Chapter 9
The chemical potential
and open systems
1
Closed system: System does not exchange matter
with surroundings.
Open system:
Quantity of matter not fixed.
Chemical Potential.
We give a brief introduction to this concept as we will
need to refer to it later in the course. As the name suggests, it is very
important for studying chemical reactions and your will encounter
this concept in a first course in physical chemistry. Just as the
temperature governs the flow of energy between two systems, the
chemical potential governs the flow of particles. When the chemical
potential of the two systems are equal, they are in diffusive
equilibrium.
We are now going to consider the introduction of matter into
a system. If we introduce, say, dn kilomoles of matter into a system,
there will be a change in energy of the system and dU will be
2
proportional to dn
For a pure substance, with a constant number of particles, we had, for
a quasi-static process: dU  đQ  PdV .......(1)
and for a reversible process, dU=TdS –PdV ………(2)
We now consider a more general case where a system may consist
of several different constituents. Furthermore, the system is no
longer necessarily closed so that the number of moles of the various
constituents may vary.
ni = number of moles of constituent i
We now generalize equation (2) to allow for the possibility of adding
of removing particles from our system. Before we had U(S,V) and now
 U 
U ( S ,V , ni ) so dU   U  dS   U  dV 


dn .....(3)



S

 V ,n



V

 S ,n

i
 n 
 i  S , V ,n j
i
In the first two derivatives, all the n are constant and in the third
derivative all the n are constant except ni
3
Comparing equations (2) and (3)
 U 

T  
 S V ,n
 U 

P  
 V  S ,n
and now we define the chemical potentials by
 U 

 i  
 n i  S ,V ,n j
Equation (3) can now be written as dU  TdS  PdV   i dni .....(4)
i
i
1
P
dS

dU

dV

dni
Solving for dS:

T
T
i T
1  S 

 
and so
T  U V ,n
P  S 

 
T  V U ,n
 S 

 
T
 ni U ,V ,n j
i
The Helmholtz function is F=U-TS
4
dF  dU  TdS  SdT  (TdS  PdV   i dni )  TdS  SdT
dF   PdV  SdT   i dni
i
i



F
(reciprocity P  
 V 
relations)

T ,n
 F 

S  
 T V ,n
and so
 F 
i   
 ni V ,T ,n j
EXAMPLE: Mixture of two ideal monatomic gases
We will take as given the Helmholtz Thermodynamic Potential.
F (V , T , n1, n2 ) From the viewpoint of thermodynamics this comes from
experiment. In statistical mechanics this is derived from a model in
which two types of molecules are present.

 2 m1k  
3
3
  
n1 ln V  n1 ln T  n1 ln( n1 N A )  n1  n1 ln 
2
2
2
 h
 

F   RT

 2 m2 k  
3
3
n2 ln V  n2 ln T  n2 ln( n2 N A )  n2  n2 ln 
 
2
2
2

 h
 5 
m=molecular mass
 F 

P  
 V T ,n
If
n2  0
If
n1  0
so
N A =Avogadro’s number h=Planck’s constant
 n1 n2 
P  RT   
PV  (n1  n2 )RT
V V 
 n1 
 
P

RT
we have
1
V 
we have
P  P1  P2
PV  nRT
n 
P2  RT  2 
V 
Dalton’s Law The pressure of a mixture of ideal gases is equal to the
sum of the partial pressures. The partial pressure of a gas is the
pressure that it would exert if it alone occupied the volume V at
temperature T.
 F 

S  
 T V ,n
F 3
S    R(n1  n2 )
T 2
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 F 

1  
 n1 V ,T ,n2

3
3  2 m1k 

1   RT ln V  ln T  ln( n1 N A )  1  1  ln
2
2
2  h



3
3  2 m1k 

1   RT ln V  ln T  ln( n1 N A )  ln
2
2
2  h


 F 3

U  F  TS  F  T   R(n1  n2 )
 T 2

U 
3
RT (n1  n2 )
2
From the fact sheet we obtain G=F+PV so
U  U1  U2
G  F  (n1  n2 )RT
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
 2 m1k  
3
3
  
n1 ln V  n1 ln T  n1 ln( n1 N A )  n1 ln 
2
2
2
 h
 

G   RT

 2 m2 k  
3
3
n2 ln V  n2 ln T  n2 ln( n2 N A )  n2 ln 
 
2
2
2

 h
 
G  G1  G2
Comparison with the expression for 1 and a similar expression for
gives G  1n1  2 n2
In general G    i ni
2
i
For a system consisting of just one constituent (one phase) we have
G  n
or
G
 g
n
We see that for such a simple system, the chemical potential
is just the Gibb’s function.
{We mentioned earlier that the Gibbs Potential was particularly
important in physical chemistry.}
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