Consider a galvanic cell composed of the SHE and a half-cell using the reaction \(\mathrm{Ag}^{+}(a q)+e^{-} \rightarrow \operatorname{Ag}(s)\). (a) Calculate the standard cell potential. (b) What is the spontaneous cell reaction under standard-state conditions? (c) Calculate the cell potential when \(\left[\mathrm{H}^{+}\right]\) in the hydrogen electrode is changed to (i) \(1.0 \times\) \(10^{-2} M\) and (ii) \(1.0 \times 10^{-5} M,\) all other reagents being held at standard-state conditions. (d) Based on this cell arrangement, suggest a design for a pH meter.

Understanding the standard cell potential is central to studying galvanic or voltaic cells, a class of electrochemical cells that generate electrical energy through spontaneous redox reactions. The standard cell potential, denoted as \(E^0_{cell}\), is a measure of the electromotive force of a cell when all species are in their standard states, typically at a concentration of 1 M and pressure of 1 atmosphere, at a temperature of 25°C (298 K).

To calculate \(E^0_{cell}\), we take the difference between the standard reduction potentials of the cathode (reduction half-cell) and the anode (oxidation half-cell). For instance, in a cell composed of a silver half-cell and a standard hydrogen electrode (SHE), the standard cell potential is found using the formula \(E^0_{cell} = E^0_{cathode} - E^0_{anode}\). With \(E^0_{Ag+/Ag}\) at +0.80V (silver half-cell) and \(E^0_{H+/H2}\) at 0V (SHE), we get a standard cell potential of +0.80V. This positive value indicates that the cell can perform work when the reaction proceeds spontaneously under standard conditions.

A spontaneous cell reaction is one that occurs without an external energy input; the cell can do electrical work on its surroundings as the reaction unfolds. The spontaneity of the reaction in a galvanic cell is indicated by the sign of the standard cell potential (\(E^0_{cell}\)).

When the standard cell potential is positive, as in our exercise's example of +0.80V, this signifies a tendency for the cell reaction to occur on its own. Essentially, this means that the reduction of silver ions to solid silver and the oxidation of molecular hydrogen to hydrogen ions are processes that release energy. Ultimately, identifying a positive standard cell potential is key in predicting that the reaction will proceed without external interference, providing necessary insight for the design and operation of electrochemical cells and devices.

The Nernst equation is a fundamental tool in electrochemistry, allowing us to compute the actual cell potential under non-standard conditions, which includes variations in temperature, pressure, or concentration of reactants and products. The equation is given by:

\(E_{cell} = E_{cell}^0 - \frac{0.0591}{n} \log Q\),

where \(E_{cell}^0\) is the standard cell potential, \(n\) is the number of moles of electrons transferred in the reaction, and \(Q\) is the reaction quotient. The reaction quotient reflects the ratio of the concentrations of products and reactants at any given point during the reaction, not to be confused with the equilibrium constant \(K\), which is the value of \(Q\) when the reaction is at equilibrium.

The application of the Nernst equation shows that as the concentrations of reactants or products change, the cell potential also changes. This is calculated using the logarithm of the reaction quotient, which means that even small changes in concentration can have a significant effect on the potential. This variable dependency allows chemists to manipulate and control the energetic output of electrochemical reactions.

The principles of electrochemistry are adeptly applied in designing pH meters, which are instruments used to measure the acidity or basicity of a solution. In our exercise, we saw that the potential of a galvanic cell changes with the hydrogen ion concentration in the hydrogen electrode, a concept utilized in pH meter technology.

A pH meter often consists of a voltmeter connected to a pair of electrodes immersed in the tested solution. One electrode is sensitive to the hydrogen ion concentration, acting similarly to the hydrogen electrode in our galvanic cell example. As the cell's potential varies with the hydrogen ion concentration, which is inversely related to pH (since \(pH = - \log [ H^+ ]\)), the voltmeter readings can directly correlate to the pH value of the solution.

To ensure precision, the pH meter is usually calibrated using standard buffer solutions with known pH values. This calibration allows for the correction of systematic errors, leading to more accurate and reliable measurements. Thus, a well-designed pH meter is capable of providing immediate, accurate assessments of solution acidity, which is indispensable in numerous scientific, industrial, and environmental applications.