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 two extra-strength aspirin tablets, containing \(500 \mathrm{mg}\) of acetylsalicylic acid each, in \(250 \mathrm{~mL}\) of water?

Short answer

The short version of the solution to the question is as follows: 1. Calculate moles of acetylsalicylic acid: Moles = (2 * 500 mg) * (1 g / 1000 mg) * (1 mol / 180.16 g) 2. Find the initial concentration of acetylsalicylic acid: C1 = moles / (250 mL * (1 L / 1000 mL)) 3. Set up an equilibrium expression with Ka: 3.3 * 10^(-4) = x^2 / (C1 - x), and approximate (C1 - x) ≈ C1 4. Solve for x: x = sqrt((3.3 * 10^(-4)) * C1) 5. Calculate pH of the solution: pH = -log10(x) Using the above steps, we can calculate the pH of a solution obtained by dissolving two extra-strength aspirin tablets in 250 mL of water.

Step-by-step solution

Calculate moles of acetylsalicylic acid

First, we will find the number of moles of acetylsalicylic acid present in the solution. We have two extra-strength aspirin tablets, each containing 500 mg of acetylsalicylic acid. The molar mass of acetylsalicylic acid is 180.16 g/mol. Moles of acetylsalicylic acid = (2 tablets * 500 mg / tablet) * (1 g / 1000 mg) * (1 mol / 180.16 g)

Calculate the initial concentration of acetylsalicylic acid

Now, we will calculate the concentration of acetylsalicylic acid in the 250 mL of water. C1 = moles of acetylsalicylic acid / volume of water in L Volume of water = 250 mL * (1 L / 1000 mL)

Set up an equilibrium expression

We will next set up an equilibrium expression representing the ionization of acetylsalicylic acid in water. HC9H7O4 (aq) <-> H+ (aq) + C9H7O4- (aq) The initial concentrations of all the species are: [HC9H7O4] = C1 [H+] = 0 [C9H7O4-] = 0 At equilibrium, the concentrations of the species will be: [HC9H7O4] = C1 - x [H+] = x [C9H7O4-] = x Now we can write the Ka expression for acetylsalicylic acid: Ka = [H+][C9H7O4-] / [HC9H7O4] By substituting the equilibrium concentrations, we get: 3.3 * 10^(-4) = x^2 / (C1 - x)

Approximate x value and solve for x

As Ka is small, we can assume that x is much smaller than C1, therefore we can approximate: (C1 - x) ≈ C1 We can now solve for x: x^2 = (3.3 * 10^(-4)) * C1 x = sqrt((3.3 * 10^(-4)) * C1)

Calculate pH of the solution

Now, we can calculate the pH of the solution using the formula: pH = -log10[H+] Substitute the value of x as [H+] and calculate the pH value. With this step-by-step solution, you should be able to calculate the pH of a solution obtained by dissolving two extra-strength aspirin tablets in a given volume of water using the provided Ka value for acetylsalicylic acid.

Key concepts

Acid-Base Equilibrium

When you dissolve acetylsalicylic acid in water, it undergoes a process known as an acid-base equilibrium. This is when the acid (acetylsalicylic acid in this case) donates a proton (H+) to the water, forming its conjugate base (the salicylate ion, C9H7O4-) and hydronium ions (H3O+ or H+ in most equations). The equilibrium is established when the rate at which the acid donates protons is equal to the rate at which the conjugate base accepts them.

It is represented by the chemical equation:HC9H7O4 (aq) <-> H+ (aq) + C9H7O4- (aq)
The equilibrium constant for this reaction, Ka, quantifies the strength of the acid in solution, and in this exercise, it's given as 3.3 × 10^-4. A large Ka value means the acid is strong and ionizes significantly in solution, while a smaller value indicates a weak acid, which only partially ionizes in solution. In calculating the pH, it's essential to understand how this equilibrium functions and how the concentrations of the acid and ions at equilibrium influence the acidity of the solution.

Molar Mass Calculation

The molar mass of a compound is the mass of one mole of that substance. It is a vital step in chemistry problems since it allows conversion between grams and moles, a necessary operation to consider quantities of substances in reactions or solutions.

To calculate the molar mass, you need to sum the atomic masses of all the atoms in the compound. For acetylsalicylic acid (HC9H7O4), you would add the masses of hydrogen (H), carbon (C), and oxygen (O) based on their respective numbers in the compound's molecular formula. The units of molar mass are grams per mole (g/mol). In our problem, the molar mass of acetylsalicylic acid is given as 180.16 g/mol, which is used to convert the mass of the acid in grams to moles, an essential step for further calculations.

pH and pKa relationship

The pH of a solution is a measure of how acidic or basic it is. The scale typically ranges from 0 to 14, with 7 being neutral, below 7 acidic, and above 7 basic. The pH level is determined by the concentration of hydrogen ions (H+) in the solution. In contrast, pKa is the negative base-10 logarithm of the acid dissociation constant (Ka) and is a measure of the strength of an acid.

The relationship between pH and pKa is crucial in understanding acid-base equilibrium. When pH equals pKa, the concentrations of the acid and its conjugate base are equal. This relationship forms the basis of the 'Henderson-Hasselbalch equation', which is often used to calculate pH when dealing with weak acids or bases. The closer the pH is to the pKa value, the stronger the buffering capacity of the solution, meaning its pH will change less with the addition of acid or base. For our exercise problem, understanding this connection is key to finding the pH of the acetylsalicylic acid solution.

Chemical Concentration Calculation

To determine the concentration of a chemical in solution, you can use the formula:C = n / V
where C represents the concentration in moles per liter (M), n is the number of moles of the solute, and V is the volume of the solution in liters. In our exercise problem with acetylsalicylic acid, you calculated the number of moles by dividing the mass of the acid by its molar mass.

Then, you used the volume of water to find the initial concentration (C1) of acetylsalicylic acid in the solution before it starts to ionize. Obtaining this initial concentration allows us to use the acid-base equilibrium concepts and the provided Ka value to estimate how many moles of ionized hydrogen will be present at equilibrium, thus enabling us to calculate the pH of the solution.

Overall, by mastering these calculations, you can address a wide range of problems involving concentrations in solutions, which is a common task in various chemistry-related fields.