Question

An aqueous solution initially contains \(0.01 \mathrm{M}-\mathrm{RNH}_{2}\left(K_{\mathrm{b}}=2.0 \times 10^{-6}\right)\) and \(10^{-4} \mathrm{M}-\mathrm{NaOH}\). The final concentration of \(\mathrm{OH}^{-}\) in the solution is about (a) \(10^{-4} \mathrm{M}\) (b) \(2.0 \times 10^{-4} \mathrm{M}\) (c) \(3.0 \times 10^{-4} \mathrm{M}\) (d) \(1.414 \times 10^{-4} \mathrm{M}\)

Short answer

\(10^{-4} \mathrm{M}\)

Step-by-step solution

Determine OH- Contribution from NaOH

Since NaOH is a strong base, it will dissociate completely in aqueous solution. The initial concentration of OH- from NaOH is equal to the initial concentration of NaOH, which is given as \(10^{-4} \mathrm{M}\).

Calculate OH- from RNH2

The weak base \(RNH_2\) will partially dissociate in water according to the equilibrium \(RNH_2 + H_2O \rightleftharpoons RNH_3^+ + OH^-\). The equilibrium expression is given by \(K_b = \frac{[RNH_3^+] [OH^-]}{[RNH_2]}\). However, since the contribution to the concentration of OH- from RNH2 is quite small compared to that from NaOH (due to the relatively small Kb and the larger amount of NaOH), it can be neglected in this approximation.

Calculate Final OH- Concentration

The final concentration of OH- is approximately the sum of the contributions from NaOH and \(RNH_2\). But since the contribution from \(RNH_2\) is negligible, the concentration of OH- in the solution is approximately equal to the concentration of OH- from NaOH. Therefore, the answer is close to \(10^{-4} \mathrm{M}\), which is choice (a).

Key concepts

Weak Base Equilibrium

Understanding weak base equilibrium is crucial in predicting the behavior of weak bases (such as amines) in water. A weak base like RNH2 does not fully ionize in solution; instead, it sets up an equilibrium where only a fraction of the base reacts with water to form its conjugate acid (RNH3+) and hydroxide ions (OH-).

The position of equilibrium is quantified by the base dissociation constant, Kb, which is an indicator of the base's strength. The smaller the value of Kb, the less the base ionizes, and the weaker the base is. In the given problem, RNH2 has a Kb of 2.0 x 10^-6, suggesting it's quite weak, thus contributing a negligible amount of OH- compared to the strong base NaOH.

When working with weak base equilibrium problems, the key is to set up an ICE table (Initial, Change, Equilibrium) in order to determine the concentrations of all species at equilibrium. However, in this specific case, due to the larger amount of strong base (NaOH) present, the equilibrium calculation for RNH2 can be skipped, simplifying our work.

Acid-Base Titration

Acid-base titration is an experimental procedure used to determine the concentration of an unknown acid or base by reacting it with a known concentration of a base or acid respectively. It involves the gradual addition of one solution to another until their reaction is complete, usually indicated by a color change of an indicator or by reaching a particular pH value.

In titrations involving weak bases, the pH at the equivalence point is greater than 7 due to the formation of a weak conjugate acid, which doesn't significantly lower the pH. The process is carefully controlled and monitored because the point at which all the weak base has reacted (the equivalence point) can involve delicate changes in pH.

While the problem at hand isn't a titration exercise, titration principles can be applied to understand how bases react with acids or water and how the equivalence point might be determined in a more complex scenario. For educational purposes, practicing titration calculations fortifies the understanding of reaction stoichiometry and equilibrium concepts.

Ionic Dissociation

Ionic dissociation is the process by which an ionic compound separates into its constituent ions when it dissolves in water. This process is crucial for conducting electricity in solution and for the various reactions that take place in aqueous solutions.

NaOH is an example of a compound that undergoes complete ionic dissociation in water, contributing a fixed concentration of hydroxide ions equal to its initial concentration. In contrast, weak electrolytes, such as weak acids and bases, dissociate only partially, leading to an equilibrium between the undissociated molecules and the ions produced.

The degree of dissociation is influenced by factors such as the temperature, the nature of the solute, and the solvent. In this example, we contrast the complete dissociation of a strong base, NaOH, with the partial dissociation of a weak base, RNH2, and see how this difference simplifies solving equilibrium problems when multiple solutes are present.