Question

Acetaminophen, \(\mathrm{HC}_{8} \mathrm{H}_{8} \mathrm{NO}_{2}(\mathrm{MM}=151.17 \mathrm{~g} / \mathrm{mol}),\) is the active ingredient in Tylenol \(^{\circledast},\) a common pain reliever. A solution is made by dissolving \(6.54 \mathrm{~g}\) of acetaminophen in enough water to make \(250.0 \mathrm{~mL}\) of solution. The resulting solution has a pH of 5.24 . What is \(K_{\mathrm{a}}\) for acetaminophen?

Short answer

Answer: To find the Ka of acetaminophen, follow these steps: 1. Calculate the concentration of acetaminophen in the solution by converting grams to moles and dividing by the volume of the solution in liters. 2. Calculate the concentration of H+ ions in the solution using the given pH value and the formula [H+] = 10^(-pH). 3. Write the Ka expression for acetaminophen and substitute the obtained values of acetaminophen concentration and H+ ions concentration. 4. Solve the expression for Ka to find the acid dissociation constant of acetaminophen.

Step-by-step solution

Calculate the concentration of acetaminophen in the solution

To find the concentration, we first need to convert the mass of acetaminophen into moles using its molar mass. \(\mathrm{MM}=151.17 \mathrm{~g} / \mathrm{mol}\) Moles of acetaminophen = \(\frac{6.54 \mathrm{~g}}{151.17 \mathrm{~g} / \mathrm{mol}}\) Next, we need to convert the volume of the solution from mL to L. Volume of solution = \(\frac{250.0 \mathrm{~mL}}{1000 \mathrm{~mL} / \mathrm{L}}\) Now, we can find the concentration by dividing moles of acetaminophen by the volume of the solution in L. Concentration of acetaminophen = \(\frac{\text{Moles of acetaminophen}}{\text{Volume of solution}}\)

Calculate the concentration of H+ ions in the solution using pH

As the pH of the solution is given, we can use the formula for pH to calculate the concentration of H+ ions in the solution. pH = -log\(_{10}\)[H+] So, [H+] = 10\(^{-\text{pH}}\) Substitute the given pH value (5.24) into the equation: [H+] = 10\(^{-5.24}\)

Write the Ka expression for acetaminophen and solve for Ka

The dissociation of acetaminophen can be represented as: HC8H8NO2(aq) \(\rightleftharpoons\) H+(aq) + C8H8NO2-(aq) The Ka expression for acetaminophen is: Ka = \(\frac{[\text{H+}][\text{C}_{8}\text{H}_{7}\text{NO}_{2}^{-}]}{[\text{HC}_{8}\text{H}_{8}\text{NO}_{2}]}\) Since we are only given one pH value, we will assume that the concentrations of H+ and C8H7NO2- ions are equal (x), and [HC8H8NO2] is the initial concentration minus x. Ka = \(\frac{x^2}{\text{acetaminophen concentration} - x}\) Substitute the values of [H+] and acetaminophen concentration obtained in step 1 and step 2: Ka = \(\frac{(10^{-5.24})^2}{\text{acetaminophen concentration} - 10^{-5.24}}\)

Solve for Ka

Now that we have the expression for Ka, we can solve for Ka by plugging in the values which we found in step 1 and step 2 to find the Ka of acetaminophen. Ka = \(\frac{(10^{-5.24})^2}{\text{acetaminophen concentration} - 10^{-5.24}}\) Finally, calculate the value of Ka for acetaminophen.