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2019年-统计热力学-PPT精选文档_图文


11-1 11-2 11-3 11-4 11-5


11-6 11-7 11-8 11-9











()



1 ()

()


2




11-1

= t + r + v + e + n
Schrdinger
()


t 8h2m (a n2 x 2 n b2 2 y n c2 z2 )
h6.62610-34 Js m a,b,c() nx,ny,nz

t 8h m 22V /3 (n x 2n2 ynz2)



3h2 t08m2V/3 gt01

t186mh22V/3 gt13



t2

9h2 8m2V/3

gt23





r



h2 82I

J(J1)

II = R02 R0 J0,1,2,......

gr = 2J + 1







(

1 2

)h

0,1,2,......

v
1 k (k ) 2

g = 1

ge0 = gn0 =



/kT ~10-19

/kT ~10-2

/kT ~10
----

11-2

1 (j) (j) (nj)
{nj}

2
(i)(i) (ni) i
{ni}
() WD
WD
D

NUV

N ni U nii

i

i

N nj U njj

j

j

VN4U8 259 (g1 =2 g3 =2)



WD N!
i

gni i
ni !

n1 , n2,, niD g1 , g2,, gi N

WD N!
i

gni i



ni !

1. NN1N 1

WD N!

2. Nn1n

n1 , n2,, ni

W D

N! ni !

i

WD N!
i

gni i



ni !

3. Ng1 , g2,, gn
n1n
n1 , n2,, ni


WD N!
i

gni i
ni !



WD
i

(ni gi -1)! ni!(gi -1)!

ni << gi

WD

i

g ni i
ni !


W D(U N V,,)
D


11-3

()()


()(PA)

n

PA



lim
m

m

m n A

PA < 1 Pi = 1


NUV
P 1

D
PD 1WDW D
WDD NUV

()

NUV()

PB



WB


NUV(N 1024)


N AB



WD



N! M!(NM)!

M = N / 2

WB



N! (N/2)(!N/2)!



2N

N=10(20)A B

M

0...4

5

6 ... 10

N-M 10 ... 6

5

4... 0

WD

1 ... 210

252 210 ... 1

PD 9.8 10-4 ... 0.20508 0.24609 0.20508 ... 9.8 10-4

M

0...9

10

11 ... 20

N-M 20 ... 11

10

9... 0

WD 1 ... 167960 184756 167960 ... 1 PD 9.5 10-7 ... 0.16018 0.17620 0.16018 ... 9.5 10-7

N! WB (N/2)(!N/2)!

2N

lim N! 1
N 2N(N/eN)

WB

2 2N N

PBW B

2 81013 N



(N/2 m N/2 + m)PB

PBW Bm 22NN(NmN ()N !!m )2!1N0.999

2

2



11-4


nj ej/kT ni giei/kT
k
ei(j)/kT

nj ej/kT ni giei/kT

N n j e j/k T e j/k T q

j

j

j

N n iie g i/k T g ie i/k T q

i

i

i

nj



N q

ej /kT

ni



N q

g ei /kT i



nj



N q

ej /kT

ni



N q

g ei /kT i



1e

- j

/

kT

j ()

gie

- i

/

kT

i

2q e j/kT giei/kT()

j

i

3ik

ni nk



g ei /kT i
g ek /kT k



WD N!
i

gni i
ni !

lnD W ln N (in l!n i lgn i!)n
i


lnN N!ln -N N



ni

N q giei/kT

Ngiei/kT gei/kT
i

i

11-5



q

e j/kT

gei/kT i

j

i



it, i r, i, ie, in,i
gi gt,g ir,gi,gie,gin,i

q g e g g g g g e i/kT

( ti , r i , i , e i , n i),/k

i

t,r i, ie ,i, n i ,i

i

i

q g te , i ti/ , k T g re , r i i/,k T g e , ii /,k T g e e , e i i /,k T g n e , n i/ i,k

i

i

i

i

i



qqtqrqqeqn





i0 i 0

q g e g e e g e i/kT ( i 0 0 )/k T 0 /kT i 0 /kT

i

i

i

i

i

i

qe0/kTq0

q0 e0/kTq

q0 e0/kTq
1

ni N qgiei/

Neeg q kT

(i00) i
0/ k T 0

/

k Tq N 0giei0/k

T

q0 e0/kTq



2

qt0

et,

q 0/kT t

qt

qr0er,/0kq Tr qr

q 0e,/0kq Teh/2kq T

qe0

e q e,0/kT e

qn0

e q n,0/kT n





t

8h2m (a n2 x 2

n2 y b2

nz2 c2

)

q ej /kT
j

q(n x,n y,n z)ex p8 h2m (a n2 x 2n b2 2 yn c2 z2)/k T

q n x 1 e x 8 p h m 2 2n x 2 k n y 1 T e x a 8 p h m 2 2n 2 yk n z 1 T e x b 8 p h m 2 2n z 2 k

qqt,xqt,yqt,z

qtx , nx 1ex p8h m 2 2knx 2T a2h mkaT
qt 2h m 2 k 3/a T 2 b c 2h m 2 k 3V /T 2ft3

ft

2hm2

kT1/2V1/3







h2 r 82I J(J1)

q r
i

g re , iri/, k T J 0 (2 1 J e)[ x 8 p h 2 I 2 k JT (1 J ])

h2 r 82 Ik



K

H2 N2 O2 CO NO HCl HBr HI Cl2 Br2 I2
r/K 85.4 2.86 2.07 2.77 2.42 15.2 12.1 9.0 0.346 0.116 0.054

qr

T r



82IkT
h2



2 1



fr qr1/2Tr1/2







( 1 )h 2

q

g e , i /kT ,i

i

exp [( 1 )h /kT ]

0

2





h
k



e h/2 kT


exp [ h/kT

]



0


H2 N2 O2 CO NO HCl HBr HI Cl2 Br2 I2 /K 6210 3340 2230 3070 2690 4140 3700 3200 810 470 310
q e /2 1 T e /2 T e h /k 2- T 1 e h /k 2T f
q 0 e /,0 k q T e h /2q k T 1 e 1 -h /k T 1 e 1 - /T





q ge ge ei,kT

e,0 kT

e

e,i

e,0

i

q ge ge ni,kT

n,k 0 T

n

n, i

n, 0

i



qe0 ge,0

qn0 gn,0

11-6



U
i

nii
i

N qgiei/kTi

q

gei/kT i

i

T q V i g ie i/k T k i T 1 2 k 1 2T i g ie i/k iT

UNk2 T q Nk 2 T lnq

q TV

TV

1 2

UNk2Tlntqrqqeqn T V

UNk2Tlntq Nk2Tlnrq Nk2Tlnq

T V

T V

T V

Nk2Tlneq Nk2Tlnnq

T V

T V

U = Ut + Ur + U + Ue + Un

3

U0 NkT2lnTq0 V UN 0UU 0
q0 e0/kTq

U 0 U t0 U r 0 U 0 U e 0 U n 0

Ut =Ur

=0 =0

Nh U 2

Ut Ur U

1Ut0

Ut0Ut Nk2T lT ntqV

qt



2hm2kT3/2V

Ut0



3 2

NkT

1mol

Ut0,m



3 2

RT

2Ur0

Ur0Ur Nk2T lT nrqV

T 82IkT

qr

r



h2

Ur0 NkT

1mol

U0 r,m



RT

3U0

U 0UN 2hNk2T lT n 0q V

q0



1 1e-/T

U0

1 N k e/T 1

>> T, U0 0 U,m0 0

<< T, U0 NkT U,m0 RT



U mU t,m U e, m U n, m 2 3R T U 0,m
U m U t, m U r, m U , m U e, m U n, m 5 2 R U T 0,m U m U t, m U r, m U , m U e, m U n, m 7 2 R U T 0,m

11-7



CV,m





Um T

V

Um

RT2lnq T V

CV,m U T mV TR2T lT nV q V

1CV,m
C V, m U T m V (U m 0 T N 0) V U T m 0 V
2
C V, mC V ,tC V, rC V,

CV,t CV,r CV,

CV,t = 3R/2 CV, r= R CV, = 0 ( T <<)

CV, = R ( T >>)

CV,m = 3R/2 CV,m = 5R/2

11-8

S = S(NUV) S = S1(N1U1V1) + S2(N2U2V2) = 1(N1U1V1) 2(N2U2V2) ln = ln1(N1U1V1) + ln2(N2U2V2) S = k ln



= 2N

WB



N! (N/2)(!N/2)!

ln = Nln2=0.693N

lnB W lnN!-2)l! n0(.N 69/23N



S = k ln WB


S = k ln






S = k ln WB
1

WD

i

g ni i
ni !

S Nkln q U Nk NT

ni



N q

g ei /kT i

S



q0 Nkln

U0



Nk

NT



WD

i

g ni i
ni !

lnD W nilni glni!n
i
ln D W n iln i n g iln i n n i
i

ni



N q

g ei /kT i

lnBW i n iln ig n il n N qgie i/kT n i

lnB W
i

nilnN qn kiiT ni

qU

Sk

l

n BW N

k ln N NT

k

2

WD N!
i

gni i
ni !

ni



N q

g ei /kT i

S Nklnq U T
S Nklnq 0 U 0 T

3

S S t S r S S e S n



St

Nkqltn0 Ut0 NT

Nk

Sr Nklnr0 qUTr0 S Nkln0 qUT0


() St, Sr, S


298.15K
CO NOH2

()

1St

St

Nkqltn0 Ut0 NT

Nk

qt0



2mk
h2

T3/2V

Ut0



3 NkT 2

St Nk2ln m h3N k 3/T V 25 2Nk

2m k3/T V 2 5

St Nklnh3N

Nk 2

1mol--

S m ,tR 2 3ln(M m 1o )/ k 5 2 llg n( T ln/(K p 2)/0 P .7 a

2Sr

Sr Nklnr0 qUTr0

qr



T r

Ur0 NkT

Sr NklTn(r)Nk
1mol
Sm ,r RlT n (r)R

3S

S Nkln0 qUT0

q0

1 1e-/T

U0

1 N k e/T 1

S N- e k - /T ) l - 1 n N T ( 1 k ( 1 / e T 1 1)

1mol

S m ,R- l e - n /T ) - ( 1 R 1 T 1 ( e /T 1 1)

11-9 (AGH)q

SNklqnUNk NT
AU-TS kTlNn /N ()q!
p ( A V T / ) N k ln T V T q()/
G A p V kT N /lN )n N ! (q k ln T V T q V )/ H U p V N2 (k ln T T V q N )/ k ln T V T q V pV = nRT


S NklnqU T
AU-TSkTlnN q
p ( A V T / ) N k ln T V T q()/
G A p V kT N lN nk q ln T V T q V )/( H U p V N2 (k ln T T V q N )/ k ln T V T q V )

(NUV) j i () gi

A. g j j i

B.

j

gj

i

C.lngj

j

i

D.

j

lng

j

i