3.012 Fundamentals of Materials Science
Fall 2003
Lecture 13: 11.17.03 Chemical and electrochemical reaction equilibria
Today:
LAST TIME .............................................................................................................................................................................................. 2
REACTIONS IN CONDENSED PHASE SOLUTIONS1 ...................................................................................................................................... 3
APPLICATION EXAMPLE: DESIGNING A BATTERY USING THE ELECTROCHEMICAL POTENTIAL ............................................................... 5
The electrochemical potential ............................................................................................................................................................ 5
The Nernst Equation .......................................................................................................................................................................... 6
The Nernst equation for a simple battery ........................................................................................................................................... 7
REFERENCES ......................................................................................................................................................................................... 10
Reading:
Supplementary Reading:
-
Lecture 13 – Chemical reactions and electrochemistry
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3.012 Fundamentals of Materials Science
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Last time

Last time we introduced models for the chemical potential of materials.

We then showed how the Gibbs condition is applied to determine chemical equilibrium for the case of a reactive
ideal gas mixture.
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Reactions in condensed phase solutions1

Suppose we wish to consider instead a reactions between solid or liquid components in our system. These may
also undergo a general reaction:
aA  bB cC  dD
(Eqn 1)

Just as the case for the ideal gas reaction, we start with the Gibbs condition for equilibrium:

(Eqn 2)
C
dG   i dX i  0
i1
cC  dD  aA  bB  0
(Eqn 3)



i (X i ,P,T)  io (P,T)  RT log ai
Thus we have:

(Eqn 5)

aA  bB  cC  dD
Recall from last lecture that the general solution model for the chemical potential is:
(Eqn 4)

or
a(Ao  RT ln aA )  a(Bo  RT ln aB )  c(Co  RT ln aC )  d(Do  RT ln aD )
Last time, we defined the standard state molar free energy change of reaction as:

products
G
o
rxn


 iG 
o
i
reac tan ts
 G
i
j
o
j
j
 cG  dG  aGAo  bGBo
o
C
(Eqn 6)
o
D
 cCo  dDo  aAo  bBo

Applying this definition to (Eqn 5) we get:

o
Grxn
 RT (c ln aC  d ln aD  aln aA  bln aB )
(Eqn 7)
e
(Eqn 8)


o
Grxn
RT
 ln
aCc aDd
 K eq (T)
aAa aBb

Lecture 13 – Chemical reactions and electrochemistry
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3.012 Fundamentals of Materials Science

Note that if we have an ideal solution (ai = Xi), the equilibrium constant reduces to the mass action law as you are
typically introduced to it in introductory chemistry:
ln
(Eqn 9)

Fall 2003
aCc aDd
XCc X Dd
nCc n Dd [C]c [D]d

ln

ln

 K eqideal (T)
a b
a
b
a b
a
b
aA aB
XA XB
n A n B [A] [B]
B
 condensed phases and gas (standard approx if there will be time).
Cover reactions between

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3.012 Fundamentals of Materials Science
Fall 2003
Application Example: Designing a Battery using the electrochemical potential

You may not have thought about it, but the function of batteries are governed by thermodynamics, and in
particular by the action of chemical potentials. Design of improved batteries, including batteries fabricated from
solid materials (as opposed to lead acid batteries that can leak their corrosive contents!) are an intense area of
current materials science & engineering research. Analysis of a battery provides an example of utilizing the
general solution model for the chemical potential, and also introduces a (yet another) new form of internal energyembodied in the electrochemical potential.
The electrochemical potential

Consider the simplest model of a battery made using two solutions containing different concentrations of a salt, as
illustrated below.2 The battery is composed of two beakers filled with solutions containing a salt, e.g. silver nitrate
(AgNO3). The beakers are connected by two sets of electrodes. One set is made of silver electrodes, the other
set, called the reference electrodes, are made of an inert metal.
o Batteries contain charged components- in the example below, the charged component is the silver ion
Ag++. Charged species are driven to partition between phases of a system by chemical forces and by
electrical forces, if an electrostatic potential is present.
Reference electrodes
Ag electrodes

Each beaker can be thought of as a two-phase system: one phase is the solution of ions, the other phase is the
surface of the solid Ag electrode. We can ignore the reference electrode because it is thermodynamically neutralit does not add or subtract ions from the system. This simple battery introduces a new type of thermodynamic
force to account for: electrostatic potential. The charged Ag++ ions in solution, if exposed to an electrostatic
potential (e.g. a buildup of negative charge on the surface of the electrode), will be driven to redistribute (e.g.
deposit on the electrode surface). Similarly, silver atoms can ionize and be driven to leave the solid surface of the
electrode if an electrostatic potential is applied that drives them into solution. Thus an applied electrostatic
potential  will modify the internal energy of the solid electrode and the silver ion solution. To account for the
responsiveness of this system to electrostatic potential, we need to introduce a new term in the fundamental
equation for the internal energy to account for this new form of electrochemical energy:
C
C
j1
i1
dU  TdS  PdV    j dN j   dqi
(Eqn 10)
Lecture 13 – Chemical reactions and electrochemistry
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
Where the second sum is over each of the charged/chargeable species in the system. The charge on each ion qi
is:
qi  zieNi
(Eqn 11)

…where zi is the valency of the ion (charge per ion), e is the unit charge of a proton, and Ni is the number of ions
of type i. Recall the Gibbs free energy is:

(Eqn 12)

Fall 2003
G  H  TS  U  PV  TS
So the differential of G, accounting for electrostatic effects, is:

(Eqn 13)
o
C
C
i1
j1
dG  SdT  VdP   i dN i   dq j
When each component in the system is ionizable, we can group together the last two sums:

(Eqn 14)
C
dG  SdT  VdP   'i dN i
i1
o
Where µ’i is called the electrochemical potential:

(Eqn 15)
'i  i  zie

The electrochemical potential contains the intrinsic chemical potential of each component, plus
the alteration in chemical potential caused by the effect of electrostatic potentials on charged
species.
  Just as the chemical potential of a species must be the same in all phases at equilibrium,
the electrochemical potential of charged components must be the same in all phases to
reach electrochemical equilibrium.
The Nernst Equation

Applying the condition for electrochemical equilibrium- equality of the chemical potentials- leads us to a famous
equation describing how electrolytes behave at equilibrium when exposed to an electrostatic potential. Suppose
we have a solution of ions that is exposed to two different electrostatic potentials. The left side of the solution
has potential L, while the right half of the solution is placed at potential R. These two regions are effectively
two different phases of the solution due to the difference in the two potentials. How will the ions distribute at
equilibrium between the two potentials?
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First, let’s write out the chemical potential for the ions in each phase. The electrochemical potential is:
o
(Eqn 16)
'
  L  ze L
(Eqn 17)
'
  R  ze R
L
R
…where we have stuck with the convention of using superscripts to denote the phase. We can make use
o
 of the general solution model to expand these expressions:

(Eqn 18)
'
  0   RT ln c L  zeL
(Eqn 19)
'
  0   RT ln c R  ze R
L
R
L
R
o Because both the left and right phases are in the same state of matter, the standard state chemical
potentials (µ0)L and (µ0)R will be equal. The condition for equilibrium at constant temperature and
pressure is:

'
L
(Eqn 20)
o
R
Substituting the above equations into the equilibrium condition we arrive at:

ln
(Eqn 21)


 '
ze L   R 
cL
ze



R
c
RT
RT
NERNST EQUATION
This is the Nernst equation, named after W.H. Nernst (1864-1941), a German chemist who was
awarded the Nobel Prize in Chemistry in 1920 for his work in chemical thermodynamics.
INTERPRET PHYSICAL MEANING OF THE RESULT- FORCES ON CHARGES.
The Nernst equation for a simple battery

Let’s return to our simple salt-solution battery and see if we can analyze the equilibrium state. The two-phase
system present in each beaker is slightly more complicated than the case of the partitioned solution described
above.
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Fall 2003
Equilibrium must be reached between the ions in solution and the solid electrode in each half-cell of the battery.
Thus we have the following equalities:
o For the left half-cell:
(Eqn 22)
'
 '
(Eqn 23)
 
 ze solid,L   0 
solid,L
0 solid,L
liquid,L
liquid,L
 RT ln c L  ze liquid,L
o Similarly for the right half-cell:
(Eqn 24)
 

o

0 solid,R
 zesolid,R   0 
liquid,R
The solid electrode is the same state of matter in each half-cell, so (µ0)solid,L = (µ0)solid,R. The purpose of
the reference electrodes is to maintain each half-cell at the same potential in the liquid phase: liquid,L =
liquid,R. Using these identities, we can subtract (Eqn 24) from (Eqn 23) and get our final answer:
(Eqn 25)
ze( solid,L   solid,R )  RT ln
(Eqn 26)
ze  RT ln


 RT ln c R  zeliquid,R

cL
cR
or
cL
cR
From this result we see that at equilibrium, if the concentration of ions is very different in the two
beakers, their equilibration with their respective electrodes will create an electrical potential
difference between the two solid Ag electrodes- a battery! The potential thus obtained can be
used to useful forms of electrical work.
 You probably haven’t used a battery made from two simple salt solutions in your
walkman. Why not?
Lecture 13 – Chemical reactions and electrochemistry
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3.012 Fundamentals of Materials Science
Lecture 13 – Chemical reactions and electrochemistry
Fall 2003
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3.012 Fundamentals of Materials Science
Fall 2003
References
1.
2.
Gaskell, D. R. Introduction to Metallurgical Thermodynamics (Hemisphere, New York, 1981) 611 pp.
Dill, K. & Bromberg, S. Molecular Driving Forces (New York, 2002) pp.
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