# Fermi Thermodynamics problems solved - ?· 1 Enrico Fermi, Thermodynamics, 1937 I. Θερµοδυναµικά…

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• 1

Enrico Fermi, Thermodynamics, 1937

I.

1. 3.12 L 4.01 L 2.34atm. .

: W = -P V = -2.34 atm 101325 Pa atm-1 (4.01 3.12) 10-3 m3 = -211 J

2. 30 g 1 m3 18C.

: ===PRT

MmPRT

MmnRTPV

atm 0.358kPa 36.3m 1

K 291.15molK J8.31447

mol g 2g 30

31 === P

3. ( ) 0C.

: ====RTPM

VmRT

MmnRTPV ( P = 1 atm)

311

1

mg 1250

K 273.15molK J 8.31447mol g 28.01348Pa 101325

=

=

kgm0.800

g 1250m 11 33

====m

Vv

4. 10 g 2 20C 1 0.3 atm.

: :

( )1

2122 lnlnln

12

1

2

1

2

1PP

MmRTPP

MmRTdP

PMmRTdP

PRT

MmPPdVW

P

P

P

P

P

P

===

==

dPPRT

MmdP

PVdV

PRT

MmV

T2=

==

: J 917atm 1atm 0.3ln

mol g 31.9988K 293.15molK J 8.31447g 10

1

1

=

=

W

.

1. 3.4 103 J 134 J.

: U = q + W = 134 J 3.4 103 J = 3266 J

2. 3 mol 5 atm 3 atm 0C;

: U = 0. U = q + W W = q

• 2

=== 1

2

1

2 lnln2

1PPnRTq

PPnRTPdVW

P

P

J 348053lnK 273.15molK J 8.31447mol 3 1 == q

3. 1 mol 291 21000 cm3 305 12700 cm3. (V, P) . .

: P1 P2

. VnRTP =

: kPa 115m 0.021

K 291molK J 8.31447mol 13

1

1 =

=

P

kPa 200m 0.0127

K 305molK J 8.31447mol 13

1

1 =

=

P

200

150

100

50

0

P (k

P)

0.0250.0200.0150.0100.0050.000V (m

3)

1

2

, , U = Q + W.

( ) ( ) J 1307m 0.0210.0127kPa2

2001152

312

212

1

=+

=+

== WVVPP

PdVWV

V

:

dVVUdT

TUdU

TV

+

= . (

) CV, 0 .

CV , U = CV T = n cV T.

cV = 3/2 R,

U = 1 mol 1.5 8.31447 J K-1 mol-1 (305 291) K = 175 J

• 3

: q = U W q = 175 J 1307 J = -1132 J

4. 1.35 . 18C. .

: Poisson: 1

1

212

122

111

==

VVTTVTVT

cV = 5/2 R cP = cV + R = 7/2 R, = 7/5 = 1.4

: ( ) C55K 3281.35K 273.1518 11.42 o==+= T

.

1. 1 mol () Carnot 400 300 . 1 L 5 L. .

: Poisson 1

1

1

212

122

111

==

TT

VVVTVT

.

cV = 3/2 R, cP = 5/2 R, = 5/3. , 1.54 L 7.70 L.

.

35

30

25

20

15

10

5

0

P (a

tm)

876543210V (L)

T = 400 K

T = 300 K

Carnot, , .

.: 2

1

2

1

TT

qq

=

==

1

2121 1 T

TqqqW

• 4

. ,

. , A

B

VV

nRTq ln= .

, kJ 5.35L 1L 5lnK 400molK J 8.31447mol 1 11-1 ==

q , , q2 = 4.01 kJ W =

1.34 kJ.

2. 400C 18C;

: 568.027340018400

2

12

2

12

2

=+

=

=

== TTT

qqq

qW

3. 1 J -18C, 38C.

: J 1.2227318

27338J 121

212

2

1

2

1 =++

=== qTT

qqTT

qq

J 0.22J 1J 1.2212 === qqW

IV.

1. 1000 g ; ( = 4.2 J g-1 K-1.

: ( )====== 12 lnln2

1

2

1

2

1

TTmcTdTmc

TdTC

dSSTdTC

TdqdS P

T

TP

T

T

PT

T

P

111

1

2 K J 1310K 273K 373lnKg J 4.2g 1000ln === S

TTmcS P

2. 1.192.1 10 TPV = . 100 L 0.42 J K-1. T V.

: dVVUdTCdV

VUdT

TUdU

TV

TV

+=

+

=

PVST

VUPdVTdSdU

TT

=

=

PdVSdTdF = Maxwell VT T

PVS

=

: 2.11.092.11.19 101.110 VTTPVTP

V

=

= .

:

• 5

( ) += dVVTVTTdTCdU V 2.11.192.11.09 10101.1dVVTdTCdU V

2.11.1810 +=

:

+=

+

= dVVSdT

TC

dVVSdT

TSdS

T

V

TV

dVVTdTTC

dS V 2.11.09101.1 +=

3. 78.3C 3580 J g-1. dP / dT .

: Clausius Clapeyron:

K 6.24atm 1K kPa 16.3

K351.45molK J 8.31447Pa 101325mol g 46g J 3580 12211

11

2 ==

=

=

dTdP

RThP

dTdP

, 1 atm.

V.

1. .

: : f = c + 2 p. 2 , . c = 2 p = 2, 2 .

2. ; ( .)

: H2O, N2, O2, . c = 3. , f = c + 2 p = 3 + 2 1 = 4 ,. 2 , . , f = 3 + 2 2 = 3, .

3. , , V: 0.924 + 0.0015 t + 0.0000061 t2, t C. 1 C 18C.

: , q = 0 U = V Q. :

dTdVTQVQU =

. , :

dTdVQTq 22

q = -1 C 291 K (0.0015+20.000006118) = -0.50 J

• 6

VI.

1. 2 2 K(T) 18C 5900. 1 atm. .

: Dalton atm 1AA2 ==+ PPP .

:

[ ]RTP

2A2A = [ ] RT

PAA =

( ) [ ][ ]

=== RTPPPRT

PP

TK 2A

A2

A

A2

2 2

AA

( )( )

++=

TPKTK

RTP 4112A

++

=

K 291molK J 8.31446Lmol 5900Pa 101325411

Lmol 59002K 291molK J 8.31447

11

1

1

11

AP

Pa 95081Pa 6244Pa 101325Pa 62442AA

=== PP

2AP ,

2AP . :

%2.3032.0

2624495081

26244

2

2A

A

A

2

==+

=+

= xPP

P

x

2. 1 -210 kJ mol-1, 19C 1 atm.

: , 19C Gibbs-Helmholz:

2, T

HTG

T nP

=

.

( ) ( ) ( ) === 200

0 lnlnlnRTH

dTTKd

RTGTKTKRTG

( ) ( )( ) ( ) ( )

=

=

=

21

0

1221

0

1

22

0 11exp11lnlnTTR

HTKTKTTR

HTKTK

TdT

RHTKd

( ) Lmol 4383K 292

1K 291

1molK J 8.31447

mol J 102.1-expLmol 590019 111-15

1

=

=CK o

• 7

++

=

K 292molK J 8.31446Lmol 4383Pa 101325411

Lmol 43832K 292molK J 8.31447

11

1

1

11

AP

Pa 94105Pa 7220Pa 101325Pa 72202AA

=== PP

%7.3037.0

2722094105

27220

2

2A

A

A

2

==+

=+

= xPP

P

x

VII.

1. 30 g aCl .

:

( )1

1 L mol 1.027L 12

mol g 35.4523.00g 301

=+===

VMm

VnC

atm 25.1bar 25.4Pa 102.54K 298mol K J 8.314m10mol 1.027 611

33 =====

RTVnP

= 298 .

T = Kf C = 1.84 K L/mol 1.027 mol/L=1.900 K, T = Ke C = 0.512 K L/mol 1.027 mol/L = 0.526 K.

2. (C6H12O6) NaCl . .

: RTMmP

1

11 = , RTM

mP2

22 2= = 21 PP

= RTMmRT

Mm

2

2

1

1 2 17.64527.3598977.22

9994.15600794.1120107.126222

1

2

1 =+

++==

MM

mm

3. .

: 2 2 . : f = c + 2 p= 2 + 2 2 = 2. .

4. ( ) . . ( . Clapeyron.)

• 8

VIII.

1. 4000 10torr. ( , .)

: [ ][ ] [ ]+

+

+=

NaNaNax

VIII 209:

310400026000

23

926000

23

92

cm mol 1012310400010931010931

===

..T.x

xn T (1),

VNn = Na.

RTnRTVN

P ii

i ==

.

ni = n (1-x), Na+ ni = n x . ( ) ( )xnRTxxxnRTP +=++= 11 ( ) 311 m mol 0.0401cmHg 76

Pa 101325K 4000molJK 8.314

cmHg 11 ===+

RTPxn . (2)

(1) (2) n

0880m mol 0.0401

cm mol 103.123

310.x ==

2. Debye 3R/2. ( .)

: (195):

( )

=TRDTC 3 ,

21

=

TD .

D (196):

( )1

3

112 1

1

0

33

=

eedxxD x

, D, . = 0.2485 D = 0.5. T = 0. 2485 .

31/1/2006 10/12/2008

1.0

0.8

0.6

0.4

0.2

0.0

D

1.00.80.60.40.20.0T/

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