Question

Ephedrine, a central nervous system stimulant, is used in nasal sprays as a decongestant. This compound is a weak organic base: \(\mathrm{C}_{10} \mathrm{H}_{15} \mathrm{ON}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{C}_{10} \mathrm{H}_{15} \mathrm{ONH}^{+}(a q)+\mathrm{OH}^{-}(a q)\) (a) What A \(0.035 \mathrm{M}\) solution of ephedrine has a pH of 11.33 . are the equilibrium concentrations of \(\mathrm{C}_{10} \mathrm{H}_{15} \mathrm{ON}\), \(\mathrm{C}_{10} \mathrm{H}_{15} \mathrm{ONH}^{+},\) and \(\mathrm{OH}^{-} ?\) (b) Calculate \(K_{b}\) for ephedrine.

Short answer

The short version of the answer is: 1. Calculate the concentration of OH- ions from the given pH: pOH = 14 - pH and [OH⁻] = $10^{-pOH}$. 2. Set up the ICE table and add the change in concentration term 'x'. 3. Plug equilibrium concentrations into the Kb expression and solve for x. 4. Use the value of x to determine the equilibrium concentrations of C10H15ON, C10H15ONH⁺, and OH⁻ ions. 5. Plug equilibrium concentrations into the Kb expression to calculate Kb for ephedrine.

Step-by-step solution

Calculate the concentration of OH- ions from the given pH

The given pH of the solution is 11.33. We can find the pOH of the solution by subtracting the pH from 14 (since pH + pOH = 14). Then, we can calculate the concentration of OH- ions using the formula for pOH: \[pOH = -\log_{10} [OH^{-}]\]

Set up the ICE table

An ICE table represents the initial, change, and equilibrium concentrations for each species in the reaction. We will set up an ICE table for ephedrine, its conjugate acid, and hydroxide ions. Initial concentrations: - [C10H15ON]: 0.035 M - [C10H15ONH+]: 0 - [OH-]: Value calculated in Step 1 Change in concentration: - C10H15ON: -x - C10H15ONH+: +x - OH-: +x Equilibrium concentrations: - [C10H15ON]: 0.035 - x - [C10H15ONH+]: x - [OH-]: Value calculated in Step 1 + x

Use the Kb expression and equilibrium concentration values to find x

The Kb expression for this reaction is: \[K_b = \frac{[C_{10}H_{15}ONH^{+}] [OH^{-}]}{[C_{10}H_{15}ON]}\] We will plug in the equilibrium concentrations from the ICE table into the Kb expression and solve for x.

Calculate the equilibrium concentrations of the species

Using the value of x, we can now calculate the equilibrium concentrations of C10H15ON, C10H15ONH+, and OH- ions by plugging the value of x into the equilibrium concentrations from the ICE table.

Determine the Kb value for ephedrine

Now that we have the equilibrium concentrations of all three species, we can plug these values into the Kb expression and solve for Kb: \[K_b = \frac{[C_{10}H_{15}ONH^{+}] [OH^{-}]}{[C_{10}H_{15}ON]}\]

Key concepts

ICE Table

The ICE table is a powerful, systematic tool used in chemistry to analyze equilibrium reactions. It helps in determining the concentrations of reactants and products at equilibrium. The three sections of the ICE table stand for Initial, Change, and Equilibrium concentrations.

• **Initial**: The initial concentrations are determined before the reaction starts. For instance, in the case of ephedrine, the concentration is given as 0.035 M, while the conjugate acid and hydroxide ions start at 0 because they form as the reaction proceeds.
• **Change**: Represents the shift in concentrations as the reaction moves toward equilibrium. These changes are usually represented by the variable \(x\) because the exact shift is not typically known at the start.
• **Equilibrium**: Expresses the concentrations at equilibrium, which are calculated by adding the changes to the initial quantities.

By inserting the values for Initial, Change, and Equilibrium, the ICE table provides a clear and organized method to track the creation and consumption of particles in a reaction. For weak bases like ephedrine, this setup is essential for identifying how much of the base remains, how much conjugate acid forms, and how many hydroxide ions are produced.

pH and pOH Calculations

pH and pOH are fundamental concepts in understanding the acidity or basicity of a solution. They are inversely related: As one increases, the other decreases. Calculating these values helps identify how acidic or basic a solution is.
For a base like ephedrine, we typically start with the pH, which was provided as 11.33. Since pH + pOH = 14, this relationship can be used to find the pOH:
\[ pOH = 14 - pH = 14 - 11.33 = 2.67 \]Given the pOH, the concentration of hydroxide ions \([OH^-]\) in the solution can be calculated with:
\[ [OH^-] = 10^{-pOH} = 10^{-2.67} \]This calculation is crucial because it puts a number on the concentration of hydroxide ions, indicating the basic nature of the solution. From there, the value can be used in further calculations, such as determining equilibrium concentrations and using the ICE table.

Equilibrium Constant (Kb)

The equilibrium constant, \(K_b\), reflects the strength of a base in solution. It measures how far the equilibrium lies to the right (towards product formation) for a weak base like ephedrine when it reacts with water to form its conjugate acid and hydroxide ions.
The expression for \(K_b\) in the context of ephedrine is:
\[ K_b = \frac{[C_{10}H_{15}ONH^+][OH^-]}{[C_{10}H_{15}ON]} \]This formula involves inserting the equilibrium concentrations of the conjugate acid, hydroxide ions, and the remaining unreacted base from the ICE table. Each concentration plays a part in indicating how extensively the base has reacted:

• The larger the value of \(K_b\), the stronger the base, which means a greater tendency to form the conjugate acid and hydroxide ions.
• Lower \(K_b\) values suggest a weaker base, with less reaction progress.

Knowing \(K_b\) is essential not only for understanding the behavior of the base in solution but also for predicting the direction of reactions and calculating related equilibrium constants.