THERMODYNAMICS OF
ELECTROCHEMICAL CELLS
1. Thermodynamic Data from Electromotive Force
Measurements
1. A. Maximum work
Recall that the change in Helmholz energy A equals the maximum work for the system
A = wmax
and that the change in Gibbs free energy G equals the maximum non-expansion work for
the system
G = wnon-pV,max
To prove the second statement, recall that
G = H  TS
= U + pV  TS
dG = dq + dw + pdV + Vdp  TdS  SdT
At constant p and T, dp = 0 and dT = 0
dG = dq + dw + pdV +  TdS
Since the maximum amount of work is produced by a reversible process, dq = TdS, which
implies
dG = dwmax + pdV
Recognizing that dw = dwnon-pV + dwpV = dwnon-pV  pdV yields
dG = dwnon-pV,max  pdV + pdV
= dwnon-pV,max
which upon integration implies
G = wnon-pV,max
1. B. Electrical work
Electrical work is defined as
work = charge  potential
Electrical work done by an electrochemical cell is therefore defined as
welec =  F  E
where  is the stoichiometric number of electrons passing through the cell circuit, a
unitless quantity determined from cell half-reactions; F is Faraday's constant, which is the
charge of a mole of electrons and equals 96,490 Coulomb·mol1; and E is the
electromotive force of the cell, usually measured in volts. Since
J = Coulomb·volt
the units of welec are J·mol1.
The work done on an electrochemical cell is therefore
welec =  F E
1. C. Thermodynamic quantities
Identifying the non-expansion work for the system as the work done on a reversible
electrochemical cell
wnon-pV = welec
gives an expression for G in terms of electromotive force of an electrochemical cell
G =  F E
To obtain expressions for S and H, recall that
 G 
 G 
 dp  
dG  
 dT
 T  p
 p  T
and
dG = Vdp  SdT
Equating coefficients implies that
 G 

  S
 T  p
or
 G 
S  

 T  p
Recalling the expression for G yields
 E 
S   F 

 T  p
Thus, S depends on the temperature dependence of the electromotive force of an
electrochemical cell.
Finally, H = G + TS at constant T and the previous results imply
 E 
H   F E   FT 

 T  p
2. Chemical System
The reaction under study is
Zn(Hg) + PbSO4 (s) = ZnSO4 (0.02 m) + Pb(Hg)
where the symbol s refers to the solid state and 0.02 m refers to the molality (moles per
kg of solvent) of ZnSO4. Zn(Hg) and Pb(Hg) are mercury amalgams of zinc and lead,
respectively.
The half-reactions for this reaction are
Zn (s) = Zn2+(aq) + 2e
2e + PbSO4 (s) = Pb (s) + SO42 (aq)
Thus, , the stoichiometric number of electrons passing through the cell, equals 2.
3. Cell Construction
3. A. Heat release
If the reactants were added directly together, the reaction would proceed irreversibly to
products. Since no gases are involved in the reaction, wpV=0, and since there would be
no electrical work, wnon-pV=0. Thus, only heat would be released by the reaction and the
measured quantity at constant p would be
qirreverible = H
In contrast, the oxidation and reduction reactions can be placed in separate
electrochemical half-cells. The electrons travel through an external circuit where they
can do electrical work. If the cell is operated reversibly, i.e., current flow is "infinitely
slow", then
qreversible = TS
and as shown above
wnon-pV,reversible = G
3. B. Electrochemical cell
The following electrochemical cell separates the oxidation (loss of electrons by Zn) and
reduction (gain of electrons by PbSO4) into separate half-cells.
Anode ()
sintered
glass disk
Cathode ()
0.02 m ZnSO4
PbSO4
6% Zn amalgam
6% Pb amalgam
[Delete above figure before converting.]
Observe that oxidation (loss of electrons) occurs at the anode and reduction (gain of
electrons) occurs at the cathode.
3. C. Experimental details
Since the same electrolyte is used through the cell, there is no junction potential which
would affect the measured E value.
The frit allows passage of ions between half-cells, but prevents direct mixing of reactants.
The purpose of the amalgam is to provide better electrical contact between the electrode
and metals (without strain induced potentials) and to provide a source/sink of metal
atoms.
Oxygen should be excluded from the cell because it is a good oxidizing agent and can
reduce water at the cathode
1/2 O2 (g) + H2O (l) + 2 e  2 OH (aq)
4. Data Analysis
Historically, electrochemical measurements were made with a potentiometer or "bridge
circuit," in which an opposing voltage was applied which prevented current flow. Today,
measurements are made with a high impedance voltmeter.
The cell electromotive force is measured as a function of temperature.
(E/T)T=298K
ET=298K
E
298K
T
The data is fit with a polynomial (which could be a straight line if appropriate), and the
values of ET=298K and (E/T)T=298K are determined from the fit. These values are related
to G, S, and H as described above.
Note that the above negative slope implies that DS is negative, i.e., that entropy decreases
even though the reaction results in conversion of solid into aqueous ions. This might
seem counter-intuitive; however, ions tend to "organize" the water around them, which
results in a net entropy decrease.
5. Relation to Standard Thermodynamic
Quantities
One is usually interested in determining standard thermodynamic quantities. These
quantities relate reactants and products in their standard states, i.e., with activities of 1 (or
concentrations of 1 M for ideal solutions). The species involved in the current
experiment are not in their standard states, since the concentration of ZnSO4 is 0.02 m.
The Nernst Equation describes the dependence of electromotove force on concentration.
E  E 
RT
ln Q
F
where Eº is the electromotive force at the standard state and Q is the reaction quotient (or
equilibrium constant K for ideal solutions)
Q
a Zn2  aSO2  a Pb
4
a PbSO4 a Zn
Observe that the form of the Nernst Equation is consistent with Le Chatelier's Principle.
If the reactant activities (concentrations) are large relative to the products, then Q is less
than one, lnQ is negative, E is more positive, G is more negative, and the reaction
proceeds spontaneously to produce products. Le Chatlier's Priciple would predict that
excess reactants would force the reaction to create more products.
The activities of the solid phases are assumed to be unity. For solutions, the activity is
defined as the product of the activity coefficient and the concentration
ai = i mi
where i is the activity coefficient and mi is the molality for species i. For ideal solutions
the activity coefficient equals unity. In contrast to the short-range interactions of neutral
species (in which the repulsive potential is proportional to r12 and the attractive potential
is proportional to r6), ionic species interact at very long distances (as the Coulomb
potential is proportional to r1!). Thus, even dilute ionic species are typically very
non-ideal.
For dilute ionic solutions the activity coefficient can be calculated from the DebyeHuckel equation (which assumes that all interactions between ions are based on
Coulombic attraction or repulsion effects)
log  =  0.509 | z+ z | I1/2
where z+ is the charge of the positive ion and z is the charge of the negative ion and I is
the ionic strength of the solution
I
1
zi2 mi

2 i
where zi is the charge for ion i.
Thus, measured values of E can be related to standard values of Eº by
E  E 
RT
ln(  Zn2  mZn2   SO2  mSO2  )
4
4
F
from which values for Gº, Sº, and Hº may be calculated.