Lecture 9: Chemical Potential and Chemical Reactions
Review
Non-bonded Interactions

Ionic

Hydrogen Bond

Hydrophobic Interactions
Today

Variation of chemical potential

Pressure

Composition

Relation between chemical potential and
chemical reactivity.

Notion of activity and standard states for
solutions
o Mole fraction
o Concentration

Activity coefficients
o Ionic species
Variation of chemical potential with Pressure
We have shown that at constant temperature G varies with
pressure as:
dG  VdP
P 
 G2  G1  nRT ln  2 
 P1 
P 
G G
 2  1  RT ln  2 
n
n
 P1 
Setting p  1atm as a s tan dard state
 ( P)   0 (1atm)  RT ln P 
Note in the above equation, P is dimensionless since it is
the ratio of P to 1atm. Now consider an ideal gas that is
mixture of two components made in a following way. We
start with components 1 and 2 at the same constant
pressure, P; confine them in two separate chambers. Then
we open the connection between the two chambers so that
the gases can mix. Since the gases are ideal the final
pressure will be P. This pressure is simply sum of the
partial pressures of components 1 and 2, i.e. P1+P2. Thus
for this isolated system we can calculate Gibbs free energy
by considering initial and final states:
Initial state
1  10  RT ln P  2   20  RT ln P
 initial total free energy G 0  n1 1  n2  2
and
After mixing
1  10  RT ln P1 and  21   20  RT ln P2
P
P 

G mix  n1 1  n2  2  G  RT  n1 ln 1  n2 ln 2   RT n1 ln X 1  n2 ln X 2 
P
P

G Mix  (n1  n2 ) RT  X 1 ln X 1  X 2 ln X 2 
Chemical Reactions
Consider following reaction and associated change in
free energy. We can show:
aA  bB  cC  dD
dG   C dnC   D dnD   A dn A   B dnB
However, in chemical reaction confined to a closed
system, the moles of A,B,C,D do not vary
independently. That is C and D production occurs at
the expense of A and B. We can quantitatively say:
dn
dn
dn
dn
 a   b  c  d  d
a
b
c
d
In other words stoichiometry decides the rate at
which a component disappears. But the normalized rate
of the reaction, dα is constant. Substituting in the above
expression:
dG   C dnC   D dnD   A dn A   B dnB
 (a A  b B )  (c C  d D )d
As discussed before the reaction will occur spontaneously,
(at constant T and P) if dG is negative, that is sum of the
chemical potentials of stoichiometrically weighed reactants
and products are positive. Now at equilibrium dG is zero.
Considering case of gaseous reaction of N2 and H2 to
produce NH3.
Fritz-Haber Process: Nitrogen fixation
N 2  3H 2  2 NH 3
G  2  NH 3   N 2  3 H 2
u sin g
 Pi 

 1atm 
 i   i 0  RT ln 
2
  PNH

 G  G  RT ln 1 33
  PN 2 PH 2
at equilibriu m
0
  P2
NH
0
G   RT ln  1 33
  PN 2 PH 2





eq

 where G 0  2 0 NH 3   0 N 2  3 0 H 2





   RT ln( K )


Where K is the equilibrium constant for the reaction. Note
it is dimensionless since each of the partial pressures used
in above equation are dimensionless. This is one of the most
common usage of thermodynamics in biology. That is if we
know the equilibrium constant we can evaluate free energy
accompanying the reaction and vice versa.
This equation can be used in the non-equilibrium situation
also, since:
2
  PNH

G  G  RT ln  1 33 
  PN 2 PH 2 
2
  PNH
  RT ln( K )  RT ln  1 33
  PN 2 PH 2
 RT ln( Q / K )
0
2

 PNH
3
 denoting Q  
1

 PN PH3

 2 2




How do we apply above equation for real gases and
solutions?
Concept of activity.
Recall, our basic equation for chemical potential:
 Pi 

1
atm


 i   i 0  RT ln 
In this equation, we used P1 to be 1atm for standard
state of a gas. Also, we used the ideal gas approximation
to get this relation. For real gases, which do not obey
ideal gas equation, we generalize the concept of
pressure with “activity”.
a  P
where gamma is the activity coefficient. The notion of
activity coefficient to correct for pressure inside the real
gas is also useful in treatment of solutions. But we still
need to define a reference state. For low pressures and
high temperature most gases behave like ideal gases.
Thus γ approaches 1 in this limit. In general, at 1atm
for most gases γ≈1 so it is common use that as the
standard state
This arbitrariness in selecting standard state does not
affect the chemical potential of the component; only μ0
gets affected. The choices of reference states are many
and are dictated by the type of problem. For a chemical
potential of solvent used in dilute solution, mole fraction
is used. For solutes, it is common to used molar/molar
concentrations. Colligative properties are used for
determination of activity coefficients.
Concept of Activity
Consider a case of solvent in solution. We choose its standard
state as pure solvent i.e. when it’s mole fraction is unity. In this
limit, the activity and activity coefficient is unity. In other
words:
Very dilute Solutions : aSolvent = 1
Dilute Solution
: aSolvent = Xsolvent
Nonideal Solutions : aSolvent = γsolventXsolvent
For solutes we use a different standard state. We prefer to
work with molar/molal solutions so we define:
aSolute=γSolutec
Where c is the molar/molal concentration. So we define the
standard state of solute corresponding to 1 molar/molal
solutions. However, for most solutes 1 molar solution behaves
nonideally, hence a virtual state is defined by extrapolating the
solute activity measured in dilute solution to 1 molar
concentration. See pages 130-137 for a through discussion of
activity and standard states.
Thus although the chemical potential or free energy of system is
fixed quantity the activity or activity related quantities can vary
with choice of the reference state.