Question

The \(\mathrm{p} K_{\mathrm{a}}\) of butyric acid (HBut) is 4.7 . Calculate \(K_{\mathrm{b}}\) for the butyrate ion (But \(^{-}\) ).

Short answer

After performing the calculation, you find that the value of \(K_{\mathrm{b}}\) for the Butyrate ion (But-\(-\))) is approximately 2 x 10^{-10}.

Step-by-step solution

Understanding pKa and pKb

The the pKa and pKb are the negative logs of the acid dissociation constant (Ka) and base dissociation constant (Kb), respectively. Here, the pKa is given, which can be used to find the Ka for Butyric Acid, which will then be used to find the Kb of Butyrate ion, using the relationship \(K_{\mathrm{a}} * K_{\mathrm{b}} = K_{\mathrm{w}}\), where \(K_{\mathrm{w}}\) is the ion product for water.

Find Ka

To find the Ka of butyric acid, use the formula \( K_{\mathrm{a}} = 10^{- pK_{\mathrm{a}}}\). Substituting \( pK_{\mathrm{a}} = 4.7\) into the formula gives \( K_{\mathrm{a}} = 10^{-4.7}\).

Calculate Kb

Use the formula \( K_{\mathrm{b}} = \frac{K_{\mathrm{w}}}{K_{\mathrm{a}}}\) to calculate Kb. The value for \( K_{\mathrm{w}}\) is typically \( 1.0 x 10^{-14}\) at 25°C. Substituting these values gives \( K_{\mathrm{b}} = \frac{ 1.0 x 10^{-14}}{10^{-4.7}}\).

Simplify Kb

Simplify the expression for Kb to get the final numerical answer.

Key concepts

pKa and pKb Relationship

Understanding the relationship between pKa and pKb is essential when dealing with acid-base equilibrium problems. Both pKa and pKb are logarithmic measures:

• pKa is the negative logarithm of the acid dissociation constant (\( K_a \)) of an acid.
• pKb is the negative logarithm of the base dissociation constant (\( K_b \)) of a base.

The key relationship between them is derived from the ion product of water (\( K_w \)). At 25°C, \( K_w = 1.0 \times 10^{-14} \).
For any conjugate acid-base pair:

• The product of \( K_a \) and \( K_b \) equals \( K_w \), i.e., \( K_a \times K_b = K_w \).
• Using logs, a useful equation is \( pK_a + pK_b = 14 \).

This equation helps in calculating the Kb for the butyrate ion if you know the pKa of butyric acid by first finding the Ka.

Dissociation Constant

The dissociation constant provides insights into how easily an acid or a base dissociates in a solution. Here, we focus on the acid version, \( K_a \):

• A large \( K_a \) value signifies a strong acid, which fully dissociates in solution.
• A small \( K_a \) value indicates a weak acid, suggesting partial dissociation.

For butyric acid, we used its pKa to find \( K_a \):
\[ K_a = 10^{-4.7} \]This shows it is a weak acid. To look at the base dissociation or \( K_b \), we rearrange the formula:
\[ K_b = \frac{K_w}{K_a} \]By substituting the values, calculating \( K_b \) becomes straightforward, enabling further understanding of the basicity of the butyrate ion.

Ion Product of Water

The ion product of water (\( K_w \)) is a critical constant in acid-base chemistry. At 25°C, \( K_w = 1.0 \times 10^{-14} \), a value derived from:

• The auto-ionization of water: \( H_2O \rightleftharpoons H^+ + OH^- \).
• Equilibrium expression: \( [H^+][OH^-] = K_w \).

This constant ties together both acid and base dissociation constants:

• For conjugate acid-base pairs: \( K_a \times K_b = K_w \).
• This relationship allows calculations of one constant when the other is known.

In the exercise, knowing \( K_w \) allows us to switch between acid and base constants efficiently, deepening comprehension of equilibrium dynamics.