Problem 73 - Chemistry The Central Science

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 0.035\(M\) solution of ephedrine has a pH of 11.33 . (a) What 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.

Step-by-Step Solution

Answer

The equilibrium concentrations are: \[[C10H15ON] = 0.0329 M\], \[[C10H15ONH^+] = 2.14 \times 10^{-3} M\], and \[[OH^-] = 2.14 \times 10^{-3} M\]. The $K_b$ value for ephedrine is $1.39 \times 10^{-4}$.

Step 1: Calculate the concentration of OH- ions

We have the pH of the solution, which is 11.33. We will use the following formulas to find the concentration of OH- ions at equilibrium: \[pH + pOH = 14\] \[pOH = -\log{[OH^-]}\] First, we will calculate the pOH of the solution: \[pOH = 14 - pH\] \[pOH = 14 - 11.33 = 2.67\] Now, we will find the concentration of OH- ions: \[OH^- = 10^{-pOH}\] \[OH^- = 10^{-2.67} = 2.14 \times 10^{-3} M\]

Step 2: Create an ICE table

Now, we will create an ICE table to examine the changes in concentrations for each compound: | | C10H15ON (aq) | H2O (l) | C10H15ONH+ (aq) | OH- (aq) | |------------|---------------|---------|-----------------|----------| | Initial | 0.035 M | - | 0 M | 0 M | | Change | -x | - | +x | +x | | Equilibrium| 0.035-x | - | x | 2.14x10^(-3)+x | We can simplify this ICE table by ignoring the initial concentration of OH- ions (0 M) since the given concentration is very small compared to the equilibrium concentration. The ICE table then becomes: | | C10H15ON (aq) | H2O (l) | C10H15ONH+ (aq) | OH- (aq) | |------------|---------------|---------|-----------------|----------| | Initial | 0.035 M | - | 0 M | - | | Change | -x | - | +x | +x | | Equilibrium| 0.035-x | - | x | 2.14x10^(-3) |

Step 3: Calculate the equilibrium concentrations

We have the equilibrium concentrations for OH- ions, so we will use that to calculate the equilibrium concentrations of the other reactants: Given: \[x = [C10H15ONH^+] = [OH^-] = 2.14 \times 10^{-3} M\] We can calculate the equilibrium concentration of C10H15ON (aq): \[[C10H15ON] = 0.035 - x = 0.035 - 2.14 \times 10^{-3} = 0.0329 M\] Now we have the equilibrium concentrations of all reactants and products: \[[C10H15ON] = 0.0329 M\] \[[C10H15ONH^+] = 2.14 \times 10^{-3} M\] \[[OH^-] = 2.14 \times 10^{-3} M\]

Step 4: Calculate the Kb value for ephedrine

We will now use the equilibrium concentrations to find the Kb value for ephedrine. The Kb expression for this reaction is: \[K_b = \frac{[C10H15ONH^+][OH^-]}{[C10H15ON]}\] Now, plug in the equilibrium concentrations: \[K_b = \frac{(2.14 \times 10^{-3})(2.14 \times 10^{-3})}{0.0329} = 1.39 \times 10^{-4}\] So, the Kb value for ephedrine is: \[K_b = 1.39 \times 10^{-4}\]

Key Concepts

Ephedrine

Ephedrine is a compound used medicinally as a decongestant and a central nervous system stimulant. It is derived from plants in the genus Ephedra and has been used in traditional Chinese medicine for centuries. Chemically, it's classified as an alkaloid and is known for its stimulant effects, which can help to relieve nasal congestion by narrowing the blood vessels in nasal passages.

As we delve into its chemical nature in solutions, understanding ephedrine's role and behavior as a weak base is crucial for students studying acid-base equilibria.

Weak Base

A weak base, in contrast to a strong base, does not fully dissociate in solution. This means that when dissolved in water, a compound like ephedrine does not completely separate into its ionic components. Weak bases have a tendency to accept protons from water to create hydroxide ions and the corresponding conjugate acid.

This partial dissociation is described quantitatively by the base dissociation constant, or Kb, which gives an idea of the extent to which the base can produce hydroxide ions and thus affect the pH of the solution. Ephedrine's behavior as a weak base is pivotally characterized by its Kb value.

pH Calculation

The pH of a solution is a measure of its acidity or alkalinity. In the context of a weak base like ephedrine, pH calculation begins by determining the concentration of hydroxide ions in the solution. The pH can then be found by first calculating the pOH, which is the negative logarithm of the hydroxide ion concentration. Since pH and pOH are related by the equation pH + pOH = 14 at 25°C, one can calculate the pH from the pOH.

Students must note that the pH scale is logarithmic, so a small change in pH corresponds to a large change in hydrogen ion concentration. Hence, accurate calculation of pH is essential for understanding the properties of the solution.

ICE Table

An ICE table, which stands for Initial, Change, and Equilibrium, is a systematic method to lay out the concentrations of reactants and products in a dynamic equilibrium. This table helps students visualize and calculate how concentrations shift from the original values to equilibrium.

Beginning with the initial concentrations, one records the changes that occur as the system moves towards equilibrium, often represented by the variable 'x'. Finally, the equilibrium concentrations are determined by applying the equilibrium condition. The ICE table is a powerful tool in solving for unknowns in acid-base equilibrium problems.

Equilibrium Concentration

The equilibrium concentration in a chemical reaction is the amount of each substance present when the rates of the forward and reverse reactions are equal, meaning the system is in a state of balance. For weak bases like ephedrine in water, the equilibrium concentration is reached when the base has partially reacted with water to form hydroxide ions and the conjugate acid.

It is essential to identify these concentrations in order to determine the strength of the base, measured by Kb, and to calculate the pH of the solution. These concentrations are pivotal for scientists and pharmacists to predict the behavior of the compound in various environments.

Kb Value

The Kb value, or the base dissociation constant, quantifies the strength of a weak base in a solution. It specifically denotes the equilibrium constant for the reaction in which the base accepts a proton from water. A higher Kb value indicates a stronger base, which dissociates more to form hydroxide ions.

In practice, knowing the Kb value is essential as it relates to the pH and therefore the potential physiological effects of the base, which is particularly important for drugs like ephedrine. Calculating the Kb in conjunction with an ICE table offers students a comprehensive understanding of acid-base equilibria.