 2 1 = U 2 U 1 = U = 2.6.2 h 10km p h p = p h + h p h = gρ h 12
ρ : g : h p h + h = p h + dp h dh dp h = gρ h dh dp dh = gρ d Q = du + pd = dh dp d Q = 0 H = U + p dh dp = 0 dh = C p d dh = dp U = C v + U 0 H = U + p = U + nr = C v + nr + U 0 = C p + U 0 13
dh = C p d = dp = dp dh dh = gρ dh d = gρ C p dh = gm C p dh M = ρ : c p = C p M : d = g C p dh c p = C p M C p C v = cr γ = C p C v = 1.41 C p = C p 1 γ = nr C p = nr 1 1 γ = γ γ 1 nr c p = C p M = γ nr γ 1 M M n = m: m = 28.8J/K mol 14
d = γ 1 gm γ R dh = 0.41 9.8m/s 2 28.8 10 3 kg/mol dh 1.41 8.31kg m 2 /s 2 /k mol = 9.7 10 3 dh K/m 100m 1K 0.6K : γ 2: : 1-2,3,9,10 2.7 Kinetic theory = = x v = v x, v y, v z v = v x, v y, v z p x = 2mv x p = v 2 x = v 2 y = v 2 z = 1 3 v2 p y = p z = 0 N 2 v x p x = N m vx 2 N = m1 v 2 : 3 15
p = 1 3 Nm v 2 = 2 3 N 1 2 m v 2 = 2 3 U : p = nr nr = 2 3 U U = 3 2 nr U = C v + U 0 C v = 3 2 nr C p = C v + nr = 5 2 nr γ = C p C v = 5 2 3 2 = 5 3 = 1.666... U = N 1 2 m v 2 = N 1 2 m vx 2 + v 2 y + v 2 3 z = 2 nr 1 2 m v 2 = 3 nr 2 N = 3 2 k B nr N = R N A = k B = 1.38 10 25 J/K : N A = N n : 1 1 2 k B O 2, N 2... 5 U = 1 2 k B 3 + 2 = 5 2 k B 16
U = N 5 2 k B = N 5 nr 2 N = 5 2 nr C v = 5 2 nr C p = 5 2 nr + nr = 7 2 nr γ = 7 2 5 2 = 1.4 3 : : 3.1 a P. Clausius b = 1. a b b a b a Q 17
a 2. b a heat engine : heat resewair R 2 Q 2 R 1 Q 1 < Q 2 Q 2 Q 1 W 3.1.1 Sadi Carnot1824 4 A B 2 1 C D 1 2 Q 2 Q 1 W = Q 2 Q 1 efficiency η = W Q 2 = Q 2 Q 1 Q 2 = 1 Q 1 Q 2 < 1 Heat Pump 3.2 3.2.1 18
3.2.2 c,c c Q 2 Q 1 = W Q 2 Q 1 = W η = W Q 2 W = ηq 2 η = W Q 2 W = ηq 2 Q 2 = Q 2 W = η Q 2 = η Q 2 W W = η Q 2 ηq 2 = η η Q 2 W W = Q 2 Q 1 Q 2 Q 1 = Q 1 Q 1 η > η Q 1 Q 1 W W b η η c η η η = η 3.2.3 = η = W Q 2 = Q 2 Q 1 Q 2 = 1 Q 1 Q 2 19
1 η c = Q 1 Q 2 = 1 2 3.2.1 3.2.4 n p = nr Q 1 Q 2 1 2 1 2 Q 2 = 2 pd = nr 2 2 = nr 2 ln 1 2 d 3 4 2 Q 1 = pd = nr 1 2 = nr 1 ln = nr 1 ln 1 4 4 3 d 3 2 3 4 1 2 γ 1 2 = 1 γ 1 3 20
1 γ 1 4 = 2 γ 1 3 2 γ 1 = 3 4 γ 1 2 = 3 4 Q 1 Q 2 = nr 1 ln nr 2 ln 3 4 2 = 1 2 3.3 η = 1 Q 1 Q 2 η c = 1 1 2 Q 1 Q 2 1 2 Q 1 1 Q 2 2 3.3.1 S = Q 2 2 = Q 1 1 3.3.2 entropy 3.3.1 Q 2 2 < Q 1 1 21
S 2 = Q 2 2 R 1 S 1 = Q 1 1 R 2 S 1 S 2 = Q 1 1 Q 2 2 > 0 3.3.1 3.3.2 Q S = Q Q=0 S = 0 3.3.3 p- 1 2 1. 1 2 3 2. 1 3 3. 3 2 S 13 = Q 13 S 32 = 0 S 12 = S 13 + S 32 = S 13 22
3.3.4 1 3 2 4 4 13 5 32 6 1 5 4 6 2 S 12 = S 15 + S 54 + S 46 + S 62 S 54 = S 62 = 0 5 3 6 4 5 S 53 + S 64 = 0 S 46 = S 64 = S 53 S 23 = S 15 + S 46 = S 15 + S 53 = S 13 = S 12 S 12 = S 13 S = Q 2 p 1 p 2 3.3.5 n 2 23
Q = 2 pd 2 1 = nr = nr ln 2 d S = Q = nr ln 2 nr ln 2 J/K 3.3.6 2 n 1 2 1 2 3 S 12 = S 13 + S 32 1 = nr ln 3 1 3 γ 1 1 γ 1 1 = 2 γ 1 3 1 γ 1 = 2 3 1 24
γ 1 ln 1 3 = ln 2 1 S 12 = nr γ 1 ln 2 1 nr = C p C v γ = C r C v nr γ 1 = C p C v C p 1 = C v C v S 12 = C v ln 2 1 3.4 ds = dq c.f. d W = pd d Q = ds 3.4.1 du = d Q + d W = ds pd 3.4.2 25
ds = du + pd 3.4.3 3.4.2 U S, = 3.4.3 : S U, U U, p = S 1 S =, U U p S = U 3.4.1 ds = du+pd du = C v d p = nr 1 nr ds = C v d + d 1 2 2 1 2 S 2 S 1 = C v d + nr 1 2 = C v ln + nr ln 1 1 1 2 d 1 0 26
2 S, = C v ln + nr ln + S 0 3.4.4 0 0 3.4.2 2 isotherms 2 isocores 2 adiabats 2 1 1 2 3.4.3 S, ds = S d + S d S ds = du + pd du = U d + U d ds = 1 [ U 1 d + U + P ] d S = 1 U = C v 3.4.5 27
U = C v S = 1 U + P 3.4.6 3.4.5 3.4.6 3.4.5 = 2 S = 1 2 U 3.4.6 = 1 U + 2 U 2 p + 1 2 2 1 U p 2 + 1 2 U = p 3.4.7 3.4.6 p p 3.4.7 = 0 p S = = p p p + p S = p 3.4.8 Maxwell 28
3.4.4 p = nr p = nr p = nr S = ds = C v nr d + d S, = C v ln + nr ln + S 0 0 0 U = nr p = p p = 0 3.4.5 van der Wals p = nr b a 2 S, U, F = a: b: S p = = nr b 29
C v ds = C v d + nr b d b S, = C v ln + nr ln + S 0 0 0 b U nr nr = b b a 2 = a 2 U U du = d + d = C v d + a d 2 U, = C v a + U 0 3.4.6 0 100g 15 0 100g 0 S 1 S 1 = Q 1 0 80cal/g 100g 4.2J/cal = 273K = 123J/K 0 15 30
ds = = du + pd dh dp = dh = C pd C p = H p C p = 1cal/g K 4.2J/cal = 420J/K 1 C p 2 1 S 2 = d = C p ln 2 273 + 15 = 420J/K ln 273 = 22.5J/K S = S 1 + S 2 = 123 + 22.5 = 145J/K 3.5 3.5.1 1 2 > 1 2 1 S = Q Q 1 2 2 1 = Q > 0 1 2 31
3.5.2 2 2 S = nr ln 2 > 0 3.5.3 p 1 > p 2 = p 1 = nr = p 2 2 S = nr ln 2 > 0 p 1 p 2 = 2 > 1 3.5.4 4 mixing entropy 2 = A + B S = S A + S B S A = Cv A ln + n A R ln 0 A 0 + S 0 S B = C B v ln + n B R ln 0 B 0 + S 0 32
S = S A + S B S A = Cv A ln + n A R ln + S 0 0 0 S B = C B v ln + n B R ln + S 0 0 0 S = S S = n A + n B R ln n A R ln 0 = n A R ln + n B R ln > 0 A B A 0 n B R ln B 0 3.5.5 : p S du = ds pd du = 0 ds pd = 0 ds = pd ds = 2 1 S 2 S 1 = 2 S 2 S 1 1 S 2 S 1 = Q 2 Q 1 33
W = Q 2 Q 1 pd = ds p S 3.5.6 : p S p, S, 2 pd = S p S = t p = S = p d S 3.6 : : W U,, N N U S U,, N = k B l n W 1900 3.6.1 N W R = N 0 N 34
N 0 : U N p 2 = p 2 2m N = U U N 2m W p W p U 2m N p 0 8N W = W R W p N = N 0 U 2m N p 0 3N p 0 ; S = k B ln W N = k B ln N 0 U 2m N p 0 3N = Nk B ln 3N U N 0 2 k N B ln 2m p 2 0 d S = d + pd 1 S = U = 3 2 Nk 1 B U U = 3 2 Nk B = 3 2 nr p S = U U 35 = Nk B 1
p = Nk B = nr 4 : : 4.1 Q S W U = Q W W = Q U S U = S 2 S 1 U 2 U 1 W U 1 S 1 U 2 S 2 F = U S4.1.1 W F 1 F 2 4.1.1 W = 2 pd = nr ln 36 2
U = 0 F 2 F 1 = F = U S = nr ln = W 2 W = F 1 F 2 W = 0 F 1 F 2 F 2 F 1 F F 4.1.2 n 1, n 2 = + 2 = F,, = F 1, + F 2, 2 + 2 = F F, = 0 F, = = F1 F1 F2 + 2 F2 2 2 = 0 37
2 = F, = U S df = du ds Sd = pd Sd F = p p 1 = p 2 p 1 = p 2 4.1.3 F = U S S df = Sd pd du = ds pd F S = F p = 38
4.2 Gibbs W = p 2 F 1 F 2 F 2 + p 2 F 1 + p G = F + p = U S + p 42.1 G 2 G 1 G G, P G 4.3 U S, du = ds pd 4.3.1 = p = U 4.3.1a S U 4.3.1b S = 2 U S = 2 U S p = S 39
S, U p = 4.3.1c S S : F = U S df = du ds Sd = Sd pd, F df = Sd pd 4.3.2 F S = 4.3.2a F p = 4.3.2b S = 2 F = 2 F p = S = p 4.3.2c : G = F + p dg = df + pd + dp = Sd 40
dg = Sd + dp4.3.3, p G S = = S p G 4.3.3a p G 4.3.3b p = 4.3.3c p : H = U + p dh = du + pd + dp = ds dh = ds + dp4.3.4 S, p H = = H 4.3.4a S p H 4.3.4b p S = 4.3.4c p S S p du = ds pd df = Sd pd dg = Sd + dp 41
dh = ds + dp F = U S G = F + p = U S + p H = U + p = G + S S p 4.3.1 : 1 f x f x x = x 0 f x 0 = v 0 y = v 0 x x 0 + f x 0 y ψ 0 ψ 0 = v 0 x 0 + f x 0 f x v = f x = df dx ψ v = f x vx f x ψ v f x envelope f x ψ v 42
4.3.2 f x = 1 2 x2 x + 1 v = df dx = x 1 x = v + 1 ψ v = f x vx ψ v = 1 2 x2 x + 1 vx = 1 2 v + 12 v + 1 + 1 v v + 1 = 1 2 v2 + v + 1 2 v 1 + 1 v2 v = 1 2 v2 v + 1 2 ψ v f x x = dψ dv = v + 1 f x = ψ v + vx = 1 2 v2 v + 1 2 + vx = 1 2 x 12 x 1 + 1 + x 1 x 2 4.3.3, S, p, 4 U, F, G, H 2 43
5 closed system open system 5.1,, 2 N,, 2 N = N 1 + N 2 N 1, N 2 F = F 1 1,, N 1 + F 2, 2, N 2 F N 1,,N F1 = N 1 = 0, + F2 N2 N 2, 2 N 1 N F1 N 1, = F2 N 2, 2 F,, N η = F 5.1.1 N, µ = η 1 = η 2 5.1.2 G = g, p = G N,p N 44
µ 1 G = µn5.1.10 F = G p = µn p 5.1.11 U = F + S = S p + µn5.1.12 d = ds pd + µdn dp = sd + vdp5.1.14, p, µ µ, p 5.1.4 µ, p S = v = µ 5.1.14a p µ 5.1.14b p dµ = S N d N dp = sd vdp s = S N : 1 v = N : 1 5.2 phase: 45
5.2.1 A = B p A = p B µ A = µ B, p µ A, p = µ B, p 5.2.1, p 2, p, p +, p + p µ A +, p + p = µ B +, p + p, p, p µa µa µ A +, p + p = µ A, p + + p + p p µa µa dµ A = d + dp p p µ A +, p + p = µ A, p + s A + v A p + µ B +, p + p = µ B, p + s B + v B p + 2 µ A, p = µ B, p s A + v A p = s B + v B p 46
v A v B p = s A s B p = s A s B v A v B dp d = s A s B v A v B = s A s B v A v B l AB = v A v B l AB : A B 1 v A v B : 1 N dp d = l AB N v A v B N L AB = A B dp d = L AB A B 5.2.2 L AB, A B p 5.2.2 1.2 1g N N = 1 N 18 A 47
L AB = l AB N = 540cal/g = 2.26 10 3 J/g A = v A N = 1 ρ A ρ A = 0.60kg/m 2 = 0.60 10 3 g/m 2 B = v B N = 1 ρ B ρ B = 0.96 10 6 g/m 2 ρ B ρ A A B p = L AB A B L AB A L ABρ A p = L AB ρ A = 2.26 103 J/g 0.60 10 3 J/m 3 273 + 100 K = 3.6 10 3 J/m 2 K = 3.6 10 3 Pa/K p = 0.2 10 5 Pa = 0.2 105 3.6 10 3 = 5.6K 48
A B L AB A A B A = nr p dp d = L AB A B L AB A = L AB nr p 2 dp p = L AB 1 nr d 2 p ln = L AB 1 p 0 nr + 10 p = p 0 e L AB 1 nr + 1 0 p = p 0e L AB nr = 373K p = 10 5 Pa p 0 7 5.3 van der Waals p = nr b a 2 49
{ v = N nr = k B N p = k B v b 0 a 0 v 2 5..3.1 { a = a 0 N 2 b = b 0 N = p v ˆ ˆ v A, C, B ˆ A, B v ˆ C v A B A B p A = p B 5.3.1 F = G p = µn pvn = µ pv N 50
f = F N = µ pv5.3.2 dµ = sd + vdp df = sd pdv5.3.3 p = f 5.3.3 v p = k B v b 0 a 0 v 2 f = f, v v f = = v 0 pdv v v 0 kb a 0 pdv v b 0 v 2 v b0 1 f, v = k B ln a 0 + h 5.3.4 v 0 b 0 v f v, double tangent v A, B ˆ A, B ˆ A, B p = f v : ˆ A, B y y f µ f = f v + ψ v = pv + ψ 51
ψ = f + pv = µ µ A = µ B µ A = f v A + p 0 v A µ B = f v B + p 0 v B µ A µ B = 0 f v B f v A + p 0 v B v A = 0 vb vb p v dv + p 0 dv = 0 v A v A vb v A p 0 p v dv = 0 p = p 0 2 f v A, B 5.4 phase rule k, mixture F,, N 1, N 2,, N k N i : i df = Sd pd + µ 1 dn 1 + µ 2 dn 2 + + µ k dn k dg = Sd dp + µ 1 dn 1 + µ 2 dn 2 + + µ k dn k G, p, N 1, N 2,, N k = µ 1 N 1 + µ 2 N 2 + + µ k N k N = N 1 + N 2 + + N k 52
c i = N i : concentration N G = Ng, p, c 1, c 2,, c k k i=1 c i = 1 c 1,, c k g = µ 1 c 1 + µ 2 c 2 + µ k c k f N j i : j i c j i = N j i : j i N j N j = N j 1 + N j 2 + + N j k, p i = 1, 2,, k k f 1, p 2 µ 1 i = µ 2 i = µ 3 i = = µ f i f k c 1 1,, c f 1 c 1 k.,, cf k c i 1 + c i 2 + + c i k = 1 k ν = 2 + f k f k f 1 = 2 f + k5.4.5 53
f: k: 5.4.1 1. 1 1 k = f = 1 ν = 2 1 + 1 = 2, p 2. 1 2 k = 1, f = 2 ν = 2 2 + 1 = 1, p 3. 1 3 k = 1, f = 3 ν = 2 3 + 1 = 0, p 4. 2 1 k = 2, f = 1 ν = 2 1 + 2 = 3, p, G 5. 2 2 k = 2, f = 2 ν = 2 2 + 2 = 2, p < 100 C 54
6. 0?k = 0, f = 1 ν = 2 1 + 0 = 1 p = c 4 : 3 5,8,13,15 7/31 BOX 55