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 pH of a solution obtained by dissolving two extra-strength aspirin tablets, containing 500 \(\mathrm{mg}\) of acetylsalicylic acid each, in 250 \(\mathrm{mL}\) of water?

Short answer

To find the pH of the aspirin solution, follow these steps: 1. Calculate the initial concentration of aspirin, \([\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial}\), by converting the mass of aspirin to moles and dividing by the volume of the solution. 2. Create an ICE table, and use the provided \(K_a\) value together with the assumption that the change in x is much smaller than the initial concentration of aspirin, to approximate and simplify the equation to: \(K_a \approx \frac{x^2}{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial}}\). 3. Solve for \([\mathrm{H}^{+}]\) by rearranging the equation: \(x = [\mathrm{H}^{+}] = \sqrt{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial} \times K_a}\). 4. Calculate the pH using \(\text{pH} = -\log_{10} {[\mathrm{H}^{+}]}\). After plugging in the numbers and performing the calculations, you will obtain the pH of the aspirin solution.

Step-by-step solution

Calculate the initial concentration of aspirin in the solution

To find the initial concentration, we will first convert the mass of aspirin to moles, as we are given the mass of aspirin (500 mg per tablet, and there are two tablets). The molecular weight of acetylsalicylic acid is $$ 1 \times 12.01 \,\text{(for C)}+ 9 \times 1.01 \,\text{(for H)}+ 4 \times 16.00 \,\text{(for O)} = 180.16 \, \mathrm{g/mol} $$. So, we can calculate the moles of aspirin as follows: $$ \text{moles of aspirin} = \frac{\text{mass of aspirin}}{\text{molecular weight}} = \frac{2 \times 500 \, \mathrm{mg}}{180.16 \, \mathrm{g/mol}} \times \frac{1 \, \mathrm{g}}{1000 \, \mathrm{mg}}$$ Now, we can find the initial concentration using the volume of the solution (250 mL): $$ \left[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}\right]_{initial} = \frac{\text{moles of aspirin}}{\text{volume of solution}} = \frac{\text{moles of aspirin}}{250 \, \mathrm{mL}} \times \frac{1000 \, \mathrm{mL}}{1 \, \mathrm{L}} $$

Create an ICE table and find the concentrations at equilibrium

To understand what happens when the acid dissociates, we can make an ICE table (Initial, Change, Equilibrium): $$\begin{array}{c|ccc} & \mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4} & \mathrm{H}^{+} & \mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^{-} \\ \hline \text{Initial} & [\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial} & 0 & 0 \\ \text{Change} & -x & +x & +x \\ \text{Equilibrium} & [\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial} - x & x & x \end{array} $$ Now, we will use the Ka value to relate the equilibrium concentrations. $$ K_a = 3.3 \times 10^{-4} = \frac{[\mathrm{H}^{+}][\mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^{-}]}{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]} = \frac{x^2}{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial} - x} $$ Since \(K_a\) is very small, we can assume that the change in x is much smaller than the initial concentration of aspirin (x << \([\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial}\])). Therefore, we can simplify the equation. $$ K_a \approx \frac{x^2}{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial}} $$

Solve for [H+] and calculate the pH

Now, we can rearrange the equation and solve for the concentration of H+ ions, which will be equal to x: $$ x = [\mathrm{H}^{+}] = \sqrt{[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}]_{initial} \times K_a} $$ Finally, we can find the pH of the solution using the hydrogen ion concentration: $$\text{pH} = -\log_{10} {[\mathrm{H}^{+}]}$$ Plug in the numbers for all the calculations above and compute the pH.

Key concepts

Acetylsalicylic Acid

Acetylsalicylic acid is a well-known substance that many people recognize as the active ingredient in aspirin. Chemically, it's a type of carboxylic acid with the formula \(\mathrm{HC}_{9} \mathrm{H}_{7} \mathrm{O}_{4}\). As an acid, it has the ability to donate a hydrogen ion (\(\mathrm{H}^{+}\)) when dissolved in water, turning into its conjugate base, salicylate \(\mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^{-}\).
This reaction is reversible and reaches a state of equilibrium in solution. The degree to which acetylsalicylic acid disassociates in water is its acid dissociation constant, \(K_{a}\). With a \(K_{a}\) value of \(3.3 \times 10^{-4}\), acetylsalicylic acid is a weak acid. This compounds' properties are crucial for understanding how it behaves in solution, which directly affects its medicinal effects and the pH of the solution when aspirin dissolves in water.
In the context of the textbook problem, understanding the properties of acetylsalicylic acid is the first step to calculating the pH of an aspirin solution.

pH Calculation

The pH scale is a measure of the acidity or basicity of an aqueous solution. A low pH indicates a high concentration of hydrogen ions (acidic), while a high pH indicates a low concentration of hydrogen ions (basic). The pH scale generally ranges from 0 to 14, with 7 being neutral.
Calculating the pH involves finding the concentration of hydrogen ions in the solution and then taking the negative logarithm to the base 10 of that concentration. Mathematically, it is expressed as \(\text{pH} = -\log_{10} {[\mathrm{H}^{+}]}\). Since the concentration of hydrogen ions is directly related to the disassociation of an acid like acetylsalicylic acid in water, pH calculation often begins with determining how much the acid disassociates in the solution. The problem hence requires an understanding of how to measure and calculate ion concentrations, which is a fundamental aspect of aqueous chemistry solutions.

ICE Table

An ICE table stands for Initial, Change, and Equilibrium. This table is a tool commonly used in chemistry to track the changes in concentrations of reactants and products during a chemical reaction that reaches equilibrium.

How to Set Up an ICE Table

For a general reaction \(\mathrm{A} \rightarrow \mathrm{B} + \mathrm{C}\), the table is divided into three rows that represent the concentrations of substances at the initial moment, the change that occurs, and the equilibrium state. It provides a systematic way to apply the equilibrium constant expression to find unknown concentrations.

• The Initial row shows the initial concentrations of reactants and products.
• The Change row shows how the concentrations change, often represented by \(x\) or \(\pm x\).
• The Equilibrium row shows the concentrations at equilibrium and is determined by combining the initial concentrations with the changes.

By applying the equilibrium constant, such as the acid dissociation constant \(K_a\), one can solve for the unknown \(x\), which represents the concentration of the ions at equilibrium. Correctly utilizing the ICE table, especially in the context of weak acid disassociation, allows students to accurately determine important solution parameters such as pH.

Acid Dissociation Constant

The acid dissociation constant, \(K_a\), is a quantitative measure of an acid's strength in a solution. It indicates the extent to which an acid disassociates into ions, specifically, the formation of \(\mathrm{H}^{+}\) ions. The \(K_a\) value is determined by the equilibrium concentrations of the reactants and products involved in the acid disassociation reaction.
A larger \(K_a\) value implies a stronger acid, which disassociates more completely in solution. Conversely, a smaller \(K_a\) indicates a weaker acid, which disassociates less fully. Acetylsalicylic acid has a \(K_a\) value of \(3.3 \times 10^{-4}\), which reflects its nature as a weak acid.
Understanding \(K_a\) is essential for predicting the behavior of acids in solution, including calculating the pH. It allows chemists to infer how shifting conditions may impact the concentrations of \(\mathrm{H}^{+}\) ions, and hence, the acidity of a solution. Calculations involving \(K_a\) often assume that the initial concentration of the acid is not significantly depleted through disassociation, a reasonable simplification for weak acids at low concentrations.