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

Similar documents
6 6.1 B A: Γ d Q S(B) S(A) = S (6.1) T (e) Γ (6.2) : Γ B A R (reversible) 6-1

// //( ) (Helmholtz, Hermann Ludwig Ferdinand von: ) [ ]< 35, 36 > δq =0 du

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

all.dvi

現代物理化学 2-1(9)16.ppt

70 5. (isolated system) ( ) E N (closed system) N T (open system) (homogeneous) (heterogeneous) (phase) (phase boundary) (grain) (grain boundary) 5. 1

untitled

i 18 2H 2 + O 2 2H 2 + ( ) 3K

30

現代物理化学 1-1(4)16.ppt

Microsoft Word - 11問題表紙（選択）.docx

5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E

i


flMŠÍ−w−î‚b

0 (Preliminary) T S pv

2 p T, Q

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

September 25, ( ) pv = nrt (T = t( )) T: ( : (K)) : : ( ) e.g. ( ) ( ): 1

Untitled

5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1

KENZOU

スライド 1

x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v

v er.1/ c /(21)

1 1 x y = y(x) y, y,..., y (n) : n y F (x, y, y,..., y (n) ) = 0 n F (x, y, y ) = 0 1 y(x) y y = G(x, y) y, y y + p(x)y = q(x) 1 p(x) q(

master.dvi

ρ(= kg m 3 ), g h P 0 C () [1] 1.3 SI Pa hpa h 100 ( : 100 ) 1m 2 1N 1Pa 1N 1kg 1m s 2 Pa hpa mb hpa 1mm 1mmHg hpa 1mmHg =

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x

kou05.dvi

0201

A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa

0 (Preliminary) F G T S pv (1)

Microsoft PowerPoint - 熱力学Ⅱ2FreeEnergy2012HP.ppt [互換モード]

2009 June 8 toki/thermodynamics.pdf ) 1

Maxwell

meiji_resume_1.PDF

( ) ( )

1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l

2,200 WEB * Ξ ( ) η ( ) DC 1.5 i


r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -

( ) ,

December 28, 2018

応力とひずみ.ppt

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2

1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC

i Γ

理想気体ideal gasの熱力学的基本関係式

18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α

genron-3

1 c Koichi Suga, ISBN

K E N Z OU

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,

Part () () Γ Part ,


A

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P

Korteweg-de Vries


pdf

DVIOUT

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3

6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2

II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re

iBookBob:Users:bob:Documents:CurrentData:flMŠÍ…e…L…X…g:Statistics.dvi

Fubini


24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x


2012 A, N, Z, Q, R, C



i

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n

B ver B

縺 縺8 縺, [ 縺 ﾁ : () () () 4 ﾁ93799; () "64": ｨｬ 9997ｨ

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math))

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

A


1 1. x 1 (1) x 2 + 2x + 5 dx d dx (x2 + 2x + 5) = 2(x + 1) x 1 x 2 + 2x + 5 = x + 1 x 2 + 2x x 2 + 2x + 5 y = x 2 + 2x + 5 dy = 2(x + 1)dx x + 1

2016 B S option) call option) put option) Chicago Board Option Exchange;CBOE) F.Black M.Scholes Option Pricing Model;OPM) B S 1

2 1 (10 5 ) 1 (10 5 ) () (1) (2) (3) (4) (1) 2 T T T T T T T T? *

A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P

P F ext 1: F ext P F ext (Count Rumford, ) H 2 O H 2 O 2 F ext F ext N 2 O 2 2

Erased_PDF.pdf

untitled

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat

第10章 アイソパラメトリック要素

2

1 1.1 [ ]., D R m, f : D R n C -. f p D (df) p : (df) p : R m R n f(p + vt) f(p) : v lim. t 0 t, (df) p., R m {x 1,..., x m }, (df) p (x i ) =

Transcription:

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) 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 = µ V U,N,, T =. (6) N U,V,, ( ) S(U, V, N) (thermodynamic potential) [ ] 2.3(44) [ ( )3/2 ( ) ] U V S(U, V )=S(U 0,V 0 )+Nk B ln U 0 V 0 U ( ) e (S S 0)/Nk B U 3/2 ( ) V = U0 V0 (7) ( ) 2/3 V0 U = U 0 e 2(S S 0)/3Nk B. (8) V T = U P = U V = V 2U 3Nk B (9) = 2U S 3V (10) 40

[ ]< 88 > (U, S, V, N, ) q p = L/ q L(q, q) H(q, p) H(q, p) = L(q, q) q q L(q, q) (11) (11) q p 41

3.3 [ ]< 88, 89 > x f(x) f(x) p(x) df = df dx p(x)dx (12) dx (Legendre transformation) x p(x) f (x) (x, f(x)) p y y g f xp (13) p g dg = df pdx xdp = xdp (14) p g p (13) g(p) =f(f 1 (p)) f 1 (p)p. (15) [ ]< 91, 92 > 2 df = f dx + f dy p(x, y)dx + q(x, y)dy (16) x y y x x g(p, y) f xp. (17) g p y dg = df xdp pdx = xdp + qdy. (18) y h(p, q) g yq. (19) h p q h = dg ydq qdy = xdp ydq. (20) 42

3.4 [ ]< 92 > (Helmholtz free energy) F (T,V,N) F = U U S = U TS. (21) df = SdT PdV + µdn + (22) S = F P = F µ = F. (23) T V,N,, V T,N,, N T,V,, F H F [ ]< 92 94 > ( T )? 2.1 (34) T δq TdS (24) 2.1 (1) δq δq = du δw W δw = PdV du + PdV TdS (25) (T = const. dv =0) d(u TS) 0 (26) ( ) df =0, F = F min (27) 43

[ ]< 95, 96 > (8) ( ) 2/3 V0 U(S, V )=U 0 e 2(S S 0)/3Nk B. V T (S, V )= U = 2U 0 V 3Nk B ( V0 V ) 2/3 ( ) 2/3 e 2(S S V0 0)/3Nk B = T 0 e 2(S S 0)/3Nk B V S [ ( )3/2 ( ) ] T V S(T,V )=S 0 + Nk B ln T 0 V 0 F (T,V )=U TS = 3 ( [ ( )3/2 ( ) ]) T V 2 Nk BT T S 0 + Nk B ln (28) T 0 V 0 S(T,V ) = F [ ( )3/2 ( ) ] T V = S T 0 + Nk B ln (29) V T 0 V 0 P (T,V ) = F = Nk BT (30) V T V 44

3.5 [ ]< 96, 97 > (enthalpy) H(S, P, N) H = U U V = U + PV. (31) V T = H [ ]< 97, 98 > dh = TdS + VdP + µdn +, (32) V = H µ = H. (33) P,N,, P S,N,, N S,P,, δw rev other dh P = d(u + PV) P = du P + PdV P = (δq P + δwother rev PdV P )+PdV P = δq rev P + δwother (34) δq P du V = δq V P ( isobaric, adiabatic) dh P = δwother rev irr δwother. (35) δw irr other =0 dh P 0 (36) [ ]< 100 > dh =0, H = H min (37) C V = δq = U, (38) dt V T V C P = δq = H. (39) dt P T P 45

3.6 ( ) [ ]< 102 > (Gibbs free energy) G(T,P,N) (free enthalpy) S = G T G = U U S U V V = U TS + PV = F + PV = H TS. (40) dg = SdT + VdP + µdn +, (41) V = G µ = G. (42) P,N,, P T,N,, N T,P,, (2 (83)) G = µn (43) 1 [ ]< 103 > (isothermal isobaric) T δq TdS (44) δq = du δw W δw = PdV du + PdV TdS (45) (T,P = const.) d(u + PV TS) 0 (46) ( ) dg =0, G = G min (47) 46

3.7 [ ]< 110, 112 > N N µdn µ U(S, V ) du = TdS PdV (48) T V = P (49) S V H(S, P ) dh = TdS + VdP (50) T P F (T,V ) = V (51) S P df = SdT PdV (52) G(T,P) V = P (53) T T V dg = SdT + VdP (54) P = V (55) T T P [ ](grand potential) N Φ=F µn = PV (56) 47

Φ(T,V,µ) F dφ = SdT PdV Ndµ (57) Φ(T,V,µ) 2 = P P, = N, V T,µ T V,µ µ T,V V T,µ N T = (58) V,µ µ T,V // //( ) (Gibbs, Josiah Willard 1839 1903) 1858 1863 Ph.D 1871 1873 34 1876 1878 10 1880 1901 [ ]< 118, 119 > ( ) (Jacobian) 2 (3 ) (u, v) J(x, y) = (x, y) = u v x x (59) u y v y 1. (u, y) (x, y) = u. (60) x y 48

2. 3. (v, u) v) = (u, (x, y) (x, y). (61) (u, v) (x, y) = (u, v) (s, t) (s, t) (x, y). (62) 4. (x, y) (u, v) = ( ) 1 (u, v). (63) (x, y) 5. d (u, v) ds (x, y) = ( du ds,v) (x, y) + ( ) u, dv ds (x, y). (64) [ ]< 113 116, 119 > U 2 = U 1 ( 1 U(T,V )=U(T 0,V 0 )+C V (T T 0 ) an 2 V 1 ) (65) V 0 U 1 = U 2 T 2 T 1 = an 2 ( 1 1 ) V2 V1 C v (66) (P 1,V 1 ) (P 2,V 2 ) U 2 U 1 = P 1 V 1 P 2 V 2 (67) (H 1 = H 2 ) ( dh = TdS + VdP = T dt + ) dp + VdP = 0 (68) T P P T 49

T T + V T V + V P = T T P H T = P C P T = V (Tα 1) C P (69) P (α ) ( ) (Tα 1) (v = V/N, k B T T ) (P + av ) (v b) =T (70) 2 v T P a Pv + b T P a T + b (71) T v T v = 2a T b (72) (70) ( v T = P a v 2 + 2ab ) 1 v 3 (73) (T ( v/ T )=v) (P a ) ( ) v 1 ( v 2 (v + b) =T = v = v P a T v 2 + 2ab ) v 3 P = 3k BT 2b + 2 2 ak B T b b (74) a b 2 (75) 50

3.8 [ ]< 119 > T bath P bath δu δs δv δq = δu + P bath δv (76) δq rev = T bath δs δq δq rev = T bath δs (76) δu + P bath δv T bath δs (77) δu T bath δs + P bath δv 0 (78) δu δu = U δs + U δv + 2 U V V S 2 (δs)2 +2 2 U V δsδv + 2 U V (δv 2 )2 + (79) U/ = T U/ V = P (78). (T T bath ) δs (P P bath ) δv + 2 U 2 (δs)2 +2 2 U V δsδv + 2 U V 2 (δv )2 0 (80) (80) δs δv 2 U 2 (δs)2 +2 2 U v δsδv + 2 U V (δv 2 )2 > 0 (81) 2 U 2 > 0 2 U V 2 > 0 2 U 2 ( U 2 ) 2 2 V U > 0 (82) 2 v 1 2 U = T 2 = T > 0 (83) V C V 2 2 U V = P 2 = 1 > 0 (84) V S Vκ S 3 ( U, ) U V = (T,P) (S, V ) (S, V ) = (T,V ) (T,P) (S, V ) (T,V ) = T P > 0 (85) C V V T 51

V > 0, < 0, T V P S V P < 0. (86) T [ - ]< 120 > - (Le Chaterier-Braun s principle) // //( ) (Le Chatelier, Henri Loius 1850 1936) 1877 1908 1884 U du = TdS PdV (+µdn) (87) H = U + PV dh = TdS + VdP (+µdn) (88) F = U TS df = SdT PdV (+µdn) (89) G = U TS + PV = µn dg = SdT + VdP (+µdn) (90) 52

2024 © docsplayer.net プライバシー | 利用規約 | フィードバック