University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
Residual Properties
Residual property defines as:
M R  M act  M ig
MR is residual property
Mact is actual property
Mig is property of ideal gas
Where M is the molar value of any extensive thermodynamic property, e.g., V, U, H, S, or
G. The residual volume, for example, is:
V R  V act  V ig  V 
Since V 
VR 
RT
P
ZRT
P
RT
( Z  1)
P
H and S are found from ideal gas and residual property by simple addition
M act  M ig  M R
dH ig  Cp ig dT
T
H
ig
H
ig

  Cp ig dT
T
ig
Cp
dP
dT  R
T
P
T
ig
Cp dT
P
 S ig  
 R ln
T
P
T
dS ig 
S ig
H act  H ig  Cpmh (T  T )  H R .....................1
T
Cpmh 
 Cp
ig
dT
T
T  T
S act  S ig  Cpms ln
T
P
 R ln  S R ...............2
T
P
T
Cpms
Cp ig dT
 T
T

T
ln
T
1
University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
Residual property from equation of sate
dH  TdS  VdP
 H 
 S 

  T
 V
 P T
 P T
 S 
 V 
but 
  
 Maxwell eq uations
 P T
 T  P
 H 
 V 

  T 
  V ............... *

P

T

T

P
 H 

1- For ideal gas 
 P T
PV  RT
PdV  VdP  RdT
R
 V 
P
  T  V  0
P
 P T
2- For real Gas ( actual )
PV  ZRT
PdV  VdP  R( ZdT  TdZ )
 V 
 Z 
P
  R[ Z  T 
 ]
 T  P
 T  P
RZ RT  Z 
 V 


 


P
P  T  P
 T  P
Apply these values on equation *
 RZ RT  Z  
 H act 

  T 


 V
P  T  P 
 P T
 P
 H act 
RT 2  Z 

  

 ......................... * *

P
P

T

P

T
 H R

 P

 H act 
 H ig
  
  

P
T 
T  P


T
2
University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
 H R
 
 P
 H R

 P

 H act 
 H ig
  
 because 

P
T 
T
 P

 RT 2
 
P
T

  0
T
 Z 


 T  P
 RT 2  Z 
dH 

 dP
P  T  P
R
 RTC2Tr2
dH 
PC Pr
R
 Z

 TC dTr

 PC dPr
 Pr
dH R
d Pr
2  Z 
0 RTC  Tr 0  Tr  Pr
Pr
HR
Pr
HR
 Z  d Pr
 Tr 2  
.................3

RTC
Tr  Pr Pr
0
Pr
Similarly another relation can be got for SR
 S 
 V 

  

 P T
 T  P
 S R

 P

 V R 
  


T
T

P
RT
VR 
( Z  1)
P
 S R 
  RT


  
( Z  1)

dT  P
P
 P T
R 
Z  1P

P dT
 S R 
R  Z
RT Z RZ  R


    T
 Z  1  



P
P
dT
P
dT
P
P


P

T
dS R  
RT  Z 
RZ
R
dP  dP

dP 
P  dT 
P
P
3
University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
SR
dP
 Z  dP
 T  
  ( Z  1)

R
dT  P 0
P
0
P
P
SR
d Pr
 Z  d Pr
 Tr  
  ( Z  1)
...............4

R
dT
Pr
Pr
 Pr
0
0
Pr
Pr

Equations 3 & 4 are depended on Pr and Tr as Z  Z  Z 
 Z  
 Z 
 Z  
   

  

 dTr  Pr  dTr  Pr
 dTr  Pr
P
 Z  
HR
 Z    d Pr
2
   
 Tr  
 
RTC
dTr
dTr

 Pr  Pr


0 
Pr

Pr

HR
d Pr
 Z   d Pr
2  Z 

 Tr  
 Tr2 

RTC
dTr
Pr
dTr

 Pr Pr


0
Pr
SR
 Tr
R
......5
Pr
 Z  
d Pr
 Z    d Pr







Z


Z

1




0  dTr 
0
Pr
 dTr  Pr  Pr
Pr


P
 
Pr
Pr
 Z 
 d Pr
  Z  
 d Pr
SR

  Z  1
   Tr 
   Tr
  Z   1
R
 dTr  Pr1
 dTr  Pr
 Pr
 Pr
0
0 
(H R )
( H R )
If the first term in equation 5 and 6 represent RTc and RTc
( S R )  ( S R )
And the second terms in equation
R
R


 HR 
HR  HR 
   
 ........... 7

RTC  RTC 
RT
 C


 SR 
SR  SR 
   
 ............. 8

R  R 
R


Equation 7 and 8 are solved by use the figures 6-6 to 6-13
4
......6