Determine \(K_{\mathrm{b}}\) for the nitrite ion, \(\mathrm{NO}_{2}^{-}\). In a \(0.10-M\) solution this base is \(0.0015 \%\) ionized.

When studying solutions, understanding the concept of percent ionization is crucial. Percent ionization refers to the fraction of the original solute that forms ions in a solution. It’s particularly important in the context of weak acids and bases, where complete dissociation does not occur.
To quantify percent ionization, you take the ratio of the concentration of ionized solute to the original concentration of the solute, then multiply by 100 to get a percentage. For the nitrite ion, \(\mathrm{NO}_{2}^{-}\), in a 0.10 M solution with a 0.0015% ionization, most of the \(\mathrm{NO}_{2}^{-}\) remains undissociated. The small percentage that does ionize reflects the weak base's ability to dissociate in water.In textbook exercises, it’s essential to convert the percentage to a decimal before using it in further calculations. This step is a crucial foundation for accurately determining the base dissociation constant (\(K_{\mathrm{b}}\)).

The equilibrium expression, an integral part of chemical equilibrium, is written for reversible reactions where reactants and products are in dynamic balance. In the case of the nitrite ion dissociating in water, the equilibrium expression is tied to the base dissociation constant, \(K_{\mathrm{b}}\).
For the dissociation of \(\mathrm{NO}_{2}^{-}\) in water, we establish \(K_{\mathrm{b}}\) as \[K_{\mathrm{b}} = \frac{{[NO_2][OH^-]}}{{[NO_2^-]}}\] where the concentrations are those at equilibrium. Here, the square brackets indicate the molar concentration of each species.In our example, we're interested in the values of these concentrations after dissociation has reached equilibrium. For weak bases, the calculation of \(K_{\mathrm{b}}\) provides insight into the strength of the base and its tendency to dissociate.

Determining the ion concentration at equilibrium is essential for calculating the dissociation constant. Knowing the percent ionization, we find the concentration of hydroxide ions \(\left[ OH^- \right]\) at equilibrium by using the initial concentration of the weak base and the converted percent ionization value.
Assuming that the dissociation of \(\mathrm{NO}_{2}^{-}\) into \(\mathrm{NO}_{2}\) and \(OH^-\) increases their concentrations by \(x\), and knowing dissociation is a small extent, the concentration of undissociated \(\mathrm{NO}_{2}^{-}\) at equilibrium can be approximated as the initial concentration.In practice, due to the low percent ionization, the change in \(\mathrm{NO}_{2}^{-}\) concentration is minimal, confirming that \(x\) is insignificant relative to the initial concentration. Calculating ion concentrations at equilibrium correctly is paramount for assessing the extent of ionization and, ultimately, solving for \(K_{\mathrm{b}}\) of the substance.