Cells at Non-Standard Conditions

Submitted by ChemPRIME Staff on Thu, 12/16/2010 - 15:36

As shown in the section on cell notation, a galvanic cell in shorthand form can be written:

Zn│Zn2+(1 M)║ Cu2+(1 M)|Cu

The parenthetical notes of (1 M) are frequently omitted because 1 M is at the standard state. However, cells can be created which use concentrations other than 1 M. In such a case, one must always indicate the concentrations as shown above. In fact, as the reaction for the cell written above takes place (that is, as the cell provides electric current), Cu2+ will be used up while Zn2+ will be generated. The reactantA substance consumed by a chemical reaction. concentations will decrease and the productA substance produced by a chemical reaction. concentrations will increase until the solutionA mixture of one or more substances dissolved in a solvent to give a homogeneous mixture. has reached a state of equilibrium. These equilibrium concentrations are not likely to be at the standard 1 M solutions indicated above.

The voltage of a cell at non-standard state is modified by the relative concentrations of the reactants and products. That is, the cell emf depends on the reaction quotient, Q. The equation for the reaction in the cell is

$\text{Zn} + \text{Cu}^{2+} \rightleftharpoons \text{Cu}+\text{Zn}^{2+}$,

We can determine the reaction quotient, Q, as follows. (Note that when denoting actual concentrations, which may or may not be equilibrium concentrations, curly brackets are used.)

$Q=\frac{\left\{\text{Zn}^{2+}\right\}}{\left\{\text{Cu}^{2+}\right\}}$.

Notice that only the aqueousDescribing a solution in which the solvent is water. states are included in the reaction quotient. Using an equation from Galvanic Cells and Free Energy, we can see that the electromotive forceThe electrical potential difference that can develop across a voltaic (galvanic) cell, used as a measure of the spontaneity of a reaction. is related to ΔG° (at standard states). (Recall that z is the number of electrons transferred and F is the Faraday constantThe electric charge carried by one mole of electrons, 9.648 670 x 104 C mol-1; abbreviated F..) This holds true even when the cell is not at standard conditions.

ΔG° = – zFE°   and   ΔG = – zFE

We will not derive it here, but there is a relationship for ΔG for non-standard conditions:

ΔG = ΔG° + RT ln Q.

Combining these equations gives a single equation for the Electromotive force of a non-standard galvanic cell: The Nernst EquationThe relationship between the standard cell potential and the cell potential under conditions of non-standard concentrations..

$E_{\text{cell}} = E_{\text{cell}}^{\circ} - \frac{RT}{zF}\ln Q$

For ease of calculation, we can change the baseIn Arrhenius theory, a substance that increases the concentration of hydroxide ions in an aqueous solution. In Bronsted-Lowry theory, a hydrogen-ion (proton) acceptor. In Lewis theory, a species that donates a pair of electrons to form a covalent bond. of the logarithm to base 10 (although most standard calculators have a natural logarithm button), the simple formula below is another commonly used form of the Nernst Equation:

$E_{\text{cell}} = E_{\text{cell}}^{\circ} - \frac{2.303 RT}{zF}\log Q$.

As most of the Galvanic cells in use will be at room temperatureA physical property that indicates whether one object can transfer thermal energy to another object. (T = 25 °C), we can substitute all of the constant values (R = 8.314 J mol–1 K–1 and F = 96,485 J V–1 mol–1), yielding an even simpler formula:

$E_{\text{cell}} = E_{\text{cell}}^{\circ} - \frac{0.0592 \text{V}}{z}\log Q$  (T = 298 K).

If you calculate Q for the standard-state concentration of 1 M (or pressureForce per unit area; in gases arising from the force exerted by collisions of gas molecules with the wall of the container. of 1 bar), note how the second term on the right-hand-side disappears and you recover $E_{\text{cell}} = E_{\text{cell}}^{\circ}$.

Although the Nernst equation is useful to predict the actual voltage of a cell under non-standard conditions, it is frequently more useful to use the measured voltage to detect the concentration of one of the species. For instance, if we use a standard H2/Pt half-cell, the detected voltage of that half reaction coupled with an unknown concentration of Fe2+ can be used to determine the concentration of Fe2+.

EXAMPLE 1 Determine the voltage measured for this galvanic cell. You may need to use the Table of Standard Reduction Potentials.

Sn2+(aq)(1 M│Sn║Ag│Ag+(0.8 M)

Solution

The strategy for solving these problems is to first find the standard electromotive force, as shown in another section, and then write the balanced chemical equationA representation of a chemical reaction in which chemical symbols represent reactants on the left side and products on the right side., from which you can derive Q and z.

$E_{\text{cell}}^{\circ} = 0.7991\text{ V} - (- 0.1375)\text{ V} = 0.9366\text{ V}$

$2\text{Ag}^{+}(aq)+\text{Sn(s)} \rarr 2\text{Ag}(\text{s}) + \text{Sn}^{2+}(aq)$

The expression for Q can be derived only from the balanced chemical equation. Thus, using the Nernst Equation, we have,

\begin{align} E_{\text{cell}} &= E_{\text{cell}}^{\circ} - \frac{0.0592 \text{ V}}{z}\log Q\ &= 0.9366\text{ V} - \frac{0.0592\text{ V}}{2}\log \frac{\left\{\text{Sn}^{2+}(aq)\right\}}{\left\{\text{Ag}^{+}(aq)\right\}^2}\ &= 0.9366\text{ V} - \frac{0.0592\text{ V}}{2}\log \frac{\left(1\text{ }M\right)}{\left(0.8\text{ }M\right)^2}\ E_{\text{cell}}&=0.9308\text{ V} \end{align}