методы вычисления энтропии смеси идеальных

536.24.01
Ⱥɥɟɤɫɚɧɞɪ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ Ʉɭɥɢɤɨɜ,
,
ltuttsu@mail.ru
Ⱥɧɚɬɨɥɢɣ Ɏɟɞɨɪɨɜɢɱ ɋɦɨɥɹɤɨɜ,
,
ɂɪɢɧɚ ɇɢɤɨɥɚɟɜɧɚ Ⱦɸɤɨɜɚ,
ɂɪɟɧɚ ȼɢɤɬɨɪɨɜɧɚ ɂɜɚɧɨɜɚ,
,
ɋɚɧɤɬ-ɉɟɬɟɪɛɭɪɝɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɥɟɫɨɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ
ɆȿɌɈȾɕ ȼɕɑɂɋɅȿɇɂə ɗɇɌɊɈɉɂɂ
ɋɆȿɋɂ ɂȾȿȺɅɖɇɕɏ ȽȺɁɈȼ
ɑȿɊȿɁ ɗɇɌɊɈɉɂɂ ɄɈɆɉɈɇȿɇɌɈȼ
ɗɧɬɪɨɩɢɹ, ɢɞɟɚɥɶɧɚɹ ɝɚɡɨɜɚɹ ɫɦɟɫɶ, ɧɟɨɛɪɚɬɢɦɨɟ ɫɦɟɲɟɧɢɟ.
Entropy, an ideal gas mix, irreversible mixture.
.
.
.
,
,
.
Ɇɟɬɨɞɢɤɚ ɢɫɫɥɟɞɨɜɚɧɢɹ.
-
.
,
1
2.
.
V
,
V1
V = V1 + V2.
m1 m2.
,
V2
.
,
1 2
Ɍ ɪ.
, -
. 1.
V
172
.
m
,
.
Ɋɢɫ. 1.
1
2
V
[1, . 107; 3, . 97],
V,
-
.
,
.
Ɍ
-
ɪ.
-
V(
,
Ɍ,
),
,
ɪ = ɪ1 + ɪ2,
ɪ1
ɪ2 –
1
(1)
2
,
.
,
:
S
 V
S T ,  –
 m
i-
 V 2  V 
 S  T ,    Si  T , i ,
 m  i  1  mi 
Vi,
,
,
ɛɚɜɤɚ (
3
,
V 
; 
m
,
3
),
/ ; Vi
 Vi 
m  –
 i
 V 
/ ; Si  T , i  –
 mi 
,
V,
mi –
/ ; S
–
(2)
-
,
– ɪɚɡɞɟɥɢɬɟɥɶɧɚɹ ɷɧɬɪɨɩɢɣɧɚɹ ɞɨi,
i-
-
/ .
173
.
,
S
–
.
,
,
(
,
. 1)
-
,
(
,
,
V).
,
.
ɫɦɟɫɢ
.
,
S
.
,
-
,
(
–
,
).
,
.
,
,
,
-
.
.
(2)
[2, . 59; 4, . 59]

2
i 1
:


 V
 V
V 
S  T ,   m  s  Ɍ ,   m cV ln T  R  ln    so  ;
 m
 m
m




 V  2
V 
 V  2
Si  T , i    mi si  Ɍ , i    mi cVi ln T  Ri ln  i   soi  ,
m
m
m
i 1
i 1
i 
i 


 i


 V
s T , 
 m
/( · ); ɫV
 V 
si  T , i  –
 mi 
ɫVi –
ii-
174
,
,
/( · ); R
/( · ); so soi –
i-
i-
(3)
(4)
,
Ri –
,
(
i-
)
Ɍ
)
(
/( · ).
(3) (4),
ɪ,
(2),
-
 g1 g 2  

  
  g1 g 2  μ1 μ 2  
 μ  μ 

  So ,
 mRo ln   1 g 2 
g2 
1

  1  μ1  1  μ 2 
 μ  μ 

  1  2

S
So  mso  m1so1  m2 so2 ,
/ ;R –
/(
· ); g1 g2 –
1 2, /
.
(5)
,
:
μ1
μ2 (
.
1
(5)
; μ1
2
,
μ2 –
S
g1
);
g2
,
,
,
.
(2)
:
2
 μ 
 Ro  ni  ln  i   So ;
μ 
i 1
S
(6)
2
g 
 Ro  ni  ln  i   So ,
 ki 
i 1
S
ni, gi, ki, μi –
i-
-
,
,
, / ,
, /
.
(2)
/
,
,
/
 V 
 V
S  T ,    Si  T , i   S
 m  i  1  mi 
(7)
;μ
–
-
:
2
,
(8)
.
(6),

 V  R  μ 
R
 V
s  T ,    mi  cVi  ln T  o  ln  i   o ln  i    mso .
μi
 m  i 1 
 mi  μ i  μ  
2
(9)
175
1
V,
1
2
V1
.
.
V2
,
V
-
(3)
(4),
,
. 2.
Ɋɢɫ. 2.
1 2
V1 V2
V
,
,
V
m
Si  mi  Ri  ln  i
 Vi
 mi
ΔS i –


V 
V 
R
  mi  o  ln    ni  Ro  ln   ,
μ
V

i
 i
 Vi 

(10)
i-
-
Vi
1 2
,
S
V,
/ .
  Si .
ΔS
1
2
1
(11)
i 1
(11),
(10),
 g1 g 2  

  
  g1 g 2  μ1 μ 2  
 μ  μ 


S 1  mRo ln   1 g 2 
g2 
1

  g1  μ1  g 2  μ 2 
 μ  μ 

  1  2

176
2
.
(12)
S
n–
1
 nn
 Ro ln  n n
 n 1n 2
 1 2
,
ɪ1
(13)
.
.
Ɍ.

 ,

ɪ 2,
.
ɧɟ ɩɪɨɢɡɨɲɥɨ.
,
V
-
,
(
. 1).
1
2
V
V1
V2 (
).
1
2
.
-
2V.
. 3.
-
.
Ɋɢɫ. 3.
1
2
,
1
ΔS
2
S
2
2
 2V
 S T ,
 m
,
S
2
 2  V 
   Si  T ,  .
 i  1  mi 
2

(14)
(4),
  1 n
n
n 
 Ro  ln  2n    n1  1  n2  2 
  n 

 S
S
:
(3)
(14)
2V
V

 Ro  ln 2n g1n1 g 2n2  So .
(15)
(16)
177
2V
V.
. 4.
Ɋɢɫ. 4.
2V
V
:
 V  
 m 
R
1
1
S 3  ms 3  mR ln      m o ln    nRo ln   ,
μ
2
2
  2V  
  m  
Δs
3
(17)
–
2V
ΔS∑123 = ΔS
1
+ ΔS
(18)
(18)
(19)
2
V,
+ ΔS
3.
(17).
-
.
(19)
ΔS
,
-
(18)
(12), (15)
S123  S
/( · ).
ΔS∑123
.
.
(8),
1
,
,
,
,
2
,
,
(
 V 2  V 
S  T ,    Si  T ,   S pi ,
 m  i  1  mi 
178
-
. 2):
(20)
 S pi
– ɪɚɡɞɟɥɢɬɟɥɶɧɚɹ ɷɧɬɪɨɩɢɣɧɚɹ ɞɨɛɚɜɤɚ,
S
ɪi ,
/ .
pi
S

,
S pi  mRo ln g1
g1/μ
1
-
Ɍ

,


g 2g2 /μ 2  So  Ro ln g1n1 g 2n2  So .
(20),
(21)
(21),


R V  R
 V
S  T ,    mi cVi ln T  o ln    o ln gi   mso .
μ i  mi  μ i
 m  i 1 

2
Ɋɟɡɭɥɶɬɚɬɵ ɢɫɫɥɟɞɨɜɚɧɢɹ.
,
(
(9)
m
1
(22)
(22)
,
-
s
)
,
-
(
):
R  V  R  μ 
 V 2 
s  T ,    gi cVi ln T  o ln  i   o ln  i    so ;
μ i  mi  μ i  μ  
 m  i 1 

R V  R
 V 2 
s  T ,    gi cVi ln T  o ln    o ln gi   so .
μ i  mi  μ i
m  i 1 


(23)
(24)
ȼɵɜɨɞɵ
,
,
,
,
,
,
.
,
,
,
,
.
,
« -
» [5].
Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣ ɫɩɢɫɨɤ
. .
1. Ɂɨɦɦɟɪɮɟɥɶɞ, Ⱥ.ȼ.
. .:
2. Ɏɟɪɦɢ, ɗ.Ⱥ.
, 1969. . 139.
[
, 1955. . 479.
/ . .
.
:
] /
.
179
3. Ʉɪɭɬɨɜ, ȼ.ɂ.
, 1991. . 384.
4. Ȼɚɡɚɪɨɜ, ɂ.ɉ.
1983. . 344.
5. ɂɝɧɚɬɨɜɢɱ, ȼ.ɇ.
/ . .
.
/ . .
/
[
. .
. 3-
.
«
», 2010. . 80.
.].
.:
.:
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-
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-
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«
-
».
***
The original method of calculation entropy mixes of ideal gases through entropy
components which after allocation from a mix can is described is in various
conditions. From the received formulas follows: if components are at temperature and
pressure of a mix their total entropy differs from entropy mixes the less, than
properties of mixing up gases less differ; if components are at temperature of a mix
and borrow the volume equal to volume of a mix their total entropy, generally, differs
from entropy mixes. These results allow to resolve a number of contradictions which
traditionally refer to as «Gibbs paradox».
180