Problem 88 - Chemistry The Central Science

Question

(a) The molecular formula of acetylsalicylic acid (aspirin), one of the most common pain relievers, is \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\). How many moles of \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\) are in a 0.500 -g tablet of aspirin? (b) How many molecules of \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\) are in this tablet? (c) How many carbon atoms are in the tablet?

Step-by-Step Solution

Answer

In the short version, the moles of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) in a 0.500-g aspirin tablet are approximately 0.002777 mol. There are approximately \(1.672 \times 10^{21}\) molecules of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) in the tablet, with around \(1.50 \times 10^{22}\) carbon atoms.

Step 1: Calculate the molar mass of acetylsalicylic acid.

First, we need to find the molar mass of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\). We can do this by multiplying the number of atoms of each element by their respective atomic masses and summing them up. Molar mass of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = (number of C atoms \(\times\) atomic mass of C) + (number of H atoms \(\times\) atomic mass of H) + (number of O atoms \(\times\) atomic mass of O) Molar mass of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = (9 \(\times\) 12.01 g/mol) + (8 \(\times\) 1.01 g/mol) + (4 \(\times\) 16.00 g/mol) = 180.17 g/mol.

Step 2: Determine the moles of acetylsalicylic acid in the tablet.

To find the moles of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\), we will use the given mass of the aspirin tablet (0.500 g) and the calculated molar mass (180.17 g/mol) using the formula: Moles of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = mass of tablet/molar mass of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) Moles of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = 0.500 g / 180.17 g/mol = 0.002777 mol

Step 3: Find the number of molecules in the tablet.

To determine the number of molecules in the tablet, we will use Avogadro's number, which is approximately \(6.022 \times 10^{23}\). Number of molecules of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = moles of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) \(\times\) Avogadro's number Number of molecules of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = 0.002777 mol \(\times\) \(6.022 \times 10^{23}\) = \(1.672 \times 10^{21}\)

Step 4: Calculate the number of carbon atoms in the tablet

The molecular formula of acetylsalicylic acid shows that it has 9 carbon atoms per molecule. Therefore, Number of carbon atoms in the tablet = number of molecules of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) \(\times\) 9 Number of carbon atoms in the tablet = \(1.672 \times 10^{21}\) \(\times\) 9 = \(15.05 \times 10^{21}\) ≈ \(1.50 \times 10^{22}\)

Key Concepts

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that allows us to count individual molecules and atoms in a given sample. It is defined as the number of atoms, ions, or molecules contained in one mole of a substance.

This number is approximately equal to \(6.022 \times 10^{23}\).

Avogadro's number is useful because it bridges the macroscopic world, which we can measure and see, with the microscopic world, which includes the atoms and molecules making up substances. For example, in the exercise, once we calculated the moles of acetylsalicylic acid, Avogadro's number enables us to convert this amount into the number of individual aspirin molecules. This conversion is crucial for understanding the number of particles involved in any chemical reaction.

Moles of a Substance

The mole is a basic unit in chemistry used to express amounts of a chemical substance. One mole contains exactly \(6.022 \times 10^{23}\) particles, which could be atoms, molecules, electrons, or other entities.

In practical terms, moles allow us to relate the mass of a substance to the number of particles it contains.

In the exercise, we use the mole concept to calculate how many moles of acetylsalicylic acid are in 0.500 grams of aspirin. This is done using the formula:

• Moles = mass of the substance / molar mass of the substance

The molar mass acts as a link between the mass we measure with scales and the number of molecules we can't directly see.

Molecular Formula

A molecular formula provides the exact number of each type of atom in a molecule of a compound.

For example, the molecular formula of acetylsalicylic acid is \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\), indicating that each molecule contains nine carbon atoms, eight hydrogen atoms, and four oxygen atoms.

Understanding a compound's molecular formula is essential because it tells us the makeup of the molecule and helps calculate the molar mass necessary for further computations, such as determining the number of moles or molecules. To find the molar mass, multiply the number of each type of atom by its atomic mass and sum these values, as demonstrated in step 1 of the solution.

Chemical Composition

Chemical composition refers to the identity and quantity of elements that make up a substance.

This gives insight into how the elements bond and react, affecting the substance's behavior and properties.

For acetylsalicylic acid, knowing its chemical composition (\(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\)) not only defines its structure but also enables us to predict its interactions in biological systems.

By understanding the chemical composition, we can delve into chemical calculations such as determining the number of carbon atoms in a sample, which is done by multiplying the number of molecules by the number of carbon atoms per molecule, as seen in the final step of the solution.