Question

Aspirin is well known as a pain reliever (analgesic) and as a fever reducer (antipyretic). It has a molar mass of \(180.2 \mathrm{~g} / \mathrm{mol}\) and a composition of \(60.0 \% \mathrm{C}, 4.48 \% \mathrm{H}\), and \(35.5 \% \mathrm{O}\). Calculate the molecular formula of aspirin.

Short answer

The molecular formula of aspirin is \(C_9H_8O_4\).

Step-by-step solution

- Determine the mass of each element in 100 grams of aspirin

Assume you have a 100 gram sample of aspirin. Based on the given percentages, 60.0 grams are Carbon (C), 4.48 grams are Hydrogen (H), and 35.5 grams are Oxygen (O).

- Convert masses to moles

Use the molar masses of each element to convert the mass into moles: \[ \text{moles of C} = \frac{60.0 \text{ grams}}{12.01 \text{ g/mol}} = 5.00 \text{ moles} \] \[ \text{moles of H} = \frac{4.48 \text{ grams}}{1.008 \text{ g/mol}} = 4.44 \text{ moles} \] \[ \text{moles of O} = \frac{35.5 \text{ grams}}{16.00 \text{ g/mol}} = 2.22 \text{ moles} \]

- Determine the simplest whole-number mole ratio

Divide all the mole values by the smallest number of moles calculated: \[ \text{C:} \frac{5.00}{2.22} = 2.25 \] \[ \text{H:} \frac{4.44}{2.22} = 2.00 \] \[ \text{O:} \frac{2.22}{2.22} = 1.00 \]

- Adjust to get whole numbers

Multiply the mole ratio by a common factor to get whole numbers (in this case, multiply by 4): \[ \text{C:} 2.25 \times 4 = 9 \]\[ \text{ H:} 2.00 \times 4 = 8 \] \[ \text{O:} 1.00 \times 4 = 4 \]

- Write the empirical formula

From the whole numbers determined, the empirical formula for aspirin is \(C_9H_8O_4\).

- Verify the molecular formula using molar mass

Calculate the molar mass of the empirical formula \(C_9H_8O_4\): \[ (9 \times 12.01) + (8 \times 1.008) + (4 \times 16.00) = 180.16 \text{ grams/mol} \] Since this matches the given molar mass of 180.2 grams/mol, the molecular formula is also \(C_9H_8O_4\).

Key concepts

molar mass

Molar mass is a key concept in chemistry. It refers to the mass of one mole of a substance. You can find the molar mass by adding up the atomic masses of all the atoms in a molecular formula. For example, the molar mass of aspirin (C_9H_8O_4) is calculated by adding the molar masses of 9 Carbon atoms, 8 Hydrogen atoms, and 4 Oxygen atoms. The calculation is: \[ (9 \times 12.01) + (8 \times 1.008) + (4 \times 16.00) \] This results in 180.16 grams per mole, very close to the given molar mass of 180.2 grams per mole. Molar mass is crucial for converting between grams and moles, making it easier to understand chemical quantities.

empirical formula

The empirical formula is the simplest whole-number ratio of atoms in a compound. To find it, you first convert the mass percentages of each element to grams, assuming you have a 100-gram sample. Then you convert these masses into moles using the molar mass of each element. From there, you determine the simplest mole ratio of the elements by dividing each number of moles by the smallest number of moles calculated. If necessary, adjust these ratios to whole numbers by multiplying by a common factor. For aspirin, the empirical formula calculated is C_9H_8O_4. This formula tells us the basic ratio of Carbon, Hydrogen, and Oxygen atoms in the compound.

mole ratio

The mole ratio is essential in finding the empirical formula. It represents the relative number of moles of each element in a compound. To find the mole ratio, you divide the number of moles of each element by the smallest number of moles present. In the case of aspirin, we divided the moles of Carbon, Hydrogen, and Oxygen by the smallest number of moles calculated (2.22 moles of Oxygen). This gave us the ratios:

• Carbon: 2.25
• Hydrogen: 2.00
• Oxygen: 1.00

To convert these ratios to whole numbers, we multiplied by 4, leading to a final ratio of C:9, H:8, O:4. This information is crucial for determining the empirical formula.

percent composition

Percent composition indicates the percentage by mass of each element in a compound. It helps in determining empirical formulas by providing the mass of each element in 100 grams of the compound. For aspirin, the percent composition is given as:

• 60.0% Carbon
• 4.48% Hydrogen
• 35.5% Oxygen

This means that in 100 grams of aspirin, there are 60 grams of Carbon, 4.48 grams of Hydrogen, and 35.5 grams of Oxygen. Knowing the percent composition allows us to convert these masses into moles and then to find the simplest whole-number mole ratio.

stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It uses the concepts of moles, molar masses, and empirical formulas. Knowing the empirical formula and molar mass of a compound, stoichiometry helps in understanding the quantitative aspects of chemical formulas and reactions. For example, once we determine the empirical formula of aspirin (C_9H_8O_4) and confirm it with the given molar mass, we can use this information to predict the amounts of reactants needed for synthesis or the amount of product formed during a reaction. Stoichiometry is a powerful tool for solving real-world chemistry problems, making it easier to perform lab calculations and ensure precise measurements.