Equilibrium
Constants
Lecture 8
The Equilibrium Constant
• Consider the reaction
aA + bB = cC + dD
• The Free Energy change of reaction is:
∆G = cµc + dµd – aµa – bµb
• At equilibrium:
∆ G = ån i µi = 0
• Expanding the right side
ån µ + RT ån ln a = 0
i
o
i
i
i
ån i µi + RT ln Õi a = 0
• or
• We define the right term as the equilibrium constant:
ni
o
i
K = ln Õ ani i
i
Free Energy and the
Equilibrium Constant
o
n
n
µ
+
RT
ln
a
åi i
Õ =0
i
i
i
• Since:
o
n
µ
å i = ∆ Gr
o
i
• then
• and
∆ Gro - RT lnK = 0
o
r
∆ G = -RT lnK
• Note of caution: our thermodynamic parameters are
additive, but because of the exponential relation
between the equilibrium constant and free energy,
equilibrium constants are multiplicative.
Manipulating Equilibrium
Constants
• Suppose we want to know the equilibrium constant for a
reaction that can be written as the sum of two reactions,
o e.g., we can sum
Feaq3+ + e+
Fe(OH )2+
aq + H
o to yield
+
Fe(OH )2+
aq + e + H
2+
Feaq
Feaq3+ + H 2O
2+
Feaq
+ H 2O
o The equilibrium constant of the net reaction would be the product of the
equilibrium constants of the individual reactions.
• For this reason and because equilibrium constants can
be very large or very small numbers, it is often
convenient to work with logs of equilibrium constants:
pK = - log K
o (we can then sum the pK’s).
Apparent Equilibrium Constants
and Distribution Coefficient
• In practice, other kinds of equilibrium constants are
used based on concentrations rather than
activities.
• Distribution Coefficient
K = Õ Xni
D
i
• Apparent Equilibrium
K appConstant
= Õ min
i
K
app
=
K eq
Kl
=
K eq
n
l
Õ i
i
Other Forms
• A ‘solubility constant’ is an equilibrium constant. For
aq
example:
aNa
aClaq
K=
+
-
aNaCls
o Since the activity of NaCl in halite = 1, then
aq
aq
K SP = aNa
a
+
Cl -
• Henry’s Law constants for describing solubility of
gases in solution (e.g., CO2 in water).
o Since Pi = hiXi
hi =
Pi
Xi
Law of Mass Action
• Important to remember our equation
K = Õ ani i
i
describes the equilibrium condition. At non-equilibrium
conditions it is called the reaction quotient, Q.
• Written for the reaction H2CO3 = HCO3- + H+
K=
aHCO- aH +
3
aH 2CO3
• We can see that addition of H+ will drive the reaction to
the left.
• “Changing the concentration of one species in a
reaction in a system at equilibrium will cause a reaction
in a direction that minimizes that change”.
Le Chatelier’s Principle
• We can generalize this to pressure and
temperature:
dG = VdP - SdT
• An increase in pressure will drive a reaction in a
direction such as to decrease volume
• An increase in temperature will drive a reaction in a
direction such as to increase entropy.
• “When perturbed, a system reacts to minimize the
effects of perturbation.”
Temperature and Pressure
Dependence
• Since ∆G˚ = ∆H˚ - T∆S˚ and ∆G˚ = -RT ln K then
∆ H ro ∆ Sro
ln K = +
RT
R
• Temperature and pressure dependencies found by
taking derivatives of this equation with respect to T
and P.
Oxidation and Reduction
Oxidation refers to processes in which atoms gain or loss
electrons, e.g., Fe2+  Fe3+
Valence and Redox
• We define valence as the charge an atom acquires
when it is dissolved in solution.
• Conventions
o
o
o
o
Valence of all elements in pure form is 0.
Sum of valences much equal actual charge of species
Valence of hydrogen is +1 except in metal hydrides when it is -1
Valence of O is -2 except in peroxides when it is -1.
• Elements generally function as either electron donors or
acceptors.
o Metals in 0 valence state are electron donors (become positively charged)
o Oxygen is the most common electron acceptor (hence the term oxidation)
• Redox
o A reduced state can be thought of as one is which the availability of electrons
is high
o An oxidized state is one in which the availability of electrons is low.