Question

If \((\text { at } 298 \mathrm{K}) \mathrm{p} K_{\mathrm{b}}\) for \(\mathrm{NH}_{3}\) is \(4.75,\) show that \(\mathrm{p} K_{\mathrm{a}}\) for \(\left[\mathrm{NH}_{4}\right]^{+}\) is 9.25.

Short answer

pKa for \(\left[\mathrm{NH}_{4}^+\right]\) is 9.25.

Step-by-step solution

Understanding the Relationship between pKa and pKb

The relationship between the ionization constants of acids and bases is given by the equation: \[\mathrm{p} K_{a} + \mathrm{p} K_{b} = 14\]This equation is valid at 298 K (25°C) for conjugate acid-base pairs. Here, \(\mathrm{NH}_{3}\) and \(\left[\mathrm{NH}_{4}^+\right]\) are a conjugate base-acid pair.

Substituting Known Values

Given \(\mathrm{p} K_{b} = 4.75\) for \(\mathrm{NH}_{3}\), substitute this value into the equation to find \(\mathrm{p} K_{a}\):\[\mathrm{p} K_{a} + 4.75 = 14\]

Solving for pKa

Rearrange the equation to solve for \(\mathrm{p} K_{a}\):\[\mathrm{p} K_{a} = 14 - 4.75\]Calculate \(\mathrm{p} K_{a}\):\[\mathrm{p} K_{a} = 9.25\]

Conclusion

The \(\mathrm{p} K_{a}\) value calculated for \(\left[\mathrm{NH}_{4}^+\right]\) is 9.25, confirming the relationship between the conjugate acid-base pair. This demonstrates the consistent behavior of ionization constants.

Key concepts

Conjugate Acid-Base Pairs

In chemistry, understanding the concept of conjugate acid-base pairs is essential for analyzing reactions and determining the behavior of acids and bases. A conjugate acid-base pair consists of two species related by the loss or gain of a proton (\(\mathrm{H}^+\)). During an acid-base reaction, an acid donates a proton to a base, transforming into its conjugate base, while the base accepts a proton, becoming its conjugate acid.
For example, in the case of \(\mathrm{NH}_3\) (ammonia) and \(\left[\mathrm{NH}_4^+\right]\) (ammonium ion), \(\mathrm{NH}_3\) acts as a base by accepting a proton to form \(\left[\mathrm{NH}_4^+\right]\), which is its conjugate acid. Conversely, \(\left[\mathrm{NH}_4^+\right]\) can act as an acid by donating a proton to revert back to \(\mathrm{NH}_3\), thus functioning as a conjugate base. These pairs are key to understanding acid-base equilibrium and dynamics in various chemical processes.

Ionization Constants

Ionization constants are fundamental in the study of acid-base chemistry. They describe the extent to which an acid or a base ionizes in an aqueous solution. This is crucial in understanding the strength of acids and bases.
- **Ka and Kb values**: The acid ionization constant (\(K_a\)) measures the strength of an acid in solution, while the base ionization constant (\(K_b\)) quantifies a base's strength.- **Equilibrium expression**: For an acid \(HA\) dissociating into \(\mathrm{H}^+\) and \(\mathrm{A}^-\), \(K_a\) is expressed as \([\mathrm{H}^+][\mathrm{A}^-] / [HA]\). Similarly, for a base \(B\) ionizing to \([B][\mathrm{OH}^-] / [BOH]\), \(K_b\) is defined.- **pKa and pKb values**: These represent the negative logarithm of \(K_a\) and \(K_b\), simplifying the expression of ionization constants by converting them into more manageable values.
The relationship \(\mathrm{p} K_{a} + \mathrm{p} K_{b} = 14\) at 298 K illustrates the interdependence of acid and base strengths in conjugate systems.

pKb for NH3

Ammonia (\(\mathrm{NH}_3\)) is a well-known base found commonly in household products and industrial processes. Its basicity can be quantified using the base ionization constant \(pK_b\). For ammonia, \(pK_b\) is experimentally determined to be 4.75 at 298 K.
This value indicates ammonia's moderate strength as a base—neither a strong base like sodium hydroxide nor a weak one like pyridine. The \(pK_b\) value provides insight into ammonia's affinity to accept a proton, converting it into its conjugate acid, the ammonium ion \(\left[\mathrm{NH}_4^+\right]\). Understanding ammonia's basicity is important for predicting its behavior in chemical reactions, environmental systems, and industrial applications.

pKa for NH4+

The ammonium ion \(\left[\mathrm{NH}_4^+\right]\) is the conjugate acid of ammonia \(\mathrm{NH}_3\). To measure its acidity, the \(pK_a\) value is employed. For \(\left[\mathrm{NH}_4^+\right]\), the \(pK_a\) is calculated to be 9.25 at 298 K.
This \(pK_a\) value reflects the ammonium ion's capacity to donate a proton, reflecting its acid strength. The relationship between \(pK_a\) and \(pK_b\) of a conjugate acid-base pair highlights that the sum of these values equals 14, revealing a reciprocal relationship in their strengths.
The calculated \(pK_a\) of 9.25 for \(\left[\mathrm{NH}_4^+\right]\) emphasizes its moderate acidity, crucial for understanding its role in buffering systems, biological processes, and chemical reactions where it participates in proton exchange.