University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
Thermodynamic properties of Fluids
PROPERTY RELATIONS FOR HOMOGENEOUS PHASES
At constant temperature and pressure G represent the part of work may be converted to
useful work; G  H  TS (where G is Gibbs free energy) While A  U  TS represent
from internal energy cab be converted to useful work at constant temperature and volume
(where A is Holmholtz free energy) .So we can express for energy H, U, P, V, G and A.Each
material has these kinds of energy and it was distributed for any body and sum of all these
energies is Enthalpy H
H  U  PV
G  H  TS
A  U  TS
U  Q W
dU  dQ  dW
 dU  TdS  PdV ...............(1)
dH  dU  d ( PV )  dU  PdV  VdP
 TdS  PdV  PdV  VdP
 dH  TdS  VdP.............(2)
dA  dU  d (TS )  TdS  PdV  TdS  SdT
 dA   PdV  SdT .........(3)
dG  dH  d (TS )  TdS  VdP  TdS  SdT
 dG  VdP  SdT ..............(4)
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University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
If F=F(x, y) then the total differential of F id define as
 F 
 F 
 dy
dF  
 dx  

x

y

y

x
or  Mdx  Ndy
 F 

 x  y
Where M  
 F 

and N  
 y  x
By further differentiation obtain
 M

 y

2F
 
 x yx
2F
 N 
 
and 
 x  y xy
U =U(S, V)
 U 
 U 
dU  
 dS  
 dV
 S V
 V  S
 U 
 U 

  T , and 
  P

S

V

V

S
H =H(S, P)
 H 
 H 
dH  
 dS  
 dP
 S  P
 P  S
 H 
 H 

  T , and 
 V
 S  P
 P  S
A =A (V, T)
 A 
 A 
dA  
 dV  
 dT
 V T
 T V
 A 
 A 

   P, and 
  S
 V T
 T V
G =G (P, T)
 G 
 G 
dG  
 dP  
 dT
 P T
 T  P
 G 
 G 

  V , and 
  S
 S T
 T  P
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University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
By apply differentiation on equations 1 through 4, obtain
 T 
 P 

  
 ......(5)
 V  S
 S V
 T 
 V 

 
 .......(6)
 P  S  S  P
 P 
 S 

 
 .......(7)
 T V  V T
 V 
 S 

  
 ........(8)
 T  P
 P T
These are known as Maxwell`s equations
Enthalpy and Entropy as Functions of T and P
To know how H and S vary with Temperature and Pressure or in other word we need to
know
H  H  S 
S 
 ,
 ,
 , and

T  P P T T  P
P T
H 
  C P ......(9)
T  P
H 
 S 
  T
 .............(10)
From equation 2,
T  P
 T  P
C
 S 
  P .....(11)
Combination equation 9 and 10, 
T
 T  P
From heat capacity definition,
 H 
 S 
Combination equation 2 and 8 
  T   V
 P T
 P T
 H 
 V 

  V T
 .....(12)
 P  T
 T  P
If H=H (T, P) and, S=S (T, P)
 dH 
 dH 
dH  
 dT  
 dP
 dT  P
 dP  T

 V  
dH  C P dT  V  T 
  dP....(13)
 T  P 

 dS 
 dS 
dS  
 dT    dP
 dT  P
 dP  T
and
dS  C P
and
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dT  V 

 dP....(14)
T  T  P
University of Babylon /College Of Engineering
Electrochemical Engineering Dept.
Second Stage /Thermodynamics
These are general equation relating the enthalpy and entropy of homogenous fluids of
constant composition to temperature and pressure.
Internal Energy as a Function of P
The pressure dependence of internal energy is obtained by differentiation U =H-PV
 U 
 H 
 V 

 
  p
 V
 P T  P T
 P T
The Ideal-Gas State
PVig = RT
 V ig 
R

 
and use this equation in equation (13) and (14) result :
 T  P P
dH ig  CPig dT
dT R
dS ig  C Pig
 dP
T
P
Alternative Forms for Liquids
By use volume expansivity (β) definition with equations 8 and 12 result
 dS ig 

   V .......(15)
dP

T
 dH ig 

  (1  T )V .......(16)
dP

T
 U 

  (P  T )V ............17
 P T
Where (κ) isothermal compressibility
These equations, incorporating β and κ, although general, are usually applied only to liquids.
However, for liquids not near the critical point, the volume itself is small, as are β and κ.
Thus at most conditions pressure has little effect on the properties of liquids.The important
special case of an incompressible fluid.
Equation (16) and (17) which required value of β and κ are usually applied only to liquids
When (δV /δT) P is replaced in equations 13 and 14
dH  CP dT  V (1  T )dP.......18
dT
dS  CP
 VdP.................19
T
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