Problem 77 - Chemistry and Chemical Reactivity

Question

Acetaminophen, whose structure is drawn below, is the active ingredient in some nonprescription pain killers. The recommended dose for an adult is two \(500-\mathrm{mg}\) caplets. How many molecules make up one dose of this drug?

Step-by-Step Solution

Answer

The one dose of acetaminophen contains approximately \(3.98 \times 10^{21}\) molecules.

Step 1: Determine the Total Mass of the Dose

Calculate the mass of the recommended dose, which consists of two caplets.Each caplet is **500 mg**, therefore, the total mass is:\[2 \times 500 \text{ mg} = 1000 \text{ mg} = 1 \text{ g}\]

Step 2: Convert Grams to Moles

The molar mass of acetaminophen (C₈H₉NO₂) is needed. Using the periodic table, calculate the molar mass:\[C: 8 \times 12.01 \, \text{g/mol} = 96.08 \, \text{g/mol}\]\[H: 9 \times 1.01 \, \text{g/mol} = 9.09 \, \text{g/mol}\]\[N: 1 \times 14.01 \, \text{g/mol} = 14.01 \, \text{g/mol}\]\[O: 2 \times 16.00 \, \text{g/mol} = 32.00 \, \text{g/mol}\]Total molar mass = 96.08 + 9.09 + 14.01 + 32.00 = 151.18 \, \text{g/mol}Now, convert from grams to moles:\[\text{Moles of acetaminophen} = \frac{1 \, \text{g}}{151.18 \, \text{g/mol}} \approx 0.00661 \, \text{mol}\]

Step 3: Convert Moles to Molecules

Use Avogadro's number to find the total number of molecules. Avogadro's number is\[6.022 \times 10^{23} \text{ molecules/mol}\]Thus, the number of molecules is:\[0.00661 \, \text{mol} \times 6.022 \times 10^{23} \text{ molecules/mol} \approx 3.98 \times 10^{21} \text{ molecules}\]

Key Concepts

Molar Mass

The molar mass is a vital concept in chemistry. It refers to the mass of one mole of a substance, typically expressed in grams per mole (g/mol). To find the molar mass of a compound, you need to add up the atomic masses of all the atoms in its molecular formula. In the case of acetaminophen, which has the formula \( \text{C}_8\text{H}_9\text{NO}_2 \), you look at the atomic mass of each element in the compound from the periodic table:

• Carbon (C): 12.01 g/mol, and there are 8 carbons, so 8 × 12.01 = 96.08 g/mol
• Hydrogen (H): 1.01 g/mol, and there are 9 hydrogens, so 9 × 1.01 = 9.09 g/mol
• Nitrogen (N): 14.01 g/mol, with 1 nitrogen, so 1 × 14.01 = 14.01 g/mol
• Oxygen (O): 16.00 g/mol, with 2 oxygens, so 2 × 16.00 = 32.00 g/mol

Adding these together gives the molar mass of acetaminophen: 151.18 g/mol. Molar mass is crucial because it allows us to convert between grams and moles, a necessary step in various chemical calculations.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry. It represents the number of constituent particles, usually atoms or molecules, that are contained in the amount of substance known as one mole. Avogadro's number is approximately \(6.022 \times 10^{23} \) per mole.

Understanding Avogadro's number is key when dealing with molecular or atomic scales. For example, if we have \(0.00661\) moles of acetaminophen, we can calculate the number of molecules by multiplying the number of moles by Avogadro's number:

• \(0.00661 \times 6.022 \times 10^{23} \approx 3.98 \times 10^{21} \) molecules

This number helps us conceptualize how incredibly small atoms and molecules are, and how many fit into even tiny amounts of substance. It’s essential for converting between moles (a macroscopic measure) and molecules (a microscopic measure).

Acetaminophen Structure

The structure of acetaminophen is crucial for differentiating it from other compounds, as well as understanding its function and characteristics. Acetaminophen, also known by its chemical name N-(4-hydroxyphenyl)acetamide, is a simple organic compound commonly used as a pain reliever and fever reducer.

The molecular formula of acetaminophen is \(\text{C}_8\text{H}_9\text{NO}_2\). Here is a deeper look into its structure:

• Aromatic Ring: It includes a benzene ring, a six-carbon ring that is a common feature in many compounds due to its stability.
• Hydroxyl Group (-OH): This group is bonded to the aromatic ring, contributing to the molecule’s reactivity and polarity.
• Acetamide Group: The NH (nitrogen-hydrogen) bond and carbonyl (C=O) group form this group, giving acetaminophen its characteristic medicinal properties.

Understanding its structural components helps in grasping how acetaminophen interacts at the molecular level, where these functional groups play crucial roles in its effectiveness as a pharmaceutical agent.