Problem 128 - Chemistry The Science in Context

Question

The common pain relievers aspirin \(\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\right),\) acetaminophen \(\left(\mathrm{C}_{8} \mathrm{H}_{9} \mathrm{NO}_{2}\right),\) and naproxen sodium \(\left(\mathrm{C}_{14} \mathrm{H}_{13} \mathrm{O}_{3} \mathrm{Na}\right)\) are all available in tablets containing \(200.0 \mathrm{mg}\) of the active ingredient. Which compound contains the greatest number of molecules per tablet? How many molecules of the active ingredient are present in each tablet?

Step-by-Step Solution

Answer

Answer: The compound with the greatest number of molecules per 200 mg tablet is Acetaminophen.

Step 1: Calculate the molar masses of the three compounds

First, calculate the molar mass (M) of each compound by adding up the atomic masses of each element in the compound. For aspirin \(\mathrm{C}_{9} \mathrm{H}_{8}\mathrm{O}_{4}\): \(M_{Aspirin} = 9 \times M_C + 8 \times M_H + 4 \times M_O = 9\times 12.01 + 8 \times 1.01 + 4 \times 16.00 = 180.16g/mol\) For acetaminophen \(\mathrm{C}_{8} \mathrm{H}_{9}\mathrm{NO}_{2}\): \(M_{Acetaminophen} = 8 \times M_C + 9 \times M_H + M_N + 2 \times M_O = 8 \times 12.01 + 9 \times 1.01 + 14.01 + 2 \times 16.00 = 151.17g/mol\) For naproxen sodium \(\mathrm{C}_{14} \mathrm{H}_{13} \mathrm{O}_{3} \mathrm{Na}\): \(M_{NaproxenSodium} = 14 \times M_C + 13 \times M_H + 3 \times M_O + M_{Na} = 14 \times 12.01 + 13 \times 1.01 + 3 \times 16.00 + 22.99 = 252.23g/mol\)

Step 2: Calculate the moles of each compound in the tablet

Next, we need to convert the mass of each compound (200 mg) to moles using the molar masses calculated above. Recall that 1 g = 1000 mg. Moles of Aspirin: \(n_{Aspirin} = \frac{200mg \times \frac{1g}{1000mg}}{180.16g/mol} = 0.001111mol\) Moles of Acetaminophen: \(n_{Acetaminophen} = \frac{200mg \times \frac{1g}{1000mg}}{151.17g/mol} = 0.001323mol\) Moles of Naproxen Sodium: \(n_{NaproxenSodium} = \frac{200mg \times \frac{1g}{1000mg}}{252.23g/mol} = 0.000793mol\)

Step 3: Calculate the number of molecules of each compound

Using Avogadro's number, convert the moles of each compound into the number of molecules. Aspirin: No. of molecules = \( n_{Aspirin} \times N_{A} = 0.001111mol \times 6.022 \times 10^{23}mol^{-1} = 6.68 \times 10^{20}\) Acetaminophen: No. of molecules = \(n_{Acetaminophen} \times N_{A} = 0.001323mol \times 6.022 \times 10^{23}mol^{-1} = 7.97 \times 10^{20}\) Naproxen Sodium: No. of molecules = \(n_{NaproxenSodium} \times N_{A} = 0.000793mol \times 6.022 \times 10^{23}mol^{-1} = 4.77 \times 10^{20}\)

Step 4: Determine which compound has the greatest number of molecules per tablet

Looking at the number of molecules calculated in step 3, we can determine which compound has the greatest number of molecules per tablet: Aspirin = \( 6.68 \times 10^{20}\) Acetaminophen = \( 7.97 \times 10^{20}\) Naproxen Sodium = \( 4.77 \times 10^{20}\) Comparing all the values, the compound with the greatest number of molecules per tablet is Acetaminophen.

Key Concepts

Molecular Formula

Understanding the molecular formula is a fundamental step in chemistry. It represents the actual number and type of atoms of each element present in a molecule. For example, aspirin has a molecular formula of \(C_9H_8O_4\), meaning it consists of 9 carbon atoms, 8 hydrogen atoms, and 4 oxygen atoms. A molecular formula provides information about the proportions of elements, which are key to determining molar mass—an essential component for various calculations including reactant-product conversion in reactions.

Moreover, knowing the molecular formula allows for the calculation of the percentage composition by mass of each element within the substance, which can be crucial for lab analysis and quality control in pharmaceuticals like aspirin, acetaminophen, and naproxen sodium.

Avogadro's Number

Avogadro's Number, denoted as \(N_A\), is a constant used to describe the number of units in a mole of any substance. Specifically, \(N_A = 6.022 \times 10^{23} mol^{-1}\), and it applies to atoms, ions, molecules, or other particles. It's named after Amedeo Avogadro, an Italian scientist who contributed to molecular theory. This number is crucial for converting between the macroscopic and microscopic scales.

For instance, calculating the number of molecules in a given amount of a substance—like the active ingredients in common pain relievers—requires the use of Avogadro's number to translate moles, a measurable quantity, into a countable number of molecules to understand quantities at the molecular level.

Stoichiometry

Stoichiometry is the study of quantitative relationships between the amounts of reactants and products in a chemical reaction. It's based on the conservation of mass where the mass of the reactants equals the mass of the products. This branch of chemistry uses the balanced equation of a chemical reaction to calculate the relative quantities of reactants and products involved.

Example of Stoichiometry

Consider producing aspirin by reacting salicylic acid with acetic anhydride. The stoichiometry of the reaction tells us how many moles of each reactant are needed and how many moles of products, including aspirin, will form. These calculations ensure that reactions are cost-effective and environmentally friendly by minimizing waste, making stoichiometry a cornerstone of green chemistry and manufacturing.

Moles to Molecules Conversion

The conversion from moles to molecules is essential in chemistry, as it links the macroscopic amounts we measure in the lab (like grams or liters) to the microscopic scale of individual molecules. This conversion is made possible by Avogadro's number.

To convert moles to molecules, you multiply the amount of substance in moles by Avogadro's number. For example, if you have 1 mole of aspirin, you would have \(6.022 \times 10^{23}\) molecules of aspirin. This relationship helps chemists determine how many molecules are in a given sample of a compound, which is fundamental for determining dosages in medication and in many other applications of chemistry in real life.