Question

Codeine, \(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{O}_{3} \mathrm{N},\) is an opiate, has analgesic and antidiarrheal properties, and is widely used. In water, codeine is a weak base. A handbook gives \(\mathrm{p} K_{\mathrm{a}}=6.05\) for protonated codeine, \(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{O}_{3} \mathrm{NH}^{+} .\) Write the reaction for \(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{O}_{3} \mathrm{NH}^{+}\) and calculate \(\mathrm{p} K_{\mathrm{b}}\) for codeine.

Short answer

The \(pK_b\) for codeine is approximately 7.95.

Step-by-step solution

Write the reaction for protonated codeine

The protonated codeine (\(\mathrm{C}_{18}\mathrm{H}_{21}\mathrm{O}_{3}\mathrm{NH}^{+}\)) works as a weak acid and can donate a proton to water in the solution, thus becoming codeine (\(\mathrm{C}_{18}\mathrm{H}_{21}\mathrm{O}_{3}\mathrm{N}\)). \n\nThe reaction would be: \(\mathrm{C}_{18}\mathrm{H}_{21}\mathrm{O}_{3}\mathrm{NH}^{+} + H_2O \leftrightarrow \mathrm{C}_{18}\mathrm{H}_{21}\mathrm{O}_{3}\mathrm{N} + H_3O^{+}\)

Formulate the definition of \(pK_b\)

The \(pK_b\) value represents the negative base-10 logarithm of the base dissociation constant, \(K_b\). Codeine, in water, behaves as a weak base, so it can accept a proton from water. However, in this problem we're not directly given \(K_b\), but we're given \(pK_a\) for the protonated version of codeine.

Calculate \(pK_b\)

Using the provided \(pK_a\), we can use the relation \(pK_w = pK_a + pK_b\). \n\nAt room temperature, the \(pK_w\) of water is approximately 14. So we can use this to calculate \(pK_b\) as follows: \(\n\n pK_b = pK_w - pK_a \)\n\nSubstitute the given and known values: \(\n\n pK_b = 14 - 6.05 = 7.95\)

Key concepts

Understanding the Relationship Between pKa and pKb

Understanding the relationship between \( pK_a \) and \( pK_b \) is essential for grasping how acids and bases behave in water. These values are logarithmic measures that tell us how easily an acid donates a proton or how easily a base accepts a proton. The crucial relationship can be represented by the equation \( pK_w = pK_a + pK_b \), where \( pK_w \) is the ion product of water, typically 14 at room temperature.
This relationship is handy, especially when you have one value (like \( pK_a \)) and need to find the other (like \( pK_b \)). By rearranging this equation, you can easily solve for the unknown, making it possible to understand the reactivity of substances like codeine in water.

• The \( pK_a \) value applies to acids and refers to the negative log of the acid dissociation constant.
• The \( pK_b \) value applies to bases and refers to the negative log of the base dissociation constant.

To find the \( pK_b \) of codeine, knowing its \( pK_a \) helps us calculate these interactions in aqueous solutions.

Unveiling the Base Dissociation Constant

The base dissociation constant, \( K_b \), helps us understand how strongly a base holds on to its protons. It's a measure of the extent to which a base dissociates in solution to form hydroxide ions. For weak bases, like codeine, \( K_b \) tends to be a small value because they don't completely dissociate.
The \( pK_b \) is derived from \( K_b \) through a logarithm: \( pK_b = -\log K_b \). This is useful because dealing with smaller numbers, like those in exponential forms (e.g., \( 1 \times 10^{-7} \)), is often cumbersome.

• A lower \( pK_b \) value indicates a stronger base, whereas a higher value indicates a weaker base.
• Codeine has a \( pK_b \) of 7.95, meaning it’s a weak base as it prefers to stay mostly protonated in solution.

Understanding \( pK_b \) allows chemists to predict the behavior of bases in various chemical environments, particularly in aqueous settings.

Chemical Reactions in Aqueous Solutions Featuring Weak Acids and Bases

Reactions in aqueous solutions often feature the interplay of acids and bases. When dealing with weak acids and bases, such as codeine, these substances only partially dissociate in water. This partial dissociation indicates a balance between the molecular form and the dissociated ions.
For example, in the case of protonated codeine, the reaction with water can be visualized as:\[\text{C}_{18}\text{H}_{21}\text{O}_{3}\text{NH}^{+} + \text{H}_2\text{O} \leftrightarrow \text{C}_{18}\text{H}_{21}\text{O}_{3}\text{N} + \text{H}_3\text{O}^{+}\]
Here, protonated codeine donates a proton, becoming codeine itself, a typical behavior in weak acids.

• Weak acids donate protons, leading to the formation of their conjugate base and \( H_3O^+ \).
• The equilibrium state of such reactions is crucial, as it determines the acidity or basicity of the solution.

The study of these reactions provides insight into various biological and chemical processes, influencing everything from drug design to environmental chemistry.