Physical Chemistry
Lecture 15
Mathematics of State Functions,
with an Emphasis on Entropy
State functions
State functions have exact differentials
dU
dH
dS
 ∂U 
= 
 dT
 ∂T V
 ∂H 
= 
 dT
 ∂T V
 ∂S 
= 
 dT
 ∂T V
 ∂U 
+ 
 dV
 ∂V T
 ∂H 
+ 
 dV
 ∂V T
 ∂S 
+ 
 dV
 ∂V T
Must evaluate derivatives to allow calculation
of changes
Partial derivatives of state
functions
Internal energy
 ∂U 


 ∂T V
= CV
 ∂H 


 ∂T  P
= CP
Enthalpy
Temperature derivatives of
entropy
General form
dS
Constant volume
CV
dU
=
dS =
dT
T
T
Constant pressure
dH
CP
=
dS =
dT
T
T
=
dqrev
T
CV
 ∂S 

 =
 ∂T V T
CP
 ∂S 

 =
 ∂T  P T
Volume derivative of entropy
Use definitions of energy changes
dU
dU
= TdS
= CV dT
− PdV

  ∂P 
+  T 
 − P dV

  ∂T V
Setting these equal and rearranging
dS
=
CV
dT
T
  ∂P 
P P
+  
 − + dV
  ∂T V T T 
Identification gives
=
 ∂S 


 ∂V T
CV
dT
T
 ∂P 
+
 dV
 ∂T V
 ∂P 
= 

 ∂T V
Pressure derivative of entropy
Use definitions of enthalpy change
dH
dH
= TdS
= C P dT
+ VdP

 ∂V  
+ V − T 
 dP
 ∂T  P 

Setting these equal and rearranging
dS
=
CP
dT
T
 V  ∂V  V 
+  − 
 − dP =
 T  ∂T  P T 
Identification gives
 ∂S 
 
 ∂P T
CP
dT
T
 ∂V 
−
 dP
 ∂T V
 ∂V 
= −

 ∂T  P
State functions
Any state function can be a function of
any two other state variables
S = S (T , V )
S
S
S
= S (T , P)
= S ( P, V )
= S (U , V )
Each state function is a natural function
of two specific variables
Calculation of entropy changes
Just like calculations of other statefunction changes
Integrate appropriate derivatives over
the range to find the change in entropy
 ∂P 
∫V  ∂T V dV
1
V2
∆S
=
∆S
 ∂V 
= − ∫
 dP
∂T  P
P1 
P2
Example calculation: entropy
of mixing of ideals gases
Process: two separated gases are allowed to
intermingle
Entropy is calculated as a sum of entropy of
each component
∆S = ∆S A + ∆S B
Entropy of mixing of ideal
gases
Temperature is constant
Consider volume change of a gas
V final
∆Si
 ∂Si 
= ∫ 
 dV
∂V 
Vinitial 
T
V final
R
dV
= ∫
V
Vinitial
T
V final
 ∂P 
= ∫ 
 dV
∂T V
Vinitial 
 V final
= R ln
 Vinitial



Entropy of mixing of ideal
gases
The entropy change for the system is
the sum of the entropy changes for the
two gases
∆S
= ∆S A
+ ∆S B
 VA + VB 
 VA + VB 
 + R ln

= R ln
 VB 
 VA 
This is a positive quantity: mixing is a
spontaneous process
Entropy change upon a phase
change
Phase changes occur at a specific
temperature, Tφ
Phase change is accompanied by a
change in the enthalpy, ∆Hφ, called the
latent enthalpy of transition
Entropy change
∆Sφ
=
∆H φ
Tφ
Examples of entropy of
transition
Melting of water


Tf = 273.15 K; ∆Hf =6.01 kJ/mole
∆Sf = 22.0 J/K-mole
Vaporization of water


Tf = 373.15 K; ∆Hf =40.66 kJ/mole
∆Sf = 109 J/K-mole
Summary
Evaluate derivatives to determine statefunction changes
Some derivatives are measured quantities

CP, CV, α, κT etc.
Others determined from equation of state
Entropy derivatives are determined from
definitions of reversible heat
Calculate changes by integration of
derivatives
Entropy changes at phase transitions
evaluated by the definition of entropy