Problem 111 - Chemistry The Central Science

Question

Aspirin has the structural formula At body temperature \(\left(37^{\circ} \mathrm{C}\right), K_{a}\) for aspirin equals \(3 \times 10^{-5}\). If two aspirin tablets, each having a mass of \(325 \mathrm{mg},\) are dissolved in a full stomach whose volume is \(1 \mathrm{~L}\) and whose \(\mathrm{pH}\) is \(2,\) what percent of the aspirin is in the form of neutral molecules?

Step-by-Step Solution

Answer

Approximately \(99.97\%\) of the aspirin is in the form of neutral molecules when two aspirin tablets, each having a mass of \(325\,\text{mg}\), are dissolved in a full stomach with a volume of \(1\,\text{L}\) and a pH of \(2\).

Step 1: Convert mass of aspirin to moles

First, we need to convert the mass of aspirin to moles. The total mass of two tablets is \(325 \times 2 = 650\,\text{mg}\). The molar mass of aspirin is \(180.16\,\text{g/mol}\). We will convert the mass to grams and then to moles: \[650\,\text{mg} \times \frac{1\,\text{g}}{1000\,\text{mg}}\times \frac{1\,\text{mol}}{180.16\,\text{g}} = 0.00361\,\text{mol}\]

Step 2: Calculate the initial concentration of H+ ions

Next, we need to calculate the initial concentration of H+ ions in the stomach. We will use the given pH value: \[\text{pH} = -\log_{10}[\text{H}^+] \Rightarrow [\text{H}^+] = 10^{-\text{pH}}\] For an initial pH of \(2\), we get: \[[\text{H}^+] = 10^{-2} = 0.01\,\text{M}\]

Step 3: Determine equilibrium concentrations of aspirin, its conjugate base, and H+ ions

Now we will set up an equilibrium expression for the reaction between aspirin and water, using the given \(K_a\) value: {\small\[K_a = \frac{[\text{Aspirin}^-][\text{H}^+]}{[\text{Aspirin}]} = 3 \times 10^{-5}\]} Let x be the change in concentration of aspirin, its conjugate base, and H+ ions at equilibrium. Therefore, we have: \[[\text{Aspirin}] = 0.00361 - x\] \[[\text{Aspirin}^-] = x\] \[[\text{H}^+] = 0.01 + x\] Substituting these values into the equilibrium expression, we get: \[\frac{x(0.01+x)}{0.00361-x} = 3 \times 10^{-5}\] Since \(K_a\) is relatively small, the change in concentration (x) will be negligible compared to the initial concentration. Thus, we can approximate: \[\frac{x \times 0.01}{0.00361} = 3 \times 10^{-5}\]

Step 4: Calculate the percentage of aspirin in the form of neutral molecules

Now, we will solve for x and calculate the percentage of aspirin in the form of neutral molecules: \[x = 0.00361 \times 3 \times 10^{-5} \div 0.01 = 1.083\times 10^{-6}\,\text{mol}\] Since x represents the concentration of aspirin's conjugate base, the concentration of neutral aspirin molecules is \(0.00361-x\): \[[\text{Aspirin}] = 0.00361 - 1.083 \times 10^{-6} = 0.003608917\,\text{mol}\] To find the percentage, we will use the following formula: \[\text{Percent Neutral Aspirin} = \frac{\text{Amount of Neutral Aspirin}}{\text{Total Amount of Aspirin}} \times 100\%\] \[\text{Percent Neutral Aspirin} = \frac{0.003608917}{0.00361} \times 100\% \approx 99.97\%\] So, approximately \(99.97\%\) of the aspirin is in the form of neutral molecules.

Key Concepts

Acid Dissociation Constant (Ka)

The acid dissociation constant, denoted as \( K_a \), is an essential parameter in understanding the strength of an acid in solution. It provides insight into how completely an acid ionizes in water. For aspirin, which is an acidic compound, \( K_a \) is defined by:\[ K_a = \frac{[\text{Aspirin}^-][\text{H}^+]}{[\text{Aspirin}]} \]In other words, \( K_a \) is the ratio of the concentration of ionized aspirin to the concentration of non-ionized aspirin at equilibrium. The smaller the \( K_a \), the weaker the acid, as a smaller \( K_a \) indicates that very few molecules donate protons. At body temperature, aspirin has a \( K_a \) value of \( 3 \times 10^{-5} \), suggesting it is a weak acid. This information helps us to predict and calculate the proportions of neutral and ionized aspirin in the stomach's acidic environment.

pH Calculations

Understanding pH is crucial when dealing with acid-base chemistry, particularly when determining the acidic environment's effect on substances like aspirin. pH, representing the 'power of hydrogen,' is calculated as:\[ \text{pH} = -\log_{10}[\text{H}^+] \]For example, a pH of 2 implies a hydronium ion concentration \([\text{H}^+]\) of \(10^{-2}\,\text{M}\). In the given scenario of aspirin in the stomach, knowing the pH allows us to calculate the initial concentration of hydrogen ions, which is vital in setting up equilibrium expressions. This helps us to determine the degree of ionization and assess how much aspirin remains as neutral molecules in such acidic conditions.

Mole Conversions

Mole conversions are a foundational skill in chemistry, critical for determining quantities in chemical reactions. It involves converting between mass, moles, and number of molecules, using molar mass as the conversion factor. In our exercise with aspirin, two tablets at \(325\,\text{mg}\) each lead to a total mass of \(650\,\text{mg}\).To find the moles of aspirin, we first convert milligrams to grams:\[ 650\,\text{mg} \times \frac{1\,\text{g}}{1000\,\text{mg}} = 0.65\,\text{g} \]Then, knowing the molar mass of aspirin is \(180.16\,\text{g/mol}\), we convert grams to moles:\[ 0.65\,\text{g} \times \frac{1\,\text{mol}}{180.16\,\text{g}} = 0.00361\,\text{mol} \]This conversion helps us understand how much aspirin is present for interactions in the stomach, thereby aiding further calculations concerning dissociation and ionization in acidic solutions.