Problem 57 - Chemistry: Principles and Reactions

Question

There are \(324 \mathrm{mg}\) of acetylsalicylic acid \((\mathrm{MM}=180.15 \mathrm{~g} / \mathrm{mol})\) per aspirin tablet. If two tablets are dissolved in water to give two ounces \(\left(\frac{1}{16}\right.\) quart) of solution, estimate the \(\mathrm{pH} . K_{\mathrm{a}}\) of acetylsalicylic acid is \(3.6 \times 10^{-4}\).

Step-by-Step Solution

Answer

Answer: The estimated pH of the resulting solution is approximately 2.51.

Step 1: Calculate the moles of acetylsalicylic acid in two tablets

To calculate the moles of acetylsalicylic acid in two tablets, we will use the given mass per tablet and molecular weight: moles of acetylsalicylic acid per tablet = \(\frac{\mathrm{mass}}{\mathrm{molecular~weight}}\) moles of acetylsalicylic acid in two tablets = 2 * \(\frac{324~\mathrm{mg}}{180.15~\mathrm{g/mol}}\) Convert mass from mg to g: moles of acetylsalicylic acid in two tablets = 2 * \(\frac{0.324~\mathrm{g}}{180.15~\mathrm{g/mol}}\) Calculate the moles: moles of acetylsalicylic acid in two tablets = 0.00360 mol

Step 2: Find the total volume of the solution

As mentioned, two tablets are dissolved in two ounces of water, and we need to convert this volume to liters: two ounces = \(\frac{1}{8}~\mathrm{quart}\) (since 1 quart = 16 ounces) Now, convert quarts to liters: two ounces = \(\frac{1}{8} \times 0.946~\mathrm{L}\) Calculate volume in liters: volume of the solution = 0.11825 L

Step 3: Determine the concentration of acetylsalicylic acid in the solution

To calculate the concentration of acetylsalicylic acid in the solution, we will use the moles determined in step 1 and the volume determined in step 2: concentration of acetylsalicylic acid = \(\frac{\mathrm{moles}}{\mathrm{volume}}\) Calculate the concentration: concentration of acetylsalicylic acid = \(\frac{0.00360~\mathrm{mol}}{0.11825~\mathrm{L}}\) Find the concentration: concentration of acetylsalicylic acid = 0.03042 M

Step 4: Using the concentration and Ka value, find the pH of the solution

Since acetylsalicylic acid is a weak acid, we can use the Ka value and the ICE table to calculate the pH of the solution: Given Ka value: \(3.6 \times 10^{-4}\) Set the concentration of acetylsalicylic acid as "initial concentration" in the ICE table. Assume that x moles of acetylsalicylic acid will ionize. After ionization, we will have: [H+] = x [A-] = x [HA] = 0.03042 - x Now, using the Ka expression: Ka = \(\frac{[\mathrm{H}^+][\mathrm{A^-}]}{[\mathrm{HA}]}\) Substitute the assumed ionization values and given Ka value: \(3.6 \times 10^{-4} = \frac{x^2}{0.03042 - x}\) Since the acid is weak, we can approximate x to be much smaller than 0.03042 and simplify the equation: \(3.6 \times 10^{-4} \approx \frac{x^2}{0.03042}\) Calculate the value of x (H+ concentration): x = 0.00310 M Lastly, we will calculate the pH using the concentration of H+: pH = -log [H+] Calculate the pH: pH = -log (0.00310) Find the pH: pH ≈ 2.51 Thus, the estimated pH of the solution is approximately 2.51.

Key Concepts

Acid Dissociation Constant (Ka)

The acid dissociation constant \textbf{(Ka)} is an essential value representing the strength of an acid in a solution. It measures the acid's tendency to donate hydrogen ions (\textbf{H+}) to the solution. For a weak acid like acetylsalicylic acid, the process can be represented as: \[\begin{equation} HA \rightleftharpoons H^+ + A^- \text{(where HA is the weak acid and A- is the conjugate base)}\text{The Ka expression for this equilibrium is:} \ Ka = \frac {[H^+][A^-]}{[HA]}\end{equation}\] Understanding Ka is crucial for predicting the behavior of the acid in a solution, including its ionization level and influence on pH. The larger the value of Ka, the stronger the acid, as it indicates more H+ ions are released into the solution. For acetylsalicylic acid (\textbf{Ka = }\(3.6 \times 10^{-4}\)), this low Ka value indicates that the substance is a weak acid that only partially ionizes in water.When the concentration of the undissociated acid is significantly higher than the concentration of H+ and A-, which is the typical case with weak acids, the denominator in the Ka expression can be approximated as the initial concentration of the acid. This approximation simplifies calculations involved in determining the pH of the solution.

Molar Mass Calculation

The molar mass of a compound is the mass in grams of one mole of that substance. It is a fundamental concept in chemistry, especially when converting between mass and moles of a substance. Calculating molar mass involves summing the atomic masses of all atoms present in a molecule. For acetylsalicylic acid, with a given formula weight of \(180.15 \text{ g/mol}\), knowing the molar mass is necessary to convert the mass of acetylsalicylic acid in tablets (324 mg per tablet) to moles, which is the starting point for all subsequent quantitative analyses in our exercise.To calculate the moles from the mass provided, we use the formula: \[ \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]Understanding how to compute molar masses correctly is imperative for accurate measurement in chemical reactions and solutions, such as the one presented in our pH estimation problem.

Weak Acid pH Calculation

Calculating the pH of a weak acid, such as acetylsalicylic acid, requires a different approach compared to strong acids that completely dissociate. The pH is a scale used to specify the acidity or basicity (alkalinity) of an aqueous solution. It is defined as the negative logarithm (base 10) of the molar concentration of hydrogen ions:\[ \text{pH} = -\log[H^+] \]For weak acids, the equilibrium concentration of \textbf{H+} ions is not the initial concentration of the acid since they do not fully dissociate. Instead, it can be found by setting up an ICE (Initial, Change, Equilibrium) table and using the Ka value. By assuming that the change (x) due to ionization is small compared to the initial concentration, the calculation simplifies, and we can solve for x to find the hydrogen ion concentration. The pH calculation is then straightforward:\[ \text{pH} = -\log(x) \]It is imperative to remember that pH is a logarithmic scale, meaning each whole pH value below 7 (which is considered neutral) is ten times more acidic than the next higher value. In our exercise, after using the Ka and the approximation, we find that the pH of the acetylsalicylic acid solution is 2.51, reflecting its acidic character.

ICE Table Equilibrium

An ICE table, which stands for Initial, Change, Equilibrium, is a valuable tool for solving equilibrium problems in chemistry. It helps keep track of the concentrations of reactants and products throughout a chemical reaction. The table is set up with rows for each substance and columns for their initial concentration, change in concentration, and equilibrium concentration.For a weak acid dissociating in water (HA → H+ + A-), we may start with a certain initial concentration of HA and have no H+ or A- present initially. When the acid dissociates by a small amount (x), we indicate this change in the table. The equilibrium concentrations can then be used in the Ka expression:\[ Ka = \frac{\text{[H+][A-]}}{\text{[HA]}} \]The ICE table assists in visualizing the shift of concentrations upon reaching equilibrium, providing a systematic approach to calculate the unknown changes. This method is integral to solving equilibrium problems and is used to deduce the pH of weak acid solutions, as shown in our acetylsalicylic acid example.