Download 12276_71996_Unit-5(Part – 1)

THERMODYNAMICS RELATIONS
Introduction
Some thermodynamic properties can be measured directly, but many others cannot.
Therefore, it is necessary to develop some relations between these two groups so that the
properties that cannot be measured directly can be evaluated. The derivations are based on the
fact that properties are point functions, and the state of a simple, compressible system is
completely specified by any two independent, intensive properties.
Some Mathematical Preliminaries
Thermodynamic properties are continuous point functions and have exact differentials.
A
property of a single component system may be written as general mathematical function z = z(x,y).
For instance, this function may be the pressure P = P(T,v). The total differential of z is written as
Fundamental Property Relations:
Apart from enthalpy, internal energy and entropy, there are other thermodynamic
properties which are necessary to apply to phase equilibria and reaction equilibria. So far we have
treated only closed systems which consist of single phase without any reaction. In order treat the
systems we need to know other energy terms involved with reaction and phase equilibria. The use
these properties will get clearer when we do the chapters on these areas. Let us derive the
fundamental property relations from basics.
Enthalpy:
From I law,
dU  dQ  dW
dU  TdS  pdV
H  U  pV
dH  dU  d(pV)
 dU  pdV  Vdp
 TdS  pdV  pdV  Vdp
dH  TdS  Vdp
Helmholtz free energy: It is defined as
F = U – TS
Differentiating, dF = dU-d(TS) = TdS – pdV – TdS – SdT, dF= – pdV – SdT
Gibb’s Free energy:
It is defined as G = H – TS
Differentiating, dG= dH –TdS – SdT = TdS +Vdp – TdS – SdT,
dG = Vdp - SdT
These four equations are called fundamental property relations as many equations are
derived from these equations.
Maxwell’s relations
As some of the thermodynamic properties are not directly measurable, we must write the
equations in terms of measurable properties. The measurable and determinable (by experiment)
properties are temperature, pressure, volume, heat capacities and in some cases even enthalpies.
These relations are derived from fundamental property relations. The derivations are based upon a
mathematical relation applied to exact differential equations.
 f 
 f 
 dx    dy
 x  y
 y  x
If z = f(x,y) then df  
This equation is also written as
df = Mdx + Ndy
For this differential equation to be exact,
 M

 y
  N 
  

 x  x  y
Applying these conditions to all the property relations, we get,
df  Mdx  Ndy
 M

 y

 N 
  

 x  x  y
dU  TdS  pdV
 T 
 p 

   
 V  S
 S V
dH  TdS  Vdp
 T   V 

  

 p  S  S  p
dA   pdV  SdT
 p 
 S 
  

 T V  V T
dG  Vdp  SdT
 S 
 V 

   
 T  p
 p T
Generating function:
Consider
 G
d
 RT
1
G
1
 RTdG  Gd ( RT )
 H  TS 

dG 
dT 
( Vdp  SdT )  
dT

2
2 2
2 
RT
RT
RT
RT

 RT 

Vdp SdT HdT SdT
V
H




dp 
dT
2
RT
RT
RT
RT
RT
RT 2
H
 G  V
d
dp 
dT

RT 2
 RT  RT
From this equation, at constant temperature and at constant pressure we can write,
 ( G / RT ) 
V



p

T RT
H
 ( G / RT ) 
;


T
RT2

p
If we know G/RT in terms of p and T, it can be used to evaluate other thermodynamic
properties and hence it is called generating function.
Entropy changes:
 Q 
 Q 
 ; CV  

 T  p
 T V
We know that CP  
CP
 S 
 S 
 
 
 T  p  T  p T
Since dQ  TdS , CP  T 
CV
 S 
 S 
 
 
 T V
 T V T
And CV  T 
2
 T2 
T 
dT
dT


Using these, ΔS p   CP
 CP ln  and ΔS V   CV
 CV ln 2 
T
T
 T1 
 T1 
T1
T1
T2
T
 S 
 S 
 dT    dp
 T  p
 p T
If S is considered as S = f (T,p) then dS  
In this equation, the first partial derivative is replaced by CP/T which is derived and second
by a Maxwell’s relation, we get, dS 
T, we get, dS 
CP
 V 
dT  
 dp Similarly by taking S as function of V and
T
 T  p .
CV
 p 
dT    dV
T
 T V
If there is any relation between three variables, from mathematics we have, a relation
 V   p   T 
 
between p, V and T as 
 
  1
 p T  T V  V  p
Using these relations, we can get, dS 
( V / T )p
CV
dT 
dV
T
( V / p )T
 ( V / T )p

dU  CV dT  T
 p dV
 ( V / p )T

  V 

 dp  pdV 

  T  p
Internal energy may also be written as dU  C P dT  T 

 V  
  dp
 T  p 
Using dH  TdS  Vdp , we get dH  C P dT  V  T 

Internal energy changes:
The internal energy changes can be expressed in terms measurable quantities such as heat
capacities, pressure, volume and temperature. We can use Maxwell’s relations to get these
expressions. The following are fundamental property relations.
dU  TdS  pdV
dH  TdS  Vdp
dA   pdV  SdT
dG  Vdp  SdT
dS 
CV
 p 
dT  
 dV
T
 T  V
( V /  T ) p
C
dS  V dT 
dV
T
( V / p )T
dS 
We have derived the following relations.
 V

 p
CP
 V 
dT  
 dp
T
 T  p
  p    T 
 
 
  1
T  T V  V  p
Let us derive the expression for internal energy in terms CP and CV.
dU  TdS  pdV
Replace dS in terms of CV,
( V / T )p
C

dU  T  V dT 
dV   pdV
( V / p )T
T

Internal energy change in terms of Cp:
 ( V / T )p

dU  CV dT  T
 p dV
 ( V / p )T

dU  TdS  pdV
Replace dS in terms of Cp,
C

 V 
dU  T  P dT  
 dp   pdV
 T  p 
 T
  V 

dU  CP dT  T 
 dp  pdV 
  T  p

Enthalpy changes:
Starting from
dH  TdS  Vdp
in the same way we can get,

 V  
dH  CP dT  V  T 
 dp
 T  p 

We can use these expressions and find the changes in enthalpy. These equations are
applicable for any gas. P-V-T relations used should be for the gas for which you are determining
the changes.
Use these relations and derive the expressions for dH and dU for ideal gas.
Effect of p, V and T on U, H and S:
 S 
 V 

  


p
 T  p

T
This is one of the Maxwell’s relations which gives the effect of pressure on entropy at
constant temperature.
C
 S 

  P
T
 T  p
This equation derived earlier gives the effect of temperature on entropy at constant
pressure.
( V / T )p 

dU  CV dT  Vdp  T
dV
( V / p )T 

dU
At constant volume,
 CV
dT
 U 

  CV
 T  V
Or
This gives the effect of temperature on internal energy at constant volume. At constant
 ( V / T )p

 U 
 pthe

   T This gives

effect of volume of internal
 V T
 ( V / p )T

temperature, since dT is zero,
energy at constant temperature.
Similarly from
we can get,

 V  
dH  CP dT  V  T 
 dp
 T  p 

At constant pressure,
 H 

  CP
 T  p
 H 
 V 

  V  T 

 T  p
  p T
At constant temperature,
These expressions give the effect of temperature and pressure on enthalpy.
Relationship between the heat capacities:
The heat capacities are related by the following expression.
 V   p 
CP  CV  T 
 

 T  p  T V
which can also be written as
2
 V   p 
CP  CV  T 
 

 T  p  V T
using the cyclic relation of partial derivatives of p, V and T.
 
1  V 

 ;
V  T  p
k 
1  V 

 are
V  p T
called
Volume
expansivity
and
Compressibility. The difference between heat capacities is written in terms of these as
CP  CV 
 2VT

Effect of pressure and volume on heat capacities:
These are given by the following equations.
 C P

 p
  2V

  T 
 T 2
T

  2V
 C P 

  T 
2
  V T
 T



p
  p 
 
  V 
T
p
 2 p 
 CV 


  T 
2 
 V T
 T V
 CV

 p
  2 p   V

 
  T 
 T 2   p
T

p 


T
isothermal