Problem 113 - Chemistry The Central Science

Question

Aspirin has the structural formula At body temperature \(\left(37^{\circ} \mathrm{C}\right), K_{a}\) for aspirin equals \(3 \times 10^{-5} .\) If two aspirin tablets, each having a mass of \(325 \mathrm{mg},\) are dissolved in a full stomach whose volume is 1 \(\mathrm{L}\) and whose \(\mathrm{pH}\) is \(2,\) what percent of the aspirin is in the form of neutral molecules?

Step-by-Step Solution

Answer

About 99.97% of the aspirin is in the form of neutral molecules when two aspirin tablets, each having a mass of 325 mg, are dissolved in a full stomach whose volume is 1 L and whose pH is 2.

Step 1: Calculate the concentration of aspirin in the stomach solution

Given two aspirin tablets, each with a mass of 325 mg, are dissolved in a solution with a volume of 1 L, we should first convert the mass of aspirin to moles and then calculate the concentration. Molecular weight of aspirin (C9H8O4) = 9(12.01) + 8(1.01) + 4(16.00) = 180.16 g/mol Total mass of aspirin in the stomach = 2 * 325 mg = 650 mg Now, convert the mass of aspirin to moles: Moles of aspirin = (650 mg) / (1,000 mg/g) * (1 mol / 180.16 g) = 0.00361 mol Now, calculate the molar concentration of aspirin: Molar concentration of aspirin = (0.00361 mol) / (1 L) = 0.00361 M

Step 2: Calculate the concentration of \(H^+\) ions using the given pH

Since the given pH of the stomach solution is 2, we can calculate the concentration of \(H^+\) ions as follows: \([H^+] = 10^{-\mathrm{pH}} = 10^{-2} = 0.01\ \mathrm{M}\)

Step 3: Write the chemical equilibrium equation for aspirin

As aspirin is an acidic compound, we can write the equilibrium equation for the dissociation of aspirin as: \(\ce{C9H8O4(aq) + H2O(l) <=> C9H7O4^{-}(aq) + H3O^{+}(aq)}\)

Step 4: Calculate the percentage of neutral aspirin molecules using the \(K_a\) value

Using the given \(K_a\) value for aspirin, we have: \(K_a = \frac{[\ce{C9H7O4^{-}}][\ce{H3O^{+}}]}{[\ce{C9H8O4}]}\) Since the total concentration of \(H^+\) and \(\ce{H3O^{+}}\) ions is 0.01M, and the concentration of the dissociated aspirin is negligible, we have: \(K_a = 3 \times 10^{-5} = \frac{[\ce{C9H7O4^{-}}]\times \left[\ce{H3^{+}}\right]}{[\ce{C9H8O4}]} = \frac{[\ce{C9H7O4^{-}}]\times 0.01\ \mathrm{M}}{[\ce{C9H8O4}]}\) We can now solve for \([\ce{C9H7O4}^{-}]\): \([\ce{C9H7O4}^{-}] = 3 \times 10^{-5} \times \frac{[\ce{C9H8O4}]}{0.01\ \mathrm{M}} = 3 \times 10^{-5}\times \frac{0.00361\ \mathrm{M}}{0.01\ \mathrm{M}} = 1.09 \times 10^{-6}\ \mathrm{M}\) Now, we can calculate the percentage of neutral aspirin molecules, which is the ratio between the concentrations of neutral and dissociated aspirin: Percentage of neutral aspirin = \(\frac{[\ce{C9H8O4}]}{[\ce{C9H7O4^{-}}]+[\ce{C9H8O4}]} \times 100 = \frac{0.00361\ \mathrm{M}}{1.09 \times 10^{-6}\ \mathrm{M} + 0.00361\ \mathrm{M}} \times 100 \approx 99.97\%\) Thus, about 99.97% of the aspirin is in the form of neutral molecules.

Key Concepts

Acid Dissociation Constant (Ka)

Understanding the acid dissociation constant (\(K_a\)) is essential for studying the behavior of weak acids like aspirin in a solution. The value of \(K_a\) represents the strength of an acid, indicating how well it dissociates into ions. For aspirin, a \(K_a\) of \(3 \times 10^{-5}\) implies that it is a weak acid because it dissociates slightly in water to form hydronium ions (\(\ce{H3O+}\)) and the corresponding aspirin anion (\(\ce{C9H7O4^-}\)).

The equilibrium formula for dissociation expresses this reaction:
\(\ce{C9H8O4(aq) + H2O(l) \rightleftharpoons C9H7O4^{-}(aq) + H3O^{+}(aq)}\). In this equation, \(\ce{C9H8O4}\) is the neutral aspirin molecule. The Ka value helps predict the ratio of dissociated to undissociated aspirin, allowing chemists to understand the extent of the reaction. A small \(K_a\) indicates that most aspirin remains as neutral molecules.

pH Calculation

The pH calculation provides insight into the acidity or basicity of a solution. It is a logarithmic scale that inversely relates to the concentration of hydrogen ions (\(H^+\)). A pH of 2, as mentioned for the stomach, is quite acidic and means the concentration of \(H^+\) ions is \(10^{-2} \rm{M}\).

We relate pH to \(H^+\) concentration with the formula \(\mathrm{pH} = -\log [H^+]\). In the context of the aspirin problem, knowing the stomach’s pH tells us how much additional acidity is present, which affects the equilibrium of aspirin dissociation. High \(H^+\) concentration pushes the equilibrium towards the non-dissociated form due to Le Chatelier's principle.

Chemical Equilibrium

In the aspect of chemical equilibrium, the system reaches a state where the rates of the forward and reverse reactions are equal, which means the concentrations of reactants and products remain constant. The concept applies to aspirin's dissociation in the stomach's aqueous environment, where the acid partially ionizes to an equilibrium point.

Chemical equilibrium is dynamic; thus, when external conditions such as pH change, the equilibrium shifts to either produce more ions or re-form the acid, following Le Chatelier’s principle. For aspirin, the high acidity of the stomach favors the equilibrium to shift towards the intact aspirin molecules (non-ionized form). Assessing the equilibrium helps determine the ratio of neutral to ionized aspirin.

Molar Mass Conversion

Molar mass conversion is a critical concept that allows the transformation of mass units to molar units, essential for stoichiometric calculations. The molecular weight of aspirin is \(180.16 \mathrm{g/mol}\), which we use to convert milligrams of aspirin to moles. This conversion is fundamental in the problem, where we start with the mass of aspirin (650 mg) and convert to moles (0.00361 mol).

This conversion enables us to understand the amount of substance in molar terms, leading to further calculations like concentration (moles per liter). Knowing the molar concentration can then assist in determining the proportion of aspirin that remains in its neutral molecular form.

Percentage Composition

Percentage composition involves determining the percentage of a compound in a particular form compared to the total possible. In the aspirin dissolution chemistry problem, we calculate the percentage of neutral aspirin molecules within the stomach solution. After several steps, including the use of the acid dissociation constant and molar concentrations, the result indicates that 99.97% of aspirin remains as neutral molecules.

This calculation highlights how the compound is partitioned between its dissociated ionized form and its intact molecular form and is pivotal in applications such as pharmaceutical drug design, where the active form of a medication must be understood.