When a pH meter was standardized with a boric acid-borate buffer with a pH of \(9.40\), the cell was \(+0.060 \mathrm{~V}\). When the buffer was replaced with a solution of unknown hydronium ion concentration, the cell potential was \(+0.22 \mathrm{~V}\). What is the \(\mathrm{pH}\) of the solution?

The Nernst equation is a fundamental principle in electrochemistry and plays an essential role in calculating the potential of an electrochemical cell, which in turn can give us insights into the concentration of ions within the solution. Put simply, it connects the measurable cell potential, the standard cell potential, and the ion concentration.

For the purpose of measuring pH, the Nernst equation is often reformulated to relate the cell potential with the concentration of hydronium ions, which is directly tied to pH levels. It assumes the following form for pH determination:
\[E = E^\circ - \frac{0.05916}{n} \cdot pH\]
In this equation, \(E\) represents the cell potential, \(E^\circ\) is the standard cell potential, and \(n\) is the number of electrons involved in the reaction. The constant 0.05916 V is specific to room temperature, reflecting the temperature coefficient at 25°C. For our calculations, since hydronium ions (\(H_3O^+\)) are involved, \(n\) is equal to 1.

Cell potential, represented as \(E\) in the Nernst equation, refers to the electrical potential difference between two electrodes of an electrochemical cell. It is a measure of the driving force behind the electrochemical reaction, often in volts (V). The cell potential can be directly measured using instruments like a voltmeter.

This value gives an immediate sense of how energetically favorable a reaction is. With relation to pH determination, a pH meter actually measures the cell potential and, through the Nernst equation, it enables us to deduce the pH of the solution in question. In the exercise, the change in cell potential from a known buffer solution to an unknown provides the basis for calculating the pH of the latter.

Hydronium ion concentration defines the number of hydronium ions, symbolized as \(H_3O^+\), in a solution. It is directly related to the acidity of a solution; the more hydronium ions present, the lower the pH and thus the more acidic the solution.

In the context of the exercise, knowing the concentration of hydronium ions allows calculation of pH, which is the negative logarithm of the hydronium ion concentration. This relationship is where the Nernst equation comes into play, allowing us to deduce the unknown concentration from the measured cell potential. The equation's power lies in translating voltage readings into meaningful chemical information.

The standard cell potential \(E^\circ\) is a pivotal part of the Nernst equation; it's a reference point used to predict the cell potential under standard conditions. It's defined as the potential difference between two half-cells when all reactants and products are at 1M concentration, the gas pressure is at 1 atm, and the temperature is 25°C (298K).

In real-world applications, such as pH meters, \(E^\circ\) can be experimentally determined using a solution with a known pH, as was performed in the buffer step of the exercise. By applying this known value, you can find the pH of other solutions with unknown hydronium ion concentrations. Understanding and correctly using \(E^\circ\) is crucial for accurate pH calculations using the Nernst equation.