Recall that the change in Helmholz energy A equals the maximum work for the system
DA = wmax
and that the change in Gibbs free energy G equals the maximum non-expansion work for the system
DG = 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
which upon integration implies
DG = wnon-pV,max
Electrical work is defined as
work = charge ´ potential
Electrical work done by an electrochemical cell is therefore defined as
welec = n F ´ E
where n 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 = -n F E
Identifying the non-expansion work for the system as the work done on a reversible electrochemical cell
wnon-pV = -welec
gives an expression for DG in terms of electromotive force of an electrochemical cell
DG = -n F E
To obtain expressions for DS and DH, recall that
dG = Vdp - SdT
Equating coefficients implies that
Recalling the expression for DG yields
Thus, DS depends on the temperature dependence of the electromotive force of an electrochemical cell.
Finally, DH = DG + TDS at constant T and the previous results imply
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, n , the stoichiometric number of electrons passing through the cell, equals 2.
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 = DH
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 = TDS
and as shown above
wnon-pV,reversible = DG
The following electrochemical cell separates the oxidation (loss of electrons by Zn) and reduction (gain of electrons by PbSO4) into separate half-cells.
Observe that oxidation (loss of electrons) occurs at the anode and reduction (gain of electrons) occurs at the cathode.
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)
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.
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 DG, DS, and DH 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.
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.
where Eº is the electromotive force at the standard state and Q is the reaction quotient (or equilibrium constant K for ideal solutions)
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, DG 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 = gi mi
where gi 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 Debye-Huckel equation (which assumes that all interactions between ions are based on Coulombic attraction or repulsion effects)
log g± = - 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
where zi is the charge for ion i.
Thus, measured values of E can be related to standard values of Eº by
from which values for DGº, DSº, and DHº may be calculated.
Find a recent article in a chemistry journal that reports an experiment that utilizes electrochemical measurements. (You might browse through the journals "Journal of Physical Chemistry" and "Analytical Chemistry" in the library.) Report the authors, title, and full reference of the article. Write a one-page summary in your own words that describes the measurement technique and discusses why this particular technique is unique, important, or relevant. (This latter information is often found in the Introduction and Discussion sections of a journal article.)