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

1. Step 1: Convert the osmotic pressure from torr to atm. Use the conversion factor 1 atm = 760 torr.
2. 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 \( T = 15 + 273.15 \text{ K} \).
3. 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 \( [OH^-] = 10^{-\text{pOH}} \).
4. 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.
5. Step 5: Simplify the expression for \( K_b \) assuming \([OH^-] \ll [B]\), which allows you to approximate \( K_b \approx \frac{[OH^-]^2}{[B]} \).