The ionization constant for water \(\left(K_{w}\right)\) is \(2.9 \times 10^{-14}\) at \(40^{\circ} \mathrm{C}\). Calculate \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right],\left[\mathrm{OH}^{-}\right], \mathrm{pH},\) and \(\mathrm{pOH}\) for pure water at \(40^{\circ} \mathrm{C}\).

The pH of a solution is a numerical scale used to specify how acidic or basic (alkaline) a water-based solution is. It is defined by the negative logarithm of the hydrogen ion concentration \( \text{pH} = -\log([H_3O^+]) \).

In the exercise, we have a scenario where the ionization constant for water at 40°C is given, and we need to compute the pH of pure water at this temperature. Because the ionization of water results in equal concentrations of hydrogen and hydroxide ions in pure water, the process of calculating pH simplifies as it solely depends on the hydrogen ion concentration, which can be derived from the ionization constant.

In a similar manner to pH, the pOH of a solution measures the alkalinity, but it is defined by the negative logarithm of the hydroxide ion concentration \( \text{pOH} = -\log([OH^-]) \).

For calculations, it is crucial to understand that in pure water at a given temperature, the product of pH and pOH equals 14 (at 25°C, but this varies with temperature), and they are inversely related. In our exercise, because the concentrations of hydrogen and hydroxide ions are equal, the pOH calculation is straightforward. By finding the negative logarithm of the hydroxide ion concentration, we reach the pOH value.

Chemical equilibrium is the state reached when the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products. The equilibrium constant \( K \) is a ratio that expresses the relative concentrations of products to reactants at equilibrium.

In the context of the autoionization of water, \(K_w \) is the equilibrium constant representing the ionization constant for water. It provides insights into the concentrations of the ionic species in water at a given temperature and is critical for calculating pH and pOH as well. Understanding \( K_w \) is vital for predicting the behavior of aqueous solutions in a variety of chemical scenarios.

Autoionization, or self-ionization, of water refers to the reaction where water molecules react with each other to form hydronium ions \( H_3O^+ \) and hydroxide ions \( OH^- \). This process is represented by the equation \( 2H_2O \rightleftharpoons H_3O^+ + OH^- \).

At equilibrium, the concentrations of \( H_3O^+ \) and \( OH^- \) ions are determined by the ionization constant for water \( K_w \), which is temperature-dependent. The autoionization process is an intrinsically balanced reaction where the concentrations of the ions in pure water remain consistent at a given temperature, making this phenomenon critical to understanding pH, pOH, and the concept of neutrality in aqueous solutions.