4 4 9............................................... 3.3......................... 4.4................. 5.5............................ 7 9..................... 9.............................3................................4..........................5.............................6........................... 3 4 3............................. 4 3............. 6 3.3...................... 7 3.4........................ 9 3 e-mail: nishitani@mtl.kyoto-u.ac.jp
( ) ( )W k B = k B ln W () exp( E i /k B T ). W N V E = N i= p i m () N N 3 6N N 6N 6N... q p q p > h (3) h W = N! ( ) N h 3 dq N dp N = N! ( ) V N h 3 dp N (4) N! N N! E E N(E) N(E + E) N(E) dn(e) E (5) de
q p : N(E) R N V N V N = (πr ) N/ NΓ(N/) (6) Γ Γ(n) =(n )! R = me N 3N V 3N = (πme)3n/ (7) 3N Γ(3N/) 4 W = N! ( V h 3 ) N (πm) 3N/ Γ(3N/) E 3N (8). (II) (I) (E T = E I + E II ) E T W (E T ) I E I W I (E I ) II E II W II (E T E I ) E I E II P (E I ) P (E I )= W I(E I )W II (E T E I ) W (E T ) (9) 3
E I P (E I ) dp (E I ) = dw I(E I ) W II (E T E I ) de I de I W (E T ) + dw II(E T E I ) W I (E I ) =. () de I W (E T ) (de I = de II ) dw I (E I ) = W I (E I ) de I dw II (E II ) () W II (E II ) de II dlnw I (E I ) de I = dlnw II(E II ) de II () dlnw II (E II ) de II = 3N I E I = k B T (3) (N 3N/ 3N/ ) E/N =3/ k B T E = T (4) W ( ) dlnw =d (5) kb = k B ln W (6).3 canonical ensemble() exp( E i /k B T ) P (E) P (E I + E II ) P (E I )P (E II ) (7) 4
(II) (I) E T = E I + E II (I) i p i (II) E II W II (E II ) E I = E i i p i p i = W i(e i )W II (E T E i ). (8) W (E T ) (9) W i E I i p i p i W II (E T E i ) (9) ln p i =ln W II (E T E i ) + const () const (II) (I) E T >> E i ln W II (E T E i ) ln W II (E T E i ) ln W II (E T ) dlnw II(E) E i () de i 3 [ p i exp dlnw ] ( II(E) E i =exp E ) i () de k B T Z p i = ( Z exp E ) i (3) k B T Z = ( exp E ) i (4) k i B T (partition function).4 ln Z dlnz dt = k B T Ei e E i/k B T e E i /k B T 5 = E (5) k B T
F = E T (6) F/T ( ) F d T = ( ) E d d T = E T dt + de d T (7) de = T d P dv (8) ( ) F d = E T T dt P dv (9) T E = T T ( F T ) V (3) 5 F = k B T ln Z (3) Z =e F/k BT (3) (E E ) = E E = ( E Z i e E i/k B T ) Ei e E i/k B T (33) Z Ē dē dt = d Ei e E i/k B T dt { Z = k B T Z E i e E i/k B T ( Z ) } Ei e E i/k B T (34) dē dt = k B T (E E ) (35) 35 6
.5 Einstein Einstein (a) (b) : (a) (b)einstein ( E n = n + ) hω (36) ω m K ω = K/m N 3 3N 3N ( E = n i + ) hω (37) i= Z = n = n 3N = e En i k BT = n i = e (n i+/) hω k B T 3N (38) n= x n = x (39) ( ) e hω/k B T 3N Z = e hω/k (4) BT 7
(3) F = k B T ln Z = 3k B TN ln ( e hω/k B T e hω/k BT (5) (35) +e hω/k BT ) (4) E = k B T dlnz dt =3N hω e hω/k (4) BT C = de ( ) hω dt =3Nk e hω/k BT B ( k B T e hω/k B T ) (43) 3 Free Energy [/3Nhw] - Temperature [k/hw] Energy [/3Nhw] Temperature [k/hw] pecific heat [/3Nk] Temperature [k/hw] 3: Einstein N A C = 3N A k B =3R (44) (Dulong-Petit) exp Debye 4: 8
( ). ( ) F F = F I (N I,T,V I )+F II (N II,T,V II ) (45) F [ ( FI ) ( ) ] (N I ) FII (N N I ) δf = δn I + = N I T,V N I T,V [ ( ) ( ) ] FI FII δn I = (46) N I N II ( ) F N T,V T,V T,V = µ (47) F (N,T,p) =F (N,T,V )+pv (48) T,p N ( ) = N T,p ( ) F + N T,p ( ) F V N,T ( ) V + p N T,p ( ) V N T,p (49) p = ( F/ V ) N,T ( ) = µ (5) N T,p 9
µ = µn (5) T,p N (αn,t,p) =α(n,t,p) (5) α (αn,t,p) = (αn,t,p) αn α αn α= α = (N,T,p) N (53) N µn (5) = µ i N i (54) i N i x i = i µ i x i (55). T p n m f = m + n (56) mn m +mn m (57) i j µ (j) i µ () = µ () = = µ (m) = µ µ () = µ () = = µ (m) = µ µ () n = µ() n = = µ(m) n = µ n (58)
n(m ) f f =+mn m n(m ) = m + n (59) f= m + n.3 I II (E I <EII ) (KI >K II ) Einstein?? I II Free Energy Phase I Ttr Temperature Phase II 5: (I) (II) Einstein I II phase transition T tr I (T tr )= II (T tr ) (6) = H T tr = (6)
H II H I = T tr ( II I ) (6) II I > I II II I (latent heat) (first-order transition) (second-order transition).4 C p H H(T )=H + T T C p (T )dt (63) H T T T C p (T ) (T )= + dt (64) T T f ncal/k/mol 8.4nJ/K/mol Richards rule n (9 J/K/g-atom) ( 4J/K/g-atom) ( 3 J/K/g-atom).5 6 I (undercooled liquid) (metastable equilibirum) Fe-Fe 3 C
I Tm[II] II Ttr Tm[stable] 6:.6 H O 7 H,O H +/O H O 7: H O - (P-T diagram) f =3 m (65) m = f = - f =3 = 3 m =3 3
f = T 3 (triple point) 3 (critical point) A,B (Clausius-Clapeyron) dp dt = V (66) d = dt + V dp (67) A dt + V A dp = B dt + V B dp (68) dp/dt P-T P-T 3 A B ( ) 3. (solution) (solid solution) segregation limit, A, B A, B A,B A,B x A,x B = x A A + x B B (69) 4
(a) (b) A T H = Ω x x < A B Free Energy ideal AB B µ B Ω= Ω> µ A Ω< x 8: (a) (b) segregation limit A,B N A,B N A,N B W = N! N A!N B! (7) (ln N! N ln N N) ln W = (N ln N N) (N A ln N A N A ) (N B ln N B N B ) = (N A + N B )lnn N A ln N A N B ln N B (7) N = N A + N B x A = N A /N, x B = N B /N = k B ln W = R(x A ln x A + x B ln x B ) (7) N R = k B N = AB = T = RT (x A ln x A + x B ln x B ) (73) (ideal solution) H (PV) (E) H = PV + E E A A, B B, A B i j N ij, e ij E AB = N AA e AA + N BB e BB + N AB e AB (74) z (coordination number) A B zx B A 5
Nx A N AB Nzx A x B A A B B N AA =/Nzx A,N BB =/Nzx B / E AB = Nzx Ae AA + Nzx Be BB + Nzx A x B e AB = Nzx A( x B )e AA + Nzx B( x A )e BB + Nzx A x B e AB = Nzx Ae AA + ( Nzx Be BB + Nzx A x B e AB e ) AA + e BB (75) segregation limit ( H = Nz e AB e ) AA + e BB x A x B =Ωx A x B (76) Ω (interaction parameter) m = x A A + x B B +Ωx A x B + RT (x A ln x A + x B ln x B ) (77) (regular solution) Ω < A B A A B B A B Ω > A B Ω = 4 8 3. (excess free energy) EX m = m ideal m =Ωx Ax B (78) Ω Ω=Ω +Ω x B (sub-regular solution) l = x A µ l A + x Bµ l B (79) 6
x =.4 A,B µ l A,µl B x = x = A µ A = A + RT ln x A (8) µ A = A + RT ln a A = A + RT ln γ Ax A (8) a A,γ A (activity) (activity coefficient) EX m = RT (x A ln γ A + x B ln γ B ) (8) a A a B aa a A a B a B x x 9: 8(b) (a)ω <, (b)ω > (b) 3.3 A?? T fa H fa = T fa fa (83) 7
T fa µ A µ A = H fa T fa (84) µ A (T )= B µ A (T )= H fa T fa (85) µ B (T )= H fb T fb (86) A,B T fa =5K, T fb =K H fa =.47KJ/mol, H fb = 8.3KJ/mol Ω = Ω =KJ/mol A B 73 m = x A µ A + x B µ B + m (87) = x Aµ A + x Bµ B + m (88) µ A (T )=µ B (T )= (a)t =5K A (b)t =3K (c) µ A = µ A µ B = µ B (89) p, q a p q p q (c) A T =K 8
(a) T=5K Free Energy [kj/mol] - (b) 5 T=3K (d) Free Energy [kj/mol] -6.5-7 -7.5 T=3K Free Energy [kj/mol] -5 - (e) -8.3 a q mole fraction B p +.5 (c) Free Energy [kj/mol] -5 - + T=K + mole fraction B Temperature [K] 6 8 4 p + () q () mole fraction B () : (a),(b),(c) (d) (b) (e).(e) (solidus) (liquidus) f = m n + m =, n = f = f = f,f.(d) f : f = q a q p : a p (9) q p (lever rule) 3.4 (Ω =6.6kJ/mol) 9
5 6 Free Energy [kj/mol] - T=6K Temperature [K] 4 8 b a c d g e f h i j k Free Energy [kj/mol] 4 - -4 T=9K e f g h Free Energy [kj/mol] T=3K 6 l m 4..4.6.8 mole fraction B 6 Free Energy [kj/mol] 4 T=753K - a 4 b T=K 4 () - i j k T=6K Free Energy [kj/mol] - -4 c d Temperature [K] 8 6 () + () + () + () () Free Energy [kj/mol] 5 l m + () 4..4.6.8 mole fraction B : 753K 3 f = (eutectic temperature) (eutectic composition) (maximum solubility limit) A,B α β 5 5 5(a) T 5(b)
: Fe-Fe 3 C