Consider the reaction below at \(25^{\circ} \mathrm{C}\) : $$ 3 \mathrm{SO}_{4}{ }^{2-}(a q)+12 \mathrm{H}^{+}(a q)+2 \mathrm{Cr}(s) \longrightarrow 3 \mathrm{SO}_{2}(g)+2 \mathrm{Cr}^{3+}(a q)+6 \mathrm{H}_{2} \mathrm{O} $$ Use Table \(18.1\) to answer the following questions. Support your answers with calculations. (a) Is the reaction spontaneous at standard conditions? (b) Is the reaction spontaneous at a \(\mathrm{pH}\) of \(3.00\) with all other ionic species at \(0.100 \mathrm{M}\) and gases at \(1.00\) atm? (c) Is the reaction spontaneous at a pH of \(8.00\) with all other ionic species at \(0.100 \mathrm{M}\) and gases at \(1.00 \mathrm{~atm}\) ? (d) At what \(\mathrm{pH}\) is the reaction at equilibrium with all other ionic species at \(0.100 \mathrm{M}\) and gases at \(1.00 \mathrm{~atm}\) ?

Understanding the standard cell potential is crucial for assessing the spontaneity of a chemical reaction under standard conditions, which include solute concentrations at 1 M, a pressure of 1 atm for gases, and a temperature of 25°C or 298 K. This potential, denoted as \(E^\text{\textdegree}_{cell}\), is a measure of the electromotive force of an electrochemical cell when all reagents are at their standard states.

To calculate \(E^\text{\textdegree}_{cell}\), one must recognize that it is the difference between the standard reduction potentials of the cathode and the anode. The standard reduction potential is a measure of how strongly a species gains electrons, with a more positive value indicating a greater tendency to be reduced. In the example given in the exercise, the reaction between sulfate ions, protons, and solid chromium to produce sulfur dioxide, trivalent chromium ions, and water has a \(E^\text{\textdegree}_{cell}\) calculated by combining the reduction potentials of each half-reaction. Since a negative \(E^\text{\textdegree}_{cell}\) suggests the reaction is non-spontaneous at standard conditions, understanding and calculating this value is the key step in predicting the behavior of the reaction without carrying it out.

While standard cell potential offers insight at standard conditions, the Nernst equation allows us to predict cell potential under non-standard conditions by accounting for temperature, concentration, and partial pressures of the reactants and products involved in a chemical reaction. The Nernst equation is expressed as:
\[ E_{cell} = E^\text{\textdegree}_{cell} - \frac{0.0592}{n} \times \text{log}Q \]
Here, \(E_{cell}\) represents the actual cell potential, \(E^\text{\textdegree}_{cell}\) is the standard cell potential, \(n\) is the number of moles of electrons exchanged in the redox reaction, and \(Q\) is the reaction quotient reflecting the ratio of the products' concentrations to the reactants' concentrations, raised to their stoichiometric coefficients. The Factor 0.0592 arises from combining constants from the gas constant, Faraday's constant, and the standard temperature (298 K). Applying the Nernst Equation makes it possible to determine the cell potential, and thus the spontaneity of a reaction, at any given pH, allowing us to peer into conditions far from standard.

The reaction quotient, \(Q\), plays a pivotal role in understanding chemical equilibria and predicting the direction of a reaction. It is calculated in a similar way to the equilibrium constant, \(K\), but with the current concentrations of the reactants and products, not necessarily those at equilibrium. The reaction quotient is given by the expression:
\[ Q = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
where square brackets denote molar concentrations, and the lowercase letters represent stoichiometric coefficients. When evaluating spontaneity at various pH levels, the hydrogen ion concentration, \([H^+]\), changes, directly influencing \(Q\) since it's part of the reaction in our example. If \(Q < K\), the reaction will proceed forward to reach equilibrium, but if \(Q > K\), the reaction will progress in the reverse direction. Therefore, by calculating the reaction quotient, we can use the Nernst Equation to determine whether a reaction will be spontaneous under varied conditions.

The concept of equilibrium pH is essential when considering reactions involving acid-base equilibria, as in the dissolution or formation of ions in water. It is the pH at which a chemical reaction achieves equilibrium, meaning the rate of the forward reaction equals that of the reverse reaction, and the net change of reactants and products is zero. At this specific pH, the reaction quotient \(Q\) equals the equilibrium constant \(K\) and the cell potential \(E_{cell}\) is zero. This is important in electrochemical reactions, as it determines the pH at which there's no driving force for the reaction to occur in either direction. As seen in the exercise, finding the equilibrium pH involves manipulating the Nernst equation to solve for the pH that satisfies this condition. This calculation is critical for understanding the acidity or basicity conditions needed for a reaction to stay at equilibrium and is valuable for process optimization in industrial applications and environmental predictions.