To examine the effect of ion removal on cell voltage, a chemist constructs two voltaic cells, each with a standard hydrogen electrode in one compartment. One cell also contains a \(\mathrm{Pb} / \mathrm{Pb}^{2+}\) half-cell; the other contains a \(\mathrm{Cu} / \mathrm{Cu}^{2+}\) half-cell. (a) What is \(E^{\circ}\) of each cell at \(298 \mathrm{~K} ?\) (b) Which electrode in each cell is negative? (c) When \(\mathrm{Na}_{2} \mathrm{~S}\) solution is added to the \(\mathrm{Pb}^{2+}\) electrolyte, solid \(\mathrm{PbS}\) forms. What happens to the cell voltage? (d) When sufficient \(\mathrm{Na}_{2} \mathrm{~S}\) is added to the \(\mathrm{Cu}^{2+}\) electrolyte, CuS forms and \(\left[\mathrm{Cu}^{2+}\right]\) drops to \(1 \times 10^{-16} \mathrm{M} .\) Find the cell voltage.

Standard electrode potential, often given the symbol \(E^\text{°}\), represents the inherent ability of a half-cell to gain or lose electrons when connected to a standard hydrogen electrode (SHE). This standardizes the comparison of different half-cells. The SHE is assigned a potential of 0 V by convention, and other half-cell potentials are measured relative to it. For instance, the standard reduction potential for the Pb/Pb²⁺ half-cell is −0.13 V, which indicates that lead is less likely to be reduced compared to hydrogen. In contrast, the Cu/Cu²⁺ half-cell has a potential of +0.34 V, showing it is more easily reduced than hydrogen.

Cell voltage, also known as electromotive force (emf), represents the difference in potential energy between two electrodes in a voltaic cell. It can be calculated using the standard electrode potentials of the two half-cells involved. The formula used to calculate cell voltage is: \[E^\text{cell} = E^\text{cathode} - E^\text{anode}\] Let's take the example from the exercise: For the Pb/Pb²⁺ cell, with the hydrogen anode (0 V) and the lead cathode (−0.13 V), the cell voltage is: \[E^\text{cell} = 0 \text{ V} - (-0.13 \text{ V}) = 0.13 \text{ V}\] Similarly, for the Cu/Cu²⁺ cell, the cell voltage calculation is: \[E^\text{cell} = 0.34 \text{ V} - 0 \text{ V} = 0.34 \text{ V}\]

The Nernst Equation is essential for understanding how concentration affects the cell voltage. It allows us to calculate the actual voltage of electrochemical cells under non-standard conditions. The equation is: \[E^\text{cell} = E^\text{cell}^\text{°} - \frac{0.0591}{n} \times \text{log}(Q)\] where \(E^\text{cell}\) is the cell potential under non-standard conditions, \(E^\text{cell}^\text{°}\) is the standard cell potential, \(n\) is the number of moles of electrons transferred, and \(Q\) is the reaction quotient. For example, when Na₂S is added to the Cu²⁺ electrolyte, CuS forms, reducing the Cu²⁺ concentration to \(1 \times 10^{-16}\) M. Plugging into the Nernst equation, with \(E^\text{cell}^\text{°} = 0.34 V\), we find: \[E^\text{cell} = 0.34 \text{ V} - \frac{0.0591}{2} \times \text{log}(1 \times 10^{16})\] Simplifying, we find: \[E^\text{cell} = 0.34 \text{ V} - 0.4728 \text{ V} = -0.1328 \text{ V}\]

In electrochemical reactions, oxidation and reduction processes occur at the electrodes. Oxidation refers to the loss of electrons and takes place at the anode, making it the negative electrode. Reduction, the gain of electrons, occurs at the cathode, which is the positive electrode in voltaic cells. For the given cells, in the Pb/Pb²⁺ cell, lead undergoes oxidation: \(\text{Pb} \rightarrow \text{Pb}^{2+} + 2e^-\). In the Cu/Cu²⁺ cell, hydrogen is oxidized: \(\text{H}_2 \rightarrow 2\text{H}^+ + 2e^-\).

Electrochemical cells are devices that convert chemical energy into electrical energy or vice versa. In voltaic cells like those in the exercise, chemical reactions generate electrical current spontaneously. The cells consist of two half-cells connected by a salt bridge or porous partition, allowing ion flow without mixing the different solutions. The two half-cells in the given problem, hydrogen and metal electrodes (Pb/Pb²⁺ or Cu/Cu²⁺), generate a voltage due to the difference in their standard electrode potentials, driving the current through the external circuit.