мет.указ. к.р. №3

3.3. .

:

p= n0m<v2 >/3; p = <v2 >/3; p =2n< >/3; < > = m<v 2>/2,

<v > - ;

> - ; n0 – ;

m – ;–

: f(v) = dN(v)/Ndv = 4 (m0/2 kT)3/2v2exp [ - m0v2/2kT],

f(v)

dN(v)/N ,

v v+dv

:

p = p0 exp [ - Mg (h – h0)/RT];

p= p0exp[ - m0g(h – h0)/kT],

0 – h u h0 , M – , m0 – , R – ,

k –

:

n = n0 exp [ -Mgh/RT]; n = n0 exp [-m0gh/kT];

n= n0exp[- W /kT] ,

n , n0 - h h0=0;

W –

:

: vb = (2RT/M)1/2 = (2kT/m0)1/2 = (2p/ )1/2

: <v > =(3RT/M)1/2= (3kT/ m0)1/2 =(3p/ )1/2

: <v> = (8RT/ M)1/2 =(8kT/ m0)1/2= (8p/ )1/2

T – ; M – ;

m0 – ; p – ;

– ;

R – ; k –

:

k> =3kT/2.

: k> = ikT/2,

i – ; k – ; T –

: i =3;

: i=5;

: i =6.

: p = nkT

3.4.

, 1 :

<Z > = d2n0 <v>,

d – ; n0 – ; <v> –

:

<l> = <v>/ <Z> = 1 2 d 2 n0

: M = – D (d /dx) t,

– , s t;/dx – ; D – D = <v><l>/3

: F = – (dv/dx) s,

F –

s; dv/dx – ;

– = <v><l>/3,

–

: Q = - (dT/dx) t,

Q ,

s t;

dT/dx – ; – :

= cv<v><l>/3,

v - .

4.

4.1. .

:

U = iRT/2 = miRT/2M = ipV/2,

– , m – ; M – ; p, V - ; i - ;

R – .

: dU = miRdT/2M

, :

A = pdV = mRdT/M.

v2 T2

: A = pdV RdT .

v1 T1

: A = p(V2-V1) = mR(T2 –T1)/M .

:

A = mRT(ln V2/V1)/M = mRT(ln p1/p2)/M = p1V1ln(V2/V1) = p2V2·ln(p1/p2)

: A = 0

; Q = U + A,

Q – , ;U – ;

A –

4.2.

: C = Q/dT,

e Q – ; dT –

: C = C/M = Q/MdT, M –

: c = C/m = Q/mdT m -

: Cv = iR/2,

i – ; R –

.

: cv = Cv/M = iR/2M

: CP = (i+2)R/2;

: cP = CP/M = (i +2)R/2M

: CP – CV = R; P – V = R/ .

(

): CP/Cv = (i +2)/i

4.3 .

:

= CP/Cv

: A = - U = mcv (T1 –T2);

A = RT1m [1 – (V1/V2) -1]/( – 1)M = p1V1[1 – (V1/V2) -1]/( –1),

T1, T2, V1, V2 –

4.4. .

): = A/Q1 = (Q1 – Q2)/Q1 = 1 – Q2/Q1,

Q1 – , ; Q2 –

, ; A – ,

:

= (T1 – T2)/T1 = 1 – T2/T1,

T1 T2 –

) ( )

4.4. . .

: TQ 0

: dS = Q/T

:

dS = (CvdT/T + RdV/V)m/M;

dS = (CPdT/T – Rdp/p)m/M;

dS = (CPdV/V + Cvdp/p)m/M.

1

: S = Si , Si –

i 1

i–

: S k ln w S

w – ; S – ; k – n

: dS > 0; Q < TdS; TdS < dU + A,

dU - ; A – ,

; dS – .

5.

:

R - , m = 1, 2,3…, n =2, 3, 4, .. n+1 -

,

: (n = 1, 2, 3….)

( ): –

- : =

H –

, ,

:

6.

: R = R0, –

: = [Zmp + (A – Z)mn – m ]c2

m = Zmp + (A – Z)mn – m

:

: N= Noe t

=

: A =

–

–

: X +a

1. ,

( .).

,

L = 0,15 .

:

L = 0,15 ; R - ?

.

:

= 2 (L/g)1/2, = 2 (J/mgd)1/2,

J – ,

;

d – , d =

(1) : L = J/md.

: J = J0 + md2,

J0 – ,

. J0 = mR2/2.

, L = (J0 + md2)/md = 3R/2.

: R = 2L/3

: R=2·0,15/3= 0,10( ).

: R=0,10

2. F t = 0,2 c

W m = 20 ,

= Asin t, =15 ; = 4 -1.

t, = /2.

: t = 0,2

m = 0,02

= Asin t

= 0,15

= 4 -1

= /2

____________________

F-? t -?

.

F = ma, = d2x/dt2 = - A 2sin t.

F = - mA 2sin t =.

W = m 2A2/2 ,

= /2, ,

/2 = sin t, sin t = 1/2, t = /6.

: F =-0,02 ..(0,15)2=-0.02 , W=0,02.16.0,0225/2= =3,55·10-2 .

t = /6 = 4,17·10-2 c.

: F = - 0,02 , W ==3,55·10-2 . t = 4,17·10-2 c.

3. 0,4

10-5 . 4

. ,

c ,

?

: L= 0,4=10-5

U=4

Q(t), U(t), I(t) - ?

W - ? W - ?

Q U C .

Q U 0C cos t Q 4 10 5 cos 0,5 103 t,

I dQ U 0C sin t dt

:

W Q 2

2C

:

: Q 4 10 5 cos 0,5 103 t, , I 2 10 2 sin 0,5 103 , W 8 10 5 cos2 0,5 103 t ,

4. ,

404,4 404,7? 3

:

=0.031 = 404,4.10-9

1 = 404,4.10-92 = 404,7.10-9 d- ?

.

:

k=1,

: d 2

d = 0,03(404,7 - 0,404,4)/404,7 = 2,2 10 6 ( )

: d = 2,2 10 6

5. : – 2

– 2.

, ?

:

1 =2·10-3 ;2 =44·10-3 ;

N1/N2 - ?

. ,

N =6,02·1023 -1.

= m/M1,

2 = m/M2.

N1 = 1NA = mNA/M1,

N2 = 2NA = mNA/M2.

N1 N2 ,

N1/N2 = M2/M1

N1/N2 = 44/2 = 22

: N1/N2 = 22

6. V = 10 1 = 1 1 = 300 . , m = 10 , 2 = 290 .

2 , ,

.

:

V = 10 = 10-2 3;

1 = 1,0 = 1,0·106 ;1 = 300 ;

m = 10 = 1,0·10-2 ;2 = 290 ;

R = 8,314 ;

= 4·10-3 ;

2 - ? U -?

. ,

(5)

U1 = m1iRT1/2M.

,

U2 = m2 iRT2/2M.

U = U1 – U2 = R(m1 – m2) (T1 – T2)/M = Rm (T1 – T2)/M.

: [U] = = .

[p]= – · · /( · · 3 ) =

=– 3 = – 3 = .

: U = 8,314.0,01(300 – 290)/0,04 = 208 ( )

2=290·106/300 – 1,0·10-2·8,314·290/(4·10-3·10-2) = 3,64·105 ( .)

: U = 208 .

7. NO =300 .

, v1 = 820 v2 = 830

:

= 300 ; v1 = 820 ;

v2 = -830 /c;

N /N - ?

. ,

, ,

v v + dv

N/N = f (v, T)dv,

f (v, T) – ;

dv – , f (v, T) = const.

,

v = v2 – v1 = 10 .

,

N/N = f (v1, N) v.

, . 6

v1

, f (v, T).

,

v = v1 v = v2 ,

v = (2kT/m0)1/2 = (2RT/M)1/2 –

(3) vb = 410 .

, (23)

f = [f(v1, T) – f (v2, T)]/f (v1, T) = 0,07 = 7%.

: f(v1,T) = 4,03·10-4 c/ , f(v2, T) = 3,75·10-4 c/ .f =(4,03·10-4 - 3,75·10-4 )/ 4,03·10-4 = 0,07 = 7%.

,

(2). ,

N/N = f (v1, T) v = 4,0 · 10-3 = 0,4%.

: N/N = 0,4%.

8.

, D =1,0·105

t = 170 .

:

=1,0·105 ;

T = 170 C =290 K; d = 3,7·10-8 ;

l -? D - ?

. <l>

D :

<l> = 1/ 21/2d2n;

D=<l>·<v>/3;

= <l>·<v> nm0/3,

n – ; <v> – ; m0 – .

:

= nkT

(4) (1) <l>

= kT( 21/2d2p ) = 6,5·10-8 .

<v> =( 8RT/ M)1/2.