Question

Which, if any, of the following statements are true? (a) The stronger the base, the smaller the \(\mathrm{p} K_{b}\). (b) The stronger the base, the larger the \(\mathrm{p} K_{b}\). (c) The stronger the base, the smaller the \(K_{b}\). (d) The stronger the base, the larger the \(K_{b}\). (e) The stronger the base, the smaller the \(\mathrm{p} K_{a}\) of its conjugate acid. (f) The stronger the base, the larger the \(\mathrm{p} K_{a}\) of its conjugate acid.

Short answer

Statements (a), (d), and (e) are true. Stronger bases have smaller \(\mathrm{p} K_{b}\) values, larger \(K_b\) values, and smaller \(\mathrm{p} K_{a}\) values of their conjugate acids.

Step-by-step solution

Statement (a)

The stronger the base, the smaller the \(\mathrm{p} K_{b}\). This statement is true. Strong bases have higher \(K_b\) values, meaning their \(\mathrm{p} K_{b}\) values will be smaller because \(\mathrm{p} K_{b} = -\log(K_b)\).

Statement (b)

The stronger the base, the larger the \(\mathrm{p} K_{b}\). This statement is false. As already stated in statement (a), stronger bases have smaller \(\mathrm{p} K_{b}\) values.

Statement (c)

The stronger the base, the smaller the \(K_{b}\). This statement is false. Strong bases have larger \(K_b\) values, implying more effective ionization.

Statement (d)

The stronger the base, the larger the \(K_{b}\). This statement is true. As already stated in statement (c), strong bases have larger \(K_b\) values.

Statement (e)

The stronger the base, the smaller the \(\mathrm{p} K_{a}\) of its conjugate acid. This statement is true. A strong base will have a weak conjugate acid. Weak acids have smaller \(K_a\) values, meaning their \(\mathrm{p} K_{a}\) values will be larger. Therefore, the stronger the base, the smaller the \(\mathrm{p} K_{a}\) of its conjugate acid.

Statement (f)

The stronger the base, the larger the \(\mathrm{p} K_{a}\) of its conjugate acid. This statement is false. As already stated in statement (e), a stronger base has a weaker conjugate acid. The weaker the acid, the smaller the \(K_a\) value, and the larger the \(\mathrm{p} K_{a}\). So, the stronger the base, the smaller the \(\mathrm{p} K_{a}\) of its conjugate acid, not larger as stated in this statement. In conclusion, statements (a), (d), and (e) are true, while the others are false.

Key concepts

pKb

In the realm of acid-base chemistry, understanding \(\mathrm{p} K_{b}\) is crucial, as it gives insight into the strength of bases. \(\mathrm{p} K_{b}\) is the negative logarithm of the base dissociation constant, \(K_b\). The formula linking them is \(\mathrm{p} K_{b} = -\log(K_b)\).
This relationship indicates that the smaller the \(\mathrm{p} K_{b}\) value, the stronger the base becomes. This is because a smaller \(\mathrm{p} K_{b}\) corresponds to a larger \(K_b\), showing a greater extent of ionization in water.
Strong bases dissociate more completely in solution, meaning they release more hydroxide ions (OH-).

• A base with a small \(\mathrm{p} K_{b}\) value like 1.75 is considered strong.
• A base with a large \(\mathrm{p} K_{b}\) value like 11.25 is considered weak.

Understanding these relationships helps predict base behavior in chemical reactions.

Kb values

The base dissociation constant, \(K_b\), acts as a yardstick for the strength of a base. It measures the extent to which a base can donate hydroxide ions \(\mathrm{(OH^-)}\) in a solution.
The larger the \(K_b\) value, the stronger the base. This means that in an aqueous solution, a stronger base will dissociate more completely to release more OH-.
For example, if a base has a \(K_b\) value of \(1.0 \times 10^{-1}\), it is stronger than a base with a \(K_b\) of \(1.0 \times 10^{-5}\).

• Strong bases such as hydroxide ions from \(\mathrm{NaOH}\) have relatively high \(K_b\) values.
• Weaker bases like ammonia (\(\mathrm{NH_3}\)) have smaller \(K_b\) values.

By comparing \(K_b\) values, chemists can predict how effectively different bases will behave in various reactions and guide the formulation of reactions involving bases.

Conjugate Acids

Within acid-base chemistry, conjugate acids are the species formed when a base accepts a proton. The strength of a conjugate acid is crucially related to the strength of the original base.
When a base is strong, it partially dissociates, resulting in a weak conjugate acid. This is because the equilibrium shifts to favor the base, making the conjugate acid less likely to donate protons.
The inverse is also true: a strong acid has a weak conjugate base.

• Weak bases result in strong conjugate acids, reflecting minimal ionization.
• The \(\mathrm{p} K_{a}\) value of the conjugate acid can be large if the original base is strong, indicating the conjugate acid’s weaker nature.

This relationship between bases and their conjugate acids is key for understanding acid-base reactions and calculating the potential outcomes and equilibrium positions of these reactions.