Cocaine is a weak organic base whose molecular formula is C17H21NO4. An aqueous solution of cocaine was found to have a pH of 8.53 and an osmotic pressure of 52.7 torr at 15 °C. Calculate the Kb for cocaine.

Verified step by step guidance

Step 1: Convert the osmotic pressure from torr to atm. Use the conversion factor 1 atm = 760 torr.

Step 2: Use the osmotic pressure formula \( \pi = iMRT \) to find the molarity (M) of the solution. Assume the van't Hoff factor \( i = 1 \) for cocaine, and use the gas constant \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \) and temperature \( T = 15 + 273.15 \text{ K} \).

Step 3: Calculate the concentration of hydroxide ions \( [OH^-] \) using the pH value. First, find the pOH by subtracting the pH from 14, then use \( [OH^-] = 10^{-\text{pOH}} \).

Step 4: Use the expression for the base dissociation constant \( K_b = \frac{[OH^-]^2}{[B] - [OH^-]} \), where \([B]\) is the initial concentration of cocaine found in Step 2.

Step 5: Simplify the expression for \( K_b \) assuming \([OH^-] \ll [B]\), which allows you to approximate \( K_b \approx \frac{[OH^-]^2}{[B]} \).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Weak Bases and Kb

A weak base is a substance that partially ionizes in solution, establishing an equilibrium between the un-ionized base and its ions. The base dissociation constant (Kb) quantifies the strength of a weak base, indicating the extent to which it can accept protons from water. A higher Kb value signifies a stronger base, while a lower value indicates a weaker base.

pH and pOH Relationship

pH is a measure of the hydrogen ion concentration in a solution, while pOH measures the hydroxide ion concentration. The relationship between pH and pOH is given by the equation pH + pOH = 14 at 25 °C. In this case, knowing the pH allows us to calculate the pOH, which is essential for determining the concentration of hydroxide ions in the solution, a key factor in calculating Kb.

Osmotic Pressure

Osmotic pressure is the pressure required to prevent the flow of solvent into a solution through a semipermeable membrane. It is directly related to the concentration of solute particles in the solution, as described by the van 't Hoff equation. In this context, osmotic pressure can be used to determine the molarity of the cocaine solution, which is necessary for calculating Kb.

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