Cell Potential, Electrical Work, and Free
Energy
Combining thermodynamics, electrochemistry, and not to mention a bit of physics!
emf and Work
 The work that can be accomplished when electrons are transferred
through a wire depends on the “push” or emf (electromotive force)
 emf is defined in terms of potential difference (in volts) between two
points in the circuit
 Recall that a volt represents a joule of work per coulomb of charge transferred
work (J)
emf = E = potential difference V =
charge (C)
0
 Thus, one joule of work is produced (or required) when one coulomb
of charge is transferred between two points in the circuit that differ by
potential of one volt
emf and Work
 Work is viewed from the point of view of the system
 Therefore, if work flows OUT of the system, it is assigned a MINUS sign
 When a cell produces a current (aka a battery), the cell potential is
positive and the current can be used to do work (like running a motor)
 Therefore, emf and work have opposite signs!
−work (J) −w
emf = E =
=
charge (C)
q
0
w = −qE
0
Whoa….what in the world is “q”?
 q is the quantity of charge in coulombs transferred
 The charge of 1 mole of electrons is a constant called
the faraday (F)
 Has the value of 96,485 coulombs of charge per mole of
electrons
 So,
C
F = 96,485
mol e−
q (in Coulombs) = # moles of e− × F = nF
Before Continuing…A Review of Spontaneity
and Gibb’s Free Energy
 In chemistry, we often refer to processes as either spontaneous or
nonspontaneous
 A spontaneous process is said to occur if it occurs without outside
intervention
 Has nothing to do with the speed of the reaction
 To explore the idea of spontaneity, consider the following physical
and chemical processes:
 A ball rolls down a hill but never spontaneously rolls back up the hill
 If exposed to air and moisture, steel rusts spontaneously. However, the
iron oxide in rust does not spontaneously change back to iron metal and
oxygen gas
 Heat flow always occurs from a hot object to a cooler one. The reverse
process never occurs spontaneously
Spontaneity and Gibb’s Free Energy
 The driving force for a spontaneous process is an increase in the
entropy of the universe
 Entropy can be viewed as a measure of molecular randomness or
disorAder
 Natural progression of things is from order to disorder (from lower entropy to
higher entropy)
 We know entropy is related to another thermodynamic quantity
called Gibb’s Free Energy (symbolized by G)
 A process at constant temperature and pressure is spontaneous in the
direction in which the free energy decreases
−∆G = +∆Suniv
More on Gibb’s Free Energy
 Gibb’s free energy is qualitatively useful by telling us
whether a process is spontaneous or not
 It is quantitatively useful because it can tell us how
much work can be done with a given process
 Thus, G is defined as the energy available in a system
that is available to do useful work
 Maximum possible work obtainable from a process at
constant temperature and pressure is equal to the change in
free energy:
wmax = ∆G
Gibb’s Free Energy and Work
 ∆G for a spontaneous process represents the energy
that is free to do useful work
 ∆G for a nonspontaneous process represents the
minimum amount of work that must be expended to
make the process occur
So, How Does This Relate to Galvanic
Cells?
 Recall that for a galvanic cell:
w=
 And:
0
−qE
q = nF
 Since:
wmax = ∆G
 We can make some substitutions to come up with a relationship between
Gibb’s Free Energy, work, and cell potential at constant temperature and
pressure:
0
0
∆G = wmax = −qE = −nFE
0
Summary of Free Energy and Cell
Potential
0
0
∆G = −nFE
G = Gibb’s Free Energy
n = number of moles of electrons
F = Faraday constant = 96, 485 coulombs per mole of electrons
 This relationship is important because it confirms that a galvanic cell
will run in the direction that gives a positive value for E0
 +E0 corresponds to a negative ∆G value (spontaneous)
 -E0 corresponds to a positive ∆G value (nonspontaneous)
Practice!
Using the Table of Standard Reduction Potentials, calculate ∆G0
for the reaction:
Cu2+ aq + Fe s → Cu s + Fe2+
Is this reaction spontaneous?
Dependence of Cell Potential on
Concentration and the Nernst Equation
Concentration and Le Chȃtelier’s
Principle
 So far, we have described galvanic cells under standard
conditions
 All solutions at 1 M, all gases at 1 atm, and 25°C (298K) for
all
 What would happen to the cell potential if the solutions were
not at 1M?
 Can be answered qualitatively in terms of Le Chȃtelier’s
Principle
Recall Le Chȃtelier’s Principle:
 If a system at equilibrium is disturbed by a
change in temperature, pressure, or the
concentration of one of the components, the
system will shift its equilibrium position so as to
counteract the effect of the disturbance
Effect of Concentration Changes on E0
 According to Le Chȃtelier’s Principle, an increase in
reactant concentration will favor the forward reaction
 Thus, the driving force on the electrons in an
electrochemical cell will increase!
 E0 will increase
 On the other hand, an increase in product
concentration will oppose the forward reaction and
favor the reverse reaction
 Thus, the driving force on the electrons in an
electrochemical cell will decrease!
 E0 will decrease
Practice!
 For the cell reaction:
2 Al s + 3 Mn2+ aq → 2 Al3+ aq + 3 Mn (s)
Predict whether E is larger or smaller than E0 for the
following cases:
 [Al3+] = 2.0 M, [Mn2+] = 1.0 M
 [Al3+] = 1.0 M, [Mn2+] = 3.0 M
Galvanic Cells NOT at Standard Conditions
 Because cell potentials
depend on concentration,
we can construct galvanic
cells where both
compartments contain
the same components but
at different
concentrations
 These electrochemical
cells are called
concentration cells
Gibb’s Free Energy at Non-Standard
Conditions
 Recall that under standard conditions, the Gibbs
Free Energy change for an electrochemical cell is
given by:
0
0
∆G = −nFE
 From thermodynamics, the Gibbs energy change
under non-standard conditions can be related to
Gibbs energy change under standard conditions via:
G  G  RT ln Q
Measuring Cell Potential at NonStandard Conditions
 After some substitution, the previous equation becomes:
 nFE  nFE   RT ln Q
 After dividing both sides by –nF, we have:
RT
0
E=E −
ln Q
nF
 The equation above is called the Nernst equation
 It is used to calculate the potential of a cell in which some or
all of the components are not in their standard states
 Remember, standard states are 1M and gases at 1 atm
Another Form of the Nernst Equation
 The Nernst equation can be simplified by
collecting all the constants together using a
temperature of 298 K
0.0591
E=E −
log Q @ 25℃ (298 K)
n
0
 Notice natural log was converted to log base 10!
Oh My Goodness…There’s So Many
Variables!
RT
E=E −
ln Q
nF
0
E=
E0
0.0591
−
log Q
n
 R = gas constant 8.315 J/K·mol
 F = Faraday constant
 96, 485 colombs per mole of electrons




E = Energy produced by reaction
T = Temperature in Kelvin
n = number of electrons exchanged in BALANCED redox equation
Q (Reaction Quotient) – Ratio of [Product] to [Reactant] NOT at
equilibrium
Relating The Nernst Equation to K
 The Nernst equation indicates that the electrical potential of
a cell depends upon the reaction quotient, Q, of the reaction
 The potential calculated is the maximum potential before any
current flow has occurred
 As the redox reaction proceeds and the cell discharges, the
concentration of reactants decreases and thus, Ecell will
decrease
 Eventually, the cell potential reaches zero
 Zero potential means reaction is at equilibrium (a dead battery)
 We already know at equilibrium that:
Q=K
and
∆G = 0
Cell EMF and Chemical Equilibrium
 So at equilibrium, the Nernst equation
becomes:
0.0592
E  E 
ln Q
n
At equilibrium E  0 and Q  K :
0.0592
0  E 
log K
n
nE 
 log K 
0.0592
Free Energy and Cell Potential
 In summary, we know the following based on the
relationship between G, Q, and K:
Q
G
E
Forward change, spontaneous
< Keq
-
+
At equilibrium
= Keq
0
0
Reverse change, spontaneous
> Keq
+
-
Practice!
More Practice!