Gary L. Bertrand
Professor of Chemistry
University of Missouri-Rolla
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Calculating   for a Reaction                                                           The Nernst Equation

A battery consists of one or more electrochemical cells. Each cell contains two metal electrodes and at least one electrolyte solution (a solution containing ions that can conduct electricity). The battery operates through electrochemical reactions called oxidation and reduction. These reactions involve the exchange of electrons between chemical species. If a chemical species loses one or more electrons, this is called oxidation. The opposite process, the gain of electrons, is called reduction. Oxidation occurs at the Anode.

Reduction occurs at the Cathode. If the reactive components of an electrochemical cell are placed in contact with each other, they will react by direct transfer of electrons (an oxidation - reduction reaction) and there is no way to harness this energy to do electrical work.  Most of the energy of the reaction is released as heat.   In most batteries, there are different materials at the two electrodes, such that they want to react -  with one material being oxidized and the other being reduced.  In the cell below, Zinc is used for the electrode on the left (the Anode) in contact with a solution of Zinc (II) ions, possibly a solution of Zinc NitrateCopper is used for the electrode on the right (the Cathode) in contact with a solution containing Copper (II) ions, perhaps Cupric Nitrate.  By keeping the materials separated, the electrons being produced by the oxidation at the Anode could be used to do electrical work as they are transferred to the Cathode where they will be consumed by the reduction process. However, the oxidation process either produces positive ions or removes negative ions from the solution at the anode (or it may change one ion to a more positive one), and the reduction process either removes positive ions or produces negative ions in the solution at the cathode.  This produces electrically charged solutions, and very quickly stops the process before a measurable number of electrons are transferred.

There must be a path for the ions to move between the two solutions in order for electrons to flow continuously through the wire.  This produces an "ion current" within the battery with cations (positively - charged ions) moving from anode to cathode, and anions (negatively - charged ions) moving from the cathode toward the anode. This path may be provided by having the two solutions in contact with each other, but this allows diffusion of all of the ions and "runs down" the battery pretty quickly.  This diffusion can be slowed down by separating the solutions with a membrane or a porous plug.  All of these can lead to a "liquid junction potential" due to differing rates of movement by the cations and anions.  A "salt bridge" can be used to separate the two solutions with a third concentrated solution of well - matched cations and anions, completely eliminating the "liquid junction potential".  In a few cases, it is possible to design a battery so that both electrodes can be placed in the same container with only one solution.

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The voltage of a cell may depend on many factors: the electrode materials, the components and concentrations of the solutions, the type of liquid junction, the temperature, and the pressure.  The voltage also depends on the electrical current being drawn from the cell.  The voltage (E) and the current (I) are related to the resistance (R) through Ohm's Law:
E = IR
The current is directly related to the rate at which electrons are pumped through the wire and any resistances in the circuit.  As the resistance is lowered to zero (a short-circuit), the current increases and the voltage of the cell decreases to zero. As the resistance is increased, the current decreases, and the voltage increases toward a limiting value.  In Chemistry, we are primarily interested in this limiting value, the maximum voltage that the electrochemical cell can deliver.  This maximum voltage or electrochemical potential is a measure of the maximum electrical work that can be obtained from the chemical reaction occurring within the cell, and this can be related to the Gibbs' Free Energy Change associated with the chemical reaction.

In the cell above, electrons are produced by metallic lead being oxidized to lead (II) ions, and they are removed by copper (II) ions being reduced to metallic copper.  Even with the ions moving across the boundary between the solutions, there is an increase in the concentration of lead ions on the left and a decrease in copper ions on the right.  This causes the voltage of the battery to decrease, and eventually the voltage will decrease to zero.  Some batteries are designed to be re-chargeable by forcing electrons to flow backwards through the cell, reversing the chemical reaction.

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Calculating the Maximum Voltage of a Cell

The Nernst Equation describes the effects of concentrations on the maximum voltage that the chemical reaction can produce by relating the voltage to the Standard Electrochemical Potential ().  This Standard Electrochemical Potential represents the maximum voltage the reaction can produce with all of the components in their standard states or at unit activity.

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Calculating
for a Reaction

In a cell shown above, there is a chemical reaction occurring in the half-cell on the left, which involves an oxidation associated with the metal electrode, which is called the anode.  If this electrode is a piece of Zinc, the oxidation is conversion of metallic zinc to zinc ions in solution:
Zn(s) Zn2+(aq) + 2 e-   (oxidation)

On the right side of this cell a reduction is occurring, associated with the metal electrode, which is called the cathode.  If this electrode is a piece of copper, the reduction is conversion of copper (II) ions in solution into metallic copper:

Cu2+(aq) + 2 e- Cu(s)  (reduction)

Each of these half-cell reactions has a Standard Electrochemical Potential.  Irrespective of whether the half-cell reaction is a reduction or an oxidation, the Standard Electrochemical Potentials are tabulated for a reduction.   Here's an abbreviated table: The  Cu/Cu2+  half-cell reaction (a reduction in this case) has Eredo = 0.339 V, and the Zn/Zn2+ half-cell reaction (an oxidation in this case) has Eoxo = - 0.763 V.    For the combined reaction,

Zn(s) + Cu2+(aq) Zn2+(aq)  + Cu(s)   .

Note that the number of electrons on the left and right hand sides of this equation must cancel out in combining the half-cell reactions.  The Standard Electrochemical Potential for the Reaction (Erxo) is calculated by subtracting the value for the oxidation (
Eoxo ) from the value for the reduction (Eredo):

Erxo = Ered-  Eoxo  = 0.339 - (-.763) = 1.102 V  .

Note that the combination of half-cell potentials to obtain
Erxo is not affected at all by the numbers of electrons in the half-cell reactions.

A positive value for this result indicates that the reaction will occur as written (if it is allowed to occur at all) as opposed to the reverse  reaction.  For the reverse reaction,

Zn2+(aq)  + Cu(s) Zn(s) + Cu2+(aq)

the  value of the Standard Electrochemical Potential is negative ,
Erxo = - 1.102 V , because the oxidation and reduction half-cells have been reversed.

The Nernst Equation and Effects of Concentrations on the Voltage

The effect of concentration on the voltage (
E) is described by the Nernst Equation:

E = - (RT/NF) ln (Q)    or    E = - 2.303(RT/NF) log (Q)

in which R is the Gas Constant (8.314 J/mol-K), T is the Absolute Temperature (K), F is the Faraday Constant (F = 96,485 C/mol), and N is the number of moles of electrons transferred per mole of reaction (equivalents/mol).  N represents the number of electrons that are cancelled out in combining the half-cell reactions to obtain the net reaction.  Voltages are often calculated for cells near room temperature (T = 298.15 K), and the Nernst Equation takes the form:

E = - (0.0257/N) ln (Q)    or    E = - (0.059/N) log (Q)

The Reaction Quotient Q is written exactly as the Equilibrium Constant for the reaction would be written, but Q can be applied to a reaction whether it is at equilibrium or not.  For a Generalized Reaction:

a A(s) + b B(g) + c C(aq) d D(aq) + e E(g) + f F(liq) ,

The value of Q may be approximated
:

Q = [D]d PEe / PBb [C]d   .

(This approximation is best at low concentrations of the ions and low pressures of any reacting gases.)

For the reaction above:
Zn(s) + Cu2+(aq) Zn2+(aq)  + Cu(s)      N = 2     Erxo = 1.102 V ,

the approximate voltage is calculated as:

E(V) = 1.102 - (0.0257/2) ln ( [Zn2+(aq)] / [Cu2+(aq)] )
or
E(V) = 1.102 - (0.059/2) log ( [Zn2+(aq)] / [Cu2+(aq)] )  .

These equations show that the voltage of this cell decreases with increasing concentration of zinc ion, and increases with increasing concentration of copper (II) ion.  This agrees with LeChatelier's Principle, such that increasing the concentration of zinc ion "pushes" the reaction to the left, and increasing the concentration of copper (II) ion "pushes" the reaction to the right.

The value of
Erxo for a reaction may be estimated experimentally by constructing the cell with the concentrations of all ions at unity (1.00 mol/L) and any gases at unit pressure (1.00 bar), and measuring the voltage E(V) of the cell.  Under these conditions,
E(V) ~ Erxo   .

For some reactions (none of the concentrations of ions in the expression for Q are raised to a power other than 1) the approximation becomes even better if all of the relevant ions are at the same (very low) concentration.

Measuring the Voltage of a Battery

Chemists use calculations with the Nernst Equation to determine the theoretical voltage of an electrochemical cell.  In doing so, they define one electrode as the anode and the other as the cathode.  They also make assumptions regarding the reactions which will occur at those electrodes.  The ultimate test of these assumptions is measurement of the voltage.  The voltage that is observed depends on the reactions which are actually occurring, and the algebraic sign of the reading depends on how the leads of the voltmeter are connected to the electrodes.
Batteries have well-marked positive (+, cathode) and negative (-, anode) electrodes, but electrochemical cells which are constructed in the laboratory do not.  When the positive lead of the voltmeter is connected to the positive pole of the battery, a positive voltage will be observed.  This means that the electrons that are released by the oxidation at the anode are travelling through the voltmeter to the cathode, where they are consumed in the reduction that occurs there.  Therefore, when a voltmeter is connected to an electrochemical cell, a positive voltage indicates that the positive lead is connected to the cathode and the negative lead is connected to the anode.

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