Consider two acids, HA and HX, with HA being a stronger acid than HX. a) Which acid has a larger value of \(K_{\mathrm{a}}\) ? b) Which conjugate base, \(\mathrm{A}^{-}\) or \(\mathrm{X}^{-}\), has a larger value of \(K_{\mathrm{b}}\) ? c) Provide a qualitative description of the relationship between the relative strength of an acid and the relative strength of its conjugate base. d) Assume that \(K_{\mathrm{a}}\) of \(\mathrm{HA}\) is \(1.0 \times 10^{-5}\) and \(K_{\mathrm{a}}\) of \(\mathrm{HX}\) is \(3.5 \times 10^{-7}\). Calculate the \(K_{\mathrm{b}}\) for \(\mathrm{A}^{-}\) and for \(\mathrm{X}^{-}\) and confirm that your answers to parts \(\mathrm{b}\) and \(\mathrm{c}\) are correct.

Acid strength refers to the propensity of a molecule to donate a proton in an aqueous solution. A strong acid is one that completely disassociates in water, releasing all of its protons. This high level of dissociation is reflected by a large value of the acid dissociation constant, denoted as \(K_a\). The stronger an acid, the more readily it donates a proton, leading to higher \(K_a\) values. Understanding acid strength is vital for predicting reaction outcomes in chemistry, as it influences the direction and equilibrium of acid-base reactions.

In the exercise, HA is considered a stronger acid than HX, which means that \(K_{\mathrm{a}}\) for HA will be larger than that of HX. This is observed as HA has a \(K_{\mathrm{a}}\) value of \(1.0 \times 10^{-5}\), which is greater than HX's \(K_{\mathrm{a}}\) value of \(3.5 \times 10^{-7}\).

The strength of a conjugate base is intimately connected to the strength of its parent acid. A conjugate base is the species that remains after an acid has donated a proton. Strong acids have weak conjugate bases because the proton donation is favored, leaving behind a species with little tendency to re-accept a proton. Conversely, a weak acid results in a relatively strong conjugate base because the incomplete transfer of protons in solution implies a higher capacity for the conjugate base to accept protons.

In our example, since HA is a stronger acid, its conjugate base, \(\mathrm{A}^{-}\), is a weaker base compared to \(\mathrm{X}^{-}\), the conjugate base of the weaker acid HX. This is why the exercise predicts that \(\mathrm{X}^{-}\) will have a larger \(K_{\mathrm{b}}\) value than \(\mathrm{A}^{-}\), as the inclination to accept protons is greater for the conjugate base of a weaker acid.

In the context of acid-base chemistry, the relationship between the acid dissociation constant (\(K_{\mathrm{a}}\)) and the base dissociation constant (\(K_{\mathrm{b}}\)) is a reflection of the inherent balance of acid-base pairs in water. The product of \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\) for a conjugate acid-base pair is constant and is equal to the ionic product of water, \(K_{\mathrm{w}}\). Mathematically, this relationship is represented as \(K_{\mathrm{w}} = K_{\mathrm{a}} \times K_{\mathrm{b}}\).

At 25 degrees Celsius, \(K_{\mathrm{w}}\) is always \(1.0 \times 10^{-14}\). This equilibrium constant represents the degree of self-ionization of water and sets a fixed point of reference for determining the strength of acids and bases. A strong acid typically has a high \(K_{\mathrm{a}}\) and, correspondingly, its conjugate base has a low \(K_{\mathrm{b}}\), which maintains the balance as required by the \(K_{\mathrm{w}}\) value.

To calculate the base dissociation constant \(K_{\mathrm{b}}\) for a conjugate base, we use the relationship highlighted previously: \(K_{\mathrm{w}} = K_{\mathrm{a}} \times K_{\mathrm{b}}\). Solving for \(K_{\mathrm{b}}\), we have \(K_{\mathrm{b}} = \frac{K_{\mathrm{w}}}{K_{\mathrm{a}}}\).

Let's apply this formula to the exercise provided. Since we know \(K_{\mathrm{a}}\) for HA is \(1.0 \times 10^{-5}\), we can calculate \(K_{\mathrm{b}}\) for \(\mathrm{A}^{-}\) as follows: \(K_{\mathrm{b}}\) for \(\mathrm{A}^{-}\) equals \(\frac{1.0 \times 10^{-14}}{1.0 \times 10^{-5}} = 1.0 \times 10^{-9}\). Similarly, for HX with \(K_{\mathrm{a}} = 3.5 \times 10^{-7}\), the \(K_{\mathrm{b}}\) for \(\mathrm{X}^{-}\) is \(2.857 \times 10^{-8}\), which confirms the earlier conclusion that the conjugate base of the weaker acid (HX) has a larger \(K_{\mathrm{b}}\) value, meaning it is a stronger base than \(\mathrm{A}^{-}\).