Question

(a) Given that \(K_{b}\) for ammonia is \(1.8 \times 10^{-5}\) and that for hydroxylamine is \(1.1 \times 10^{-8}\), which is the stronger base? (b) Which is the stronger acid, the ammonium ion or the hydroxylammonium ion? (c) Calculate \(K_{a}\) values for \(\mathrm{NH}_{4}^{+}\) and \(\mathrm{H}_{3} \mathrm{NOH}^{+}\).

Short answer

(a) Ammonia is the stronger base since its \(K_b = 1.8 \times 10^{-5}\) is greater than the \(K_b\) of hydroxylamine (\(1.1 \times 10^{-8}\)). (b) The hydroxylammonium ion is the stronger acid because it is the conjugate acid of the weaker base, hydroxylamine. (c) The \(K_a\) values for \(\mathrm{NH}_{4}^{+}\) and \(\mathrm{H}_{3}\mathrm{NOH}^{+}\) are \(5.56 \times 10^{-10}\) and \(9.09 \times 10^{-7}\), respectively.

Step-by-step solution

(Step 1): Determine the stronger base

The \(K_b\) values are given for ammonia and hydroxylamine. A higher \(K_b\) value indicates a stronger base. - Ammonia, \(K_b = 1.8 \times 10^{-5}\) - Hydroxylamine, \(K_b = 1.1 \times 10^{-8}\) Since the \(K_b\) value of ammonia is greater than that of hydroxylamine, ammonia is the stronger base.

(Step 2): Determine the stronger acid

The conjugate acids of the bases in question are the ammonium ion (\(\mathrm{NH}_{4}^{+}\)) and the hydroxylammonium ion (\(\mathrm{H}_{3}\mathrm{NOH}^{+}\)). The stronger base has a weaker conjugate acid. Since ammonia is the stronger base, its conjugate acid, the ammonium ion, will be the weaker acid. That means the hydroxylammonium ion is the stronger acid.

(Step 3): Calculate the \(K_a\) values

To obtain the \(K_a\) values, we will first need the equilibrium constant for water (\(K_w\)), which is \(1.0 \times 10^{-14}\) at 25°C. We can use the formula $$K_a \times K_b = K_w$$ to calculate \(K_a\) values for each ion. For the ammonium ion, \(K_b\) = 1.8 x 10^{-5}: $$K_a \times (1.8 \times 10^{-5}) = 1.0 \times 10^{-14}$$ $$K_a = \frac{1.0 \times 10^{-14}}{1.8 \times 10^{-5}}$$ $$K_a = 5.56 \times 10^{-10}$$ For the hydroxylammonium ion, \(K_b\) = 1.1 x 10^{-8}: $$K_a \times (1.1 \times 10^{-8}) = 1.0 \times 10^{-14}$$ $$K_a = \frac{1.0 \times 10^{-14}}{1.1 \times 10^{-8}}$$ $$K_a = 9.09 \times 10^{-7}$$ So, the \(K_a\) values for \(\mathrm{NH}_{4}^{+}\) and \(\mathrm{H}_{3}\mathrm{NOH}^{+}\) are: - \(K_a (\mathrm{NH}_4^+) = 5.56 \times 10^{-10}\) - \(K_a (\mathrm{H}_{3}\mathrm{NOH}^{+}) = 9.09 \times 10^{-7}\)

Key concepts

Ammonia

Ammonia, a compound composed of nitrogen and hydrogen with the formula \(\text{NH}_3\), is a well-known weak base. It has a characteristic pungent odor and is commonly found in household and industrial cleaning products. Ammonia's basicity stems from its ability to accept a proton (\(\text{H}^+\)) due to the presence of a lone pair of electrons on the nitrogen atom.
In the context of acid-base equilibrium, ammonia has a base dissociation constant \(K_b\) of \(1.8 \times 10^{-5}\). This value indicates how readily ammonia accepts protons from water to form its conjugate acid, the ammonium ion (\(\text{NH}_4^+\)). The higher the \(K_b\), the stronger the base, which signifies ammonia's relatively higher strength as a base compared to hydroxylamine.
When ammonia dissolves in water, it forms a dynamic equilibrium described by the reaction:

• \(\text{NH}_3 + \text{H}_2\text{O} \rightleftharpoons \text{NH}_4^+ + \text{OH}^-\)

This reaction showcases ammonia's ability to form hydroxide ions (\(\text{OH}^-\)), responsible for its basic nature. Its interactions in aqueous solutions are crucial for various applications, from fertilizers to refrigeration systems.

Hydroxylamine

Hydroxylamine (\(\text{H}_3\text{NO}\)) is a compound similar to ammonia but with an additional oxygen atom. It is used in various chemical synthesis processes and as a stabilizing agent. Despite its structural similarity to ammonia, hydroxylamine has different chemical properties, which include being a weaker base. This is reflected in its much lower base dissociation constant \(K_b\) of \(1.1 \times 10^{-8}\).
The weak basic nature of hydroxylamine arises because its extra chemical structure can interfere with its ability to accept protons as effectively as ammonia does. The equilibrium reaction in water can be represented as:

• \(\text{H}_3\text{NO} + \text{H}_2\text{O} \rightleftharpoons \text{H}_3\text{NOH}^+ + \text{OH}^-\)

Here, hydroxylamine also forms hydroxide ions (\(\text{OH}^-\)) and its corresponding conjugate acid, the hydroxylammonium ion (\(\text{H}_3\text{NOH}^+\)).
Although both ammonia and hydroxylamine can participate in proton transfer reactions, the lower \(K_b\) value of hydroxylamine suggests it is less efficient in accepting protons, making it a weaker base. This property makes hydroxylamine useful in specific applications where a lower pH increase is desired.

Equilibrium Constant

The equilibrium constant \(K\) is a fundamental concept in chemistry that determines the extent to which a chemical reaction proceeds. For acid-base reactions, specific constants such as \(K_a\) and \(K_b\) are used to describe the strength of acids and bases, respectively.
In this exercise, the equilibrium constants for ammonia and hydroxylamine illustrate their relative strengths as bases. The higher \(K_b\) value of ammonia means it is a stronger base than hydroxylamine. Conversely, when you calculate the acid dissociation constants \(K_a\) of their conjugate acids (ammonium and hydroxylammonium ions), the stronger base has a weaker conjugate acid. This relationship is critical for understanding acid-base equilibria.
You can determine \(K_a\) from \(K_b\) through the formula:

• \(K_a \times K_b = K_w\), where \(K_w = 1.0 \times 10^{-14}\) at 25°C

By rearranging this formula, you find the \(K_a\) values. For the ammonium ion \(\text{NH}_4^+\), the calculated \(K_a\) is \(5.56 \times 10^{-10}\), whereas for the hydroxylammonium ion \(\text{H}_3\text{NOH}^+\), it is \(9.09 \times 10^{-7}\).
These calculations elucidate how an understanding of \(K\) values informs the behavior and comparative strength of acids and bases in solution. They also highlight the importance of equilibrium constants in predicting the direction and extent of chemical reactions.