Problem 62 - Chemistry: The Central Science in SI Units, Global Edition

Question

The active ingredient in aspirin is acetylsalicylic acid \(\left(\mathrm{HC}_{9} \mathrm{H}_{7} \mathrm{O}_{4}\right),\) a monoprotic acid with \(K_{a}=3.3 \times 10^{-4}\) at \(25^{\circ} \mathrm{C}\) What is the \(\mathrm{pH}\) of a solution obtained by dissolving one regular aspirin tablet, containing \(100 \mathrm{mg}\) of acetylsalicylic acid, in \(200 \mathrm{~mL}\) of water?

Step-by-Step Solution

Answer

The pH of a solution obtained by dissolving one regular aspirin tablet containing 100 mg of acetylsalicylic acid in 200 mL of water is approximately 3.53.

Step 1: Convert mass to moles

To convert the mass of acetylsalicylic acid (\(100 \mathrm{mg}\)) into moles, we need the molar mass of acetylsalicylic acid, which can be calculated as follows: Molar mass of C\(_9\)H\(_8\)O\(_4\) = 9 (Molar mass of C) + 8 (Molar mass of H) + 4 (Molar mass of O) = 9(12.01 g/mol) + 8(1.01 g/mol) + 4(16.00 g/mol) = 180.16 g/mol Now, convert the mass to moles: 100 mg = 0.1 g (1 mg = 0.001 g) Moles of acetylsalicylic acid = mass/molar mass = 0.1 g / 180.16 g/mol ≈ 0.000555 mol

Step 2: Find the molar concentration of acetylsalicylic acid

To find the molar concentration of acetylsalicylic acid, we will divide the moles of acetylsalicylic acid by the volume of the solution in liters: Volume of the solution = 200 mL = 0.2 L (1 mL = 0.001 L) Molar concentration of acetylsalicylic acid = Moles / Volume = 0.000555 mol / 0.2 L ≈ 0.00278 M

Step 3: Use \(K_a\) to find the concentration of \(\mathrm{H}^{+}\) ions

The dissociation of acetylsalicylic acid can be represented by the following equation: \(\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}\rightleftharpoons\mathrm{H}^{+}+\mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^{-}\) Since \(K_a\) of acetylsalicylic acid is given (\(3.3 \times 10^{-4}\)), we can write: \[K_a = \frac{[\mathrm{H}^{+}][\mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^{-}]}{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]}\] Let the concentration of \(\mathrm{H}^{+}\) ions at equilibrium be x: \[\begin{array}{l} 3.3 \times 10^{-4} = \frac{x^{2}}{(0.00278 - x)} \end{array}\] However, since the value of \(K_a\) is small, we can assume that x is much smaller compared to 0.00278. Therefore, we can approximate the equation as follows: \[\begin{array}{l} 3.3 \times 10^{-4} = \frac{x^{2}}{0.00278} \end{array}\] Solve for x: \[x = \sqrt{3.3 \times 10^{-4} \times 0.00278} \approx 0.000295\]

Step 4: Calculate the pH

Now that we have found the concentration of \(\mathrm{H}^{+}\) ions, we can calculate the pH of the aspirin solution using the pH formula: \[pH = -\log_{10}[\mathrm{H}^{+}]\] pH = -log10(0.000295) ≈ 3.53 The pH of the aspirin solution is approximately 3.53.

Key Concepts

Acid dissociation constant

An acid dissociation constant, known as \( K_a \), is a crucial parameter in chemistry that quantifies the strength of an acid in solution. It provides insight into how well an acid can donate its proton (\( \mathrm{H}^+ \)) to the surrounding solvent.
When an acid dissolves in water, it partially ionizes into hydrogen ions and its conjugate base. This ionization can be depicted as follows:

• \( \mathrm{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^- \)

In this equation, \( \mathrm{HA} \) represents the acid, while \( \mathrm{A}^- \) is its conjugate base. The \( K_a \) expression is:

• \( K_a = \frac{[\mathrm{H}^+][\mathrm{A}^-]}{[\mathrm{HA}]} \)

In this exercise, we are dealing with acetylsalicylic acid, where \( K_a = 3.3 \times 10^{-4} \). Such a small \( K_a \) value suggests that acetylsalicylic acid is a weak acid, only slightly dissociating in solution. This approximation allows further simplifications in our calculations, making it easier to work with equilibrium expressions.

Equilibrium calculations

Equilibrium calculations are essential in chemistry as they help us determine the concentrations of reactants and products in a reversible reaction at equilibrium. When dealing with weak acids, we often need equilibrium calculations to find out how much the acid dissociates in water.
For acetylsalicylic acid, the dissociation in water yields:

• \( \mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4} \rightleftharpoons \mathrm{H}^+ + \mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^- \)

To calculate equilibrium concentrations, we use the given \( K_a \) and set up an equation:

• \( K_a = \frac{[\mathrm{H}^+][\mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^-]}{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]} \)

Since the initial concentration of \( \mathrm{H}^+ \) ions is zero, the changes in concentration will influence each species involved in the equilibrium.
By assuming \( x \), the concentration of ions formed, is small compared to the initial concentration of the acid, we approximate:

• \( K_a \approx \frac{x^2}{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_\text{initial}} \)

Solving for \( x \) provides insight into how the molecules are distributed at equilibrium.

pH calculation

pH is a measure of the acidity or basicity of an aqueous solution. It is important for understanding the chemical behavior of substances when dissolved in water. The pH scale typically ranges from 0 (very acidic) to 14 (very basic), with 7 being neutral.
To calculate the pH of the solution, we use the concentration of hydrogen ions \( [\mathrm{H}^+] \), which we deduced from our equilibrium calculations:

• \( pH = -\log_{10}[\mathrm{H}^+] \)

In this exercise, after determining \( [\mathrm{H}^+] \approx 0.000295 \), we apply the pH formula:

• \( pH = -\log_{10}(0.000295) \approx 3.53 \)

This result indicates that the aspirin solution is slightly acidic, as expected for a solution of a weak monoprotic acid. The calculated pH provides valuable insights into the potential effects of the solution, whether it's benign or requires caution when handled.

Molar concentration

Molar concentration, often referred to as molarity, is a key concept in chemistry that describes the number of moles of a solute present in one liter of solution. Understanding molarity is vital for performing accurate quantitative chemistry.
To find the molar concentration of acetylsalicylic acid in our solution, we start by converting the given mass of the solute to moles, using its molar mass:

• \( \text{Moles} = \frac{\text{mass}}{\text{molar mass}} \)
• \( \text{Mass} = 0.1 \text{ g}, \text{Molar mass} = 180.16 \text{ g/mol}, \) yields about \( 0.000555 \text{ moles} \)

Next, we calculate molarity by dividing the number of moles by the volume of the solution in liters:

• \( \text{Volume} = 0.2 \text{ L} \), thus \( \text{Molarity} = \frac{0.000555}{0.2} \approx 0.00278 \text{ M} \)

This process of converting mass to molarity is crucial as it allows chemists to understand and predict the reactivity and behavior of the solution in various chemical processes.