Question

Given that the $K_{\mathrm{a}}$ value for acetic acid is $1.8\times {10}^{-5}$ and the $K_{\mathrm{a}}$ value for hypochlorous acid is $3.5\times {10}^{-8},$ which is the stronger base, ${\mathrm{OCl}}^{-}$ or ${\mathrm{C}}_2{\mathrm{H}}_3{\mathrm{O}}_2^{-}?$

Short answer

Given that the $K_{\mathrm{a}}$ values for acetic acid and hypochlorous acid are $1.8\times {10}^{-5}$ and $3.5\times {10}^{-8}$, respectively, we can calculate the $K_{\mathrm{b}}$ values for their conjugate bases, ${\mathrm{OCl}}^{-}$ and ${\mathrm{C}}_2{\mathrm{H}}_3{\mathrm{O}}_2^{-}$, using the relationship $K_{\mathrm{a}}\cdot K_{\mathrm{b}}=K_{\mathrm{w}}$. The calculated $K_{\mathrm{b}}$ values are $2.86\times {10}^{-7}$ for ${\mathrm{OCl}}^{-}$ and $5.56\times {10}^{-10}$ for ${\mathrm{C}}_2{\mathrm{H}}_3{\mathrm{O}}_2^{-}$. Since the $K_{\mathrm{b}}$ value of ${\mathrm{OCl}}^{-}$ is larger than the $K_{\mathrm{b}}$ value of ${\mathrm{C}}_2{\mathrm{H}}_3{\mathrm{O}}_2^{-}$, ${\mathrm{OCl}}^{-}$ is the stronger base.

Step-by-step solution

Calculate the Kb values of the conjugate bases of acetic acid and hypochlorous acid.

In order to determine the stronger base, we first need to calculate the Kb values for OCl- and C2H3O2-. We can do this using the relationship between Ka, Kb, and Kw: $K_{\mathrm{a}}\cdot K_{\mathrm{b}}=K_{\mathrm{w}}$ where Ka is the acid dissociation constant, Kb is the base dissociation constant, and Kw is the ion product of water, which is equal to $1.0\times {10}^{-14}$ at 25°C. For acetic acid: $K_{\mathrm{a}}({\mathrm{C}}_2{\mathrm{H}}_4{\mathrm{O}}_2)=1.8\times {10}^{-5}$ So, $K_{\mathrm{b}}({\mathrm{C}}_2{\mathrm{H}}_3{\mathrm{O}}_2^{-})=\frac{K_{\mathrm{w}}}{K_{\mathrm{a}}({\mathrm{C}}_2{\mathrm{H}}_4{\mathrm{O}}_2)}=\frac{1.0\times {10}^{-14}}{1.8\times {10}^{-5}}=5.56\times {10}^{-10}$ For hypochlorous acid: $K_{\mathrm{a}}(\mathrm{H}\mathrm{O}\mathrm{Cl})=3.5\times {10}^{-8}$ So, $K_{\mathrm{b}}(\mathrm{O}{\mathrm{Cl}}^{-})=\frac{K_{\mathrm{w}}}{K_{\mathrm{a}}(\mathrm{H}\mathrm{O}\mathrm{Cl})}=\frac{1.0\times {10}^{-14}}{3.5\times {10}^{-8}}=2.86\times {10}^{-7}$

Compare the Kb values to determine the stronger base.

Now that we have the Kb values for both OCl− and C2H3O2−, we can compare them to determine which is the stronger base: $K_{\mathrm{b}}(\mathrm{O}{\mathrm{Cl}}^{-})=2.86\times {10}^{-7}$ $K_{\mathrm{b}}({\mathrm{C}}_2{\mathrm{H}}_3{\mathrm{O}}_2^{-})=5.56\times {10}^{-10}$ Since the Kb value of OCl− is larger than the Kb value of C2H3O2−, OCl− is the stronger base.

Key concepts

Ka and Kb relationship

Understanding the relationship between the acid dissociation constant ($K_a$) and the base dissociation constant ($K_b$) is crucial in acid-base equilibria. These constants are indicators of the strength of acids and bases in a solution.
In essence:

• $K_a$ measures how well an acid donates its protons in an aqueous solution.
• $K_b$ measures how well a base accepts protons in an aqueous solution.

The relationship between them, known as the water ion-product constant $K_w$, is always present in equilibrium equations and tends to be $1.0\times {10}^{-14}$ at 25°C. The formula connecting these constants is:$$K_a\cdot K_b=K_w$$This means that if you know either $K_a$ or $K_b$ for a substance, you can find the other using the $K_w$ value. Consequently, if an acid has a high $K_a$ value, its conjugate base will have a low $K_b$ value, implying that strong acids form weak bases and vice versa.

Conjugate bases

Conjugate bases are formed when an acid loses a proton. Understanding their role in equilibrium can help you determine the strength of a base.
For example:

• Acetic acid ($C{H}_{3}COOH$) donates a proton to become acetate ($C_2{H}_3{O}_2^{-}$), its conjugate base.
• Hypochlorous acid ($HOCl$) loses a proton to form hypochlorite ($OC{l}^{-}$), its conjugate base.

The strength of a conjugate base can be tied back to its corresponding weak acid through the formula mentioned earlier, $K_a\cdot K_b=K_w$.When comparing conjugate bases, their $K_b$ values can be directly calculated from the $K_a$ of their respective acids. The base with the higher $K_b$ value is stronger because it has a higher ability to accept protons.

Strength of Bases

The strength of a base is determined by how readily it can accept protons in aqueous solution. Bases with higher $K_b$ values are considered stronger because they are more likely to accept protons. In our given example:

• The $K_b$ of hypochlorite ($OC{l}^{-}$) is $2.86\times {10}^{-7}$.
• The $K_b$ of acetate ($C_2{H}_3{O}_2^{-}$) is $5.56\times {10}^{-10}$.

The higher $K_b$ value for $OC{l}^{-}$ indicates a stronger base because it implies that $OC{l}^{-}$ is more effective at accepting protons compared to $C_2{H}_3{O}_2^{-}$. Therefore, when choosing which base is stronger in a chemical setting, compare the $K_b$ values straightforwardly. This allows for clear identification of their relative strengths.