Question

Although the acid-dissociation constant for phenol $\left({\mathrm{C}}_6{\mathrm{H}}_5\mathrm{OH}\right)$ is listed in Appendix $\mathrm{D}$, the base - $\mathrm{d}$ issociation constant for the phenolate ion $\left({\mathrm{C}}_6{\mathrm{H}}_5\mathrm{O}\right)$ is not. (a) Explain why it is not necessary to list both $K_a$ for phenol and $K_b$ for the phenolate ion. (b) Calculate $K_b$ for the phenolate ion. (c) Is the phenolate ion a weaker or stronger base than ammoria?

Short answer

The base-dissociation constant (Kb) for the phenolate ion can be calculated using the relationship between Ka and Kb: $K_b=\frac{K_w}{K_a}$. Given the Ka value for phenol as $1.3\times {10}^{-10}$, the Kb value for the phenolate ion is approximately $7.7\times {10}^{-5}$. Since this is greater than the Kb value for ammonia ($1.8\times {10}^{-5}$), the phenolate ion is a stronger base than ammonia.

Step-by-step solution

Understand the relationship between Ka and Kb

For any conjugate acid-base pair, the following relation holds: $$K_a\times K_b=K_w$$ where Kw is the ion-product constant of water, which is equal to $1.0\times {10}^{-14}$ at 25°C. In this case, the conjugate acid is phenol (C6H5OH) and the conjugate base is the phenolate ion (C6H5O-).

Determine Ka of phenol

We are given that the acid-dissociation constant (Ka) for phenol is listed in Appendix D. Look up the value of Ka for phenol, which should be: $K_a=1.3\times {10}^{-10}$

Calculate Kb for the phenolate ion

Using the relationship between Ka and Kb, we can calculate the base-dissociation constant (Kb) for the phenolate ion: $$K_b=\frac{K_w}{K_a}$$ Plug in the values of Kw and Ka: $K_b=\frac{1.0\times {10}^{-14}}{1.3\times {10}^{-10}}$ Now, simplify and calculate the value of Kb: $K_b\approx 7.7\times {10}^{-5}$

Compare the basicity of phenolate ion to ammonia

To determine whether the phenolate ion is a weaker or stronger base than ammonia, we need to compare their Kb values. The Kb value for ammonia is: $K_b(\text{ammonia})=1.8\times {10}^{-5}$ Since the Kb value for the phenolate ion ($7.7\times {10}^{-5}$) is greater than the Kb value for ammonia ($1.8\times {10}^{-5}$), the phenolate ion is a stronger base than ammonia.

Key concepts

Acid-Dissociation Constant (Ka)

Understanding the acid-dissociation constant, often represented as Ka, is crucial for grasping acid-base equilibrium. Ka measures the strength of an acid, quantifying its tendency to donate a proton to a base, consequently forming its conjugate base. This is presented in the equilibrium equation for the dissociation of an acid (HA) in water:
$$HA(aq)+H_{2}O(l)\rightleftharpoons H_3{O}^{+}(aq)+A^{-}(aq)$$
Where 'HA' represents the acid, 'H3O+' the hydronium ion, and 'A-' the conjugate base of the acid. A higher Ka value indicates a stronger acid, implying a greater proportion of acid molecules donating protons. Thus, when given the Ka, one can predict the degree of ionization of an acid in a solution.

Base-Dissociation Constant (Kb)

The base-dissociation constant, denoted as Kb, is synonymous to Ka but for bases. Kb assesses a base's strength by its propensity to accept a proton from water, forming its conjugate acid. The base-dissociation equilibrium can be represented as follows:
$$B(aq)+H_{2}O(l)\rightleftharpoons B{H}^{+}(aq)+O{H}^{-}(aq)$$
Here, 'B' stands for the base, 'BH+' the conjugate acid formed, and 'OH-' the hydroxide ion. Just as with acids, the larger the value of Kb, the stronger the base, indicating a higher concentration of OH- ions in the solution. Knowledge of Kb is valuable, as it helps students predict the outcome of reactions involving bases.

Conjugate Acid-Base Pairs

Diving deeper into the acid-base relationships brings us to conjugate acid-base pairs. When an acid donates a proton, it forms a conjugate base, while the acceptance of a proton by a base forms its conjugate acid. These two are known as a conjugate acid-base pair and are linked by the reverse nature of their formation.
For example, if we have acetic acid (CH3COOH) reacting with water, acetic acid (acid) donates a proton to form acetate (CH3COO-) (conjugate base). As they are directly related, understanding one member of the pair provides insight into the properties of the other.
This concept is important as it underpins the calculation of Kb from Ka (and vice versa) through the ion-product constant of water, Kw. This concept is fundamental as it ensures that the list of either Ka or Kb values in textbooks is sufficient to determine both the acidic and basic properties of a substance.

Ion-Product Constant of Water (Kw)

Lastly, the ion-product constant of water, Kw, is a fundamental concept in understanding acid-base chemistry. This constant represents the equilibrium constant for the self-ionization of water:
$$H_{2}O(l)+H_{2}O(l)\rightleftharpoons H_3{O}^{+}(aq)+O{H}^{-}(aq)$$
Where Kw is the product of the concentrations of the hydronium ions, [H3O+], and hydroxide ions, [OH-], at equilibrium:
$$K_w=[H_3{O}^{+}][O{H}^{-}]$$
At 25°C, Kw is always equal to $1.0\times {10}^{-14}$. This constant is fundamental because it provides the link between the acid-dissociation constant (Ka) and the base-dissociation constant (Kb) across conjugate pairs. Recognizing that the product of Ka and Kb for a conjugate pair equals Kw allows one to calculate the Kb of a base if the Ka of its conjugate acid is known, and vice versa.