Problem 106 - General Chemistry

Question

Codeine, \(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{NO}_{3},\) is an alkaloid \(\left(K_{b}=6.2 \times\right.\) \(10^{-9}\) ) used as a painkiller and cough suppressant. A solution of codeine is acidified with hydrochloric acid to \(\mathrm{pH}\) 4.50. What is the ratio of the concentration of the conjugate acid of codeine to that of the base codeine?

Step-by-Step Solution

Answer

The ratio of the conjugate acid to the base concentration is approximately 0.051.

Step 1: Write the Expression for pH

First, write down the expression for pH, which relates the concentration of hydronium ions in a solution:\[pH = -\log[H^+]\]Given that the \( pH = 4.50 \), the concentration of \( H^+ \) is:\[[H^+] = 10^{-4.50}\]

Step 2: Calculate Hydronium Ion Concentration

Convert the pH value of 4.50 to the hydrogen ion concentration:\[[H^+] = 10^{-4.50} = 3.16 \times 10^{-5} \, \text{M}\]

Step 3: Use the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation for a weak base given with its conjugate acid is:\[pOH = pK_b + \log \left(\frac{[\text{Base}]}{[\text{Conjugate Acid}]\right)}\]We can find the value of \(pK_b\) as follows:\[pK_b = -\log K_b = -\log (6.2 \times 10^{-9}) = 8.21\]Convert \(pH\) to \(pOH\) since we're dealing with a basic solution:\[pOH = 14 - pH = 14 - 4.50 = 9.50\]

Step 4: Solve for the Ratio of Concentrations

Rearrange the Henderson-Hasselbalch equation to solve for the ratio:\[9.50 = 8.21 + \log \left(\frac{[\text{Base}]}{[\text{Conjugate Acid}]\right)}\]This gives:\[1.29 = \log \left(\frac{[\text{Base}]}{[\text{Conjugate Acid}]\right)}\]Taking the antilog:\[\frac{[\text{Base}]}{[\text{Conjugate Acid}]} = 10^{1.29} \approx 19.5\]Therefore, the ratio of concentrations is:\[\frac{[\text{Conjugate Acid}]}{[\text{Base}]} = \frac{1}{19.5}\]\[\text{Conjugate Acid to Base Ratio} = 0.051\]

Step 5: Final Calculation and Interpretation

With all values calculated, we conclude that the concentration of the conjugate acid is significantly lower than that of the base. Thus, the final ratio of the concentration of the conjugate acid to the concentration of the codeine base is approximately \(0.051\).

Key Concepts

pH Calculation

Understanding pH calculation is key when dealing with acids and bases. The pH of a solution tells us how acidic or basic it is. It's calculated by taking the negative logarithm of the hydrogen ion \(H^+\) concentration. This is represented mathematically as:

• \(pH = -\log[H^+]\)

When given a pH value, we can reverse this calculation to find the \(H^+\) concentration. For instance, if the pH is 4.50, the hydrogen ion concentration would be: \[ [H^+] = 10^{-4.50} = 3.16 \times 10^{-5} \, \text{M} \]
This calculation is important in solutions where precise hydrogen ion amounts need to be known, especially when dealing with weak bases and their conjugate acids. Once you have the hydrogen ion concentration, you can analyze the equilibrium behavior of these systems more effectively.

Weak Base Equilibrium

Weak bases do not fully dissociate in solution, which is why their equilibrium needs to be carefully analyzed. The equilibrium constant for a base is denoted as \(K_b\), and it shows the extent of dissociation in water. The expression involves concentrations of the base and its conjugate acid:

• \( \text{Base} + H_2O \rightleftharpoons \text{Conjugate Acid} + OH^- \)
• \(K_b = \frac{[\text{Conjugate Acid}][OH^-]}{[\text{Base}]} \)

The value of \(K_b\) helps to predict the behavior of a weak base in water. For instance, codeine, with \(K_b = 6.2 \times 10^{-9}\), is a weak base that slightly dissociates in water.
To find the \(pK_b\), which is useful in various calculations, you apply the formula: \[ pK_b = -\log(K_b) \] Thus, knowing \(K_b\) and how to calculate \(pK_b\) allows you to predict how much of the base and its conjugate remain in solution, which is vital for further calculations like using the Henderson-Hasselbalch equation.

Conjugate Acid-Base Pairs

In weak base solutions like those containing codeine, understanding conjugate acid-base pairs is crucial. A conjugate pair consists of two species that transform into each other by gain or loss of a proton. For codeine, the base \( \text{C}_{18} \text{H}_{21} \text{NO}_{3} \) becomes its conjugate acid upon gaining a proton.
The Henderson-Hasselbalch equation is a tool that relates the pH and the concentration ratio of a weak base to its conjugate acid:

• \( pOH = pK_b + \log \left(\frac{[\text{Base}]}{[\text{Conjugate Acid}]}\right) \)

This equation helps determine the balance between a base and its conjugate acid in solution. By reorganizing the equation, you can find the ratio of these concentrations, which provides insight into the solution's chemistry at a given \(pH\).
Applying this method to our example with a calculated \(pOH\) of 9.50, the conjugate acid to base concentration ratio is found to be approximately 0.051, meaning that in solution, the codeine base is significantly more abundant than its conjugate acid.