# 1.9: Cells at Non-Standard Conditions


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

$\text{Zn}│\text{Zn}^{2+} ~(1 M)~║~ \text{Cu}^{2+} ~~(1 M)~|~ \text{Cu}$

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 reactant concentrations will decrease and the product concentrations will increase until the solution 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, indicated as []. The equation for the reaction in the cell is

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

To calculate the electromotive force of a non-standard galvanic cell we use The Nernst Equation

$E_{\text{cell}} = E_{\text{cell}}^{\circ} - \dfrac{0.0592}{n}\log {[products]/[reactants]}$ (T = 298 \text{ K})\]

This equation is valid at 298ºK, which the temperature at which most of the Galvanic cells are used.  Only the soluble species (or gases) affect the cell potential, not the solid electrode. The n value refers to the number of electrons exchanged (in our reaction above, n=2). If you calculate the [product]/[reactants] ratio for the standard-state concentration of 1 M, note how the second term on the right-hand-side disappears and you recover $$E_{\text{cell}} = E_{\text{cell}}^{\circ}$$.

This expression for the Nernst equation is a simplified version that applies only when the number of electrons gained at the anode is identical to the number of electrons lost at the cathode. In other words, this expression for the Nernst equation is only valid when multiplication is not necessary to balance the number of electrons in the overall cell reaction. When the number of electrons involved at the cathode and anode is not the same, the expression is more complicated and it involves the concept of reaction quotient (Q), which we will learn later in the semester.

Example $$\PageIndex{1}$$ : Voltage of a Galvanic Cell

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

$\text{Cu}^{+} (aq) (1 M)│\text{Cu}║\text{Ag}│\text{Ag}^{+} (0.8 M) \nonumber$

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 equation, from which you can derive reactants and products. Remember that solids do not participate in the Nernst equation.

$E_{\text{cell}}^{\circ} = 0.7991\text{ V} - (0.520)\text{ V} = 0.2791\text{ V} \nonumber$

$\text{Ag}^{+}(aq)+\text{Cu(s)} \rightarrow \text{Ag}(\text{s}) + \text{Cu}^{+}(aq) \nonumber$

The expression for [products]/[reactants] can be derived only from the balanced overall cell reaction. Thus, using the Nernst Equation, we have,

\begin{align*}E_{\text{cell}} &= E_{\text{cell}}^{\circ} - \dfrac{0.0592 \text{ V}}{z}\log Q\\&= 0.2791\text{ V} - \dfrac{0.0592\text{ V}}{1}\log \dfrac{\left\{\text{Cu}^{+}(aq)\right\}}{\left\{\text{Ag}^{+}(aq)\right\}}\\&= 0.2791\text{ V} - \dfrac{0.0592\text{ V}}{1}\log \dfrac{\left(1\text{ }M\right)}{\left(0.8\text{ }M\right)}\\E_{\text{cell}}&=0.2734\text{ V}\end{align*}