Problem 143 - Chemistry

Question

Codeine \(\left(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{NO}_{3}\right)\) is a derivative of morphine that is used as an analgesic, narcotic, or antitussive. It was once commonly used in cough syrups but is now available only by prescription because of its addictive properties. If the \(\mathrm{pH}\) of a \(1.7 \times 10^{-3} M\) solution of codeine is \(9.59\), calculate \(K_{\mathrm{b}}\).

Step-by-Step Solution

Answer

The base dissociation constant (Kb) for codeine is approximately \(8.9 \times 10^{-8}\).

Step 1: Calculate the pOH of the solution

Given the pH of the solution is 9.59, we can find the pOH using the relationship between pH and pOH: pH + pOH = 14 Substituting the given pH value, we get: 9.59 + pOH = 14 Now solve for pOH: pOH = 14 - 9.59 = 4.41

Step 2: Calculate the Hydroxide ion concentration

Using the formula to find the concentration of hydroxide ions from pOH: \[OH^- = 10^{-\text{pOH}}\] Substitute the pOH value we found: \[OH^- = 10^{-4.41}\] \[OH^- = 3.9 \times 10^{-5} M\]

Step 3: Set up the base dissociation equation

Codeine (\(C_{18}H_{21}NO_{3}\)) is a derivative of morphine and acts as a weak base, so it can accept a proton from water: \[C_{18}H_{21}NO_{3} + H_2O \rightleftharpoons C_{18}H_{22}NO_{3}^+ + OH^-\] The base dissociation constant equation is given by: \[K_b = \frac{[C_{18}H_{22}NO_{3}^+][OH^-]}{[C_{18}H_{21}NO_{3}]}\]

Step 4: Calculate the concentrations

Assuming that the initial concentration of codeine in the solution is equal to the equilibrium concentration, we can say that the amount of codeine that has turned into its conjugate acid is extremely small and can be neglected, so: \[[C_{18}H_{21}NO_{3}] = 1.7 \times 10^{-3} M\] Since the concentration of hydroxide ions (\(OH^-\)) is equal to the concentration of codeine ions that have gained a proton (\(C_{18}H_{22}NO_{3}^+\)): \[[C_{18}H_{22}NO_{3}^+] = [OH^-] = 3.9 \times 10^{-5} M\]

Step 5: Calculate the base dissociation constant (Kb)

Now we can substitute the concentrations into the Kb equation: \[K_b = \frac{(3.9 \times 10^{-5})(3.9 \times 10^{-5})}{1.7 \times 10^{-3}}\] Finally, we calculate the value: \[K_b \approx 8.9 \times 10^{-8}\] The base dissociation constant (Kb) for codeine is approximately 8.9 x 10^-8.

Key Concepts

Codeine

Codeine, scientifically known as \(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{NO}_{3}\), is a compound that stems from morphine. It is frequently utilized in the medical field for its analgesic, narcotic, and antitussive capabilities. Once a common ingredient in cough syrups, its addictive nature has led to its availability being restricted to prescription only.
As a chemical entity, codeine serves as a weak base in solutions, meaning it can act as a proton acceptor during chemical reactions.
Understanding its behavior in solutions, like calculating its base dissociation constant, is crucial to grasp its chemical properties and aid in pharmaceutical applications.

pH and pOH Relationship

In chemistry, the pH and pOH of a solution provide essential information about its acidity and basicity. The relationship between them is expressed by the equation:\[\text{pH} + \text{pOH} = 14\] This relationship is fundamental when studying solutions because it helps us understand the concentration of hydrogen ions \((H^+)\) compared to hydroxide ions \((OH^-)\).
In the case provided, a solution with a pH of 9.59 indicates a basic environment since its pH exceeds 7. By applying the pH and pOH relationship, we can determine that:- The pOH will be 14 - 9.59, yielding a pOH value of 4.41.- This helps in subsequently finding the hydroxide ion concentration, which further contributes to calculating the base dissociation constant \(K_b\).

Weak Base

A weak base is characterized by its limited ability to accept protons compared to strong bases. Unlike strong bases, which completely dissociate in water, weak bases like codeine only partially dissociate. This partial dissociation is evident in the equilibrium that exists between codeine and its conjugate acid: \[ \mathrm{C}_{18} \mathrm{H}_{21} \mathrm{NO}_{3} + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{C}_{18} \mathrm{H}_{22} \mathrm{NO}_{3}^+ + \mathrm{OH}^- \]
Weak bases have an important implication in calculating the base dissociation constant \(K_b\), as it quantifies the extent to which the base can dissociate to liberate hydroxide ions \((OH^-)\).
This property helps in assessing the "strength" of the base in solutions, which is essential in understanding chemical reactions and designing pharmaceutical therapies.

Hydroxide Ion Concentration

Hydroxide ion concentration \((OH^-)\) is a crucial parameter in determining the nature and strength of a basic solution. It measures the amount of hydroxide ions present in a solution and provides insight into its basicity. We can calculate the concentration of hydroxide ions originating from the given pOH. The formula used is:\[ OH^- = 10^{- ext{pOH}} \] For the specific solution in question, with a pOH of 4.41, the hydroxide ion concentration comes out to:\[ OH^- = 10^{-4.41} \approx 3.9 \times 10^{-5} \space M \]
This concentration helps in analyzing how codeine, as a weak base, is capable of accepting protons and contributing to its dissociation in water. These dynamics facilitate the calculation of the base dissociation constant \(K_b\). Knowing the hydroxide ion concentration is key to understanding not just the basicity, but also the extent of partial dissociation in weak bases.