Colloid chemistry chapter 4 thermodynamics of surface edit

COLLOID CHEMISTRY

Chapter 4 – Thermodynamics of Surface
Dr. Ngo Thanh An


1. Surface excess

• The presence of an interface influences
generally all thermodynamic parameters of a
system
• To consider the thermodynamic of a system
with an interface, we divide that system into
three parts: The two bulk phases with volumes
Vα and Vβ and the interface σ


1. Surface excess

Gibbs dividing plane
• In Gibbs convention the two phases are thought
to be separated by an infinitesimal thin boundary
layer, the Gibbs dividing plane (this is of course
an idealization)
• Gibbs dividing plane also called an ideal
interface
• In Gibbs model the interface is ideally thin (V σ =
0) and the total volume is
V = Vα + Vβ

1. Surface excess

Gibbs dividing plane
∞

𝑧𝑒

𝑧𝑒

0

Γ =∫ [ 𝜌 ( 𝑧 ) − 𝜌 𝑏 ] 𝑑𝑧 −∫ [ 𝜌 𝑎 − 𝜌 ( 𝑧 ) ] 𝑑𝑧 =0

•

In Gibbs convention the two phases α and β are separated by an ideal interface σ which is infinitely thin:
Guggenheim explicitly treated an extended interphase with a volume


1. Surface excess

Interfacial excess

N i  N i  ciV   ci V 

i

N
i 
A

V  V  V 

i


i




i


i



N Ni  c V  c  c V




1. Surface excess

1


 c V  c


1


1


1


2


2


2


V

N  N1  c V  c  c V

2

N N 2

Multiply both sides of equation (1) by

c



(1)



(2)

 c2  c2 
 

 
 c1  c1 












c2  c2
c2  c2
c2  c2 



  N1  
  c1 V  

N1  
 
 
 
 c1  c1 
 c1  c1 
 c1  c1 



c2  c2 




 c1  c1 V  
 
 c1  c1 





(3)


1. Surface excess

Equation (2) – equation (3)

 c2  c2 





N  N  

N

c
V


c

c

V
2
2
2
2
 
c

c
1 

 1
 c2  c2  
 c2  c2 
  c1 V  

 N1  
 
 
 c1  c1 
 c1  c1 

2


1

 c2  c2 

 c  c V  
 
 c1  c1 

1


1













c

c
c

c




2
2
2
2
  N 2  c2 V   N1  c1 V  

N 2  N1  
 
 
 c1  c1 
 c1  c1  




 







c

c
c

c

 i


i
i
i
N i  N1 

N

c
V


(
N

c
V
)
i
i
1
1
c1  c1
c1  c1


1. Surface excess
Finally, dividing through by the interfacial area A, gives:


 c2  c2   
N 2 N1  c2  c2  1 


   N 2  c2 V   N1  c1 V  


 
 
  
A

A  c1  c1  A 
 c1  c1   




 







  
N 2 N1  c2  c2  1 
c

c


2
2
   N 2  c2 V   N1  c1 V  


 
 
  
A

A  c1  c1  A 
 c1  c1   




 

2(1)



c

c
2  1 2 1
c1  c1

i(1)



c

c
i  1 i i
c1  c1



1. Surface excess

Interfacial excess
• The right side of the equation does not depend on the position of the Gibbs
dividing plane and thus, also the left side is invariant. We divide this quantity by the
surface area and obtain the invariant quantity

i(1)



c

c
i  1 i i
c1  c1

 It is called relative adsorption of component i with respect to
component 1.
 This is an important quantity because it can be determined
experimentally and it can be measured by determinig the surface
tension of liquid versus the concentration of the solute


2. Fundamentals of thermodynamic relations

a. Internal energy and Helmholtz energy


V V  V



dV dV   dV 



dV dV  dV

U U   U   U 

i


i

N i N  N  N




S S  S  S




i



2. Fundamentals of thermodynamic relations

a. Internal energy and Helmholtz energy

dU TdS  PdV   i dN i  dW
dU dU   dU   dU 
dU TdS   P dV    i dN i
 TdS   P  dV    i dN i
 TdS    dN  dA



i


i


2. Fundamentals of thermodynamic relations

a. Internal energy and Helmholtz energy







dU T dS  dS  dS



 P







dV  P dV



  i dN i   i dN i   i dN i  dA










dU TdS  P dV  P  P dV



  i dN i   i dN i   i dN i  dA


2. Fundamentals of thermodynamic relations

a. Internal energy and Helmholtz energy





dF  SdT  P dV  P   P dV 
   dN    dN    dN  dA

i


i


i


i


i



i


2. Fundamentals of thermodynamic relations

b. Equilibrium conditions

i


i

N i N  N  N


i

dN i 0
dN i  dN i  dN i









dF  P   P dV   dA   i  i dN i





     dN

i


i


i


2. Fundamentals of thermodynamic relations

b. Equilibrium conditions


2. Fundamentals of thermodynamic relations

b. Equilibrium conditions





dF  P  P dV  dA   i dN i







2. Fundamentals of thermodynamic relations

c. Gibbs energy
dG  SdT  V  dP  V  dP    i dN i  dA

dG  SdT  VdP   i dN i  dA

G

A T , P , N i


2. Fundamentals of thermodynamic relations

d. Free surface energy
dU  TdS    i dN i  dA
U  TS    i N i  A
F A   i N



i

dF   S  dT   i dN i  dA

S 

A

T , N i



T

A, N i


2. Fundamentals of thermodynamic relations

e. Eurler’s theorem
A two-variable homogenous function: f = f(x,y)
Exact differential for f:
Integrate both sides of equation:
Thefore, we have:


2. Fundamentals of thermodynamic relations

e. Eurler’s theorem
Application of Euler’s theorem
On the interface, we have:
Integrate both sides of the above equation: