Question

What is the mass in grams of \(5.0 \times 10^{20}\) molecules of aspirin \(\left(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\right) ?\) (Hint: Using Avogadro's number, convert the number of molecules to moles.)

Short answer

The mass of \(5.0 \times 10^{20}\) molecules of aspirin is approximately 0.149 grams.

Step-by-step solution

Understand the Problem

We need to find the mass in grams of a given number of aspirin molecules \(5.0 \times 10^{20}\). Aspirin is represented by the chemical formula \(\text{C}_9 \text{H}_8 \text{O}_4\). To solve this, we will first convert the number of molecules to moles using Avogadro's number, \(6.022 \times 10^{23}\, \text{molecules/mole}.\)

Convert Molecules to Moles

Use Avogadro's number to convert the molecules into moles. The formula is: \(\text{moles} = \frac{\text{number of molecules}}{\text{Avogadro's number}}\). Plug in the numbers: \[\text{moles} = \frac{5.0 \times 10^{20}}{6.022 \times 10^{23}}\approx 8.30 \times 10^{-4}\, \text{moles}.\]

Find Molar Mass of Aspirin

Calculate the molar mass of \(\text{C}_9 \text{H}_8 \text{O}_4\). Add together the atomic masses of all the atoms: \(\text{C} = 12.01\, \text{g/mol}, \text{H} = 1.008\, \text{g/mol}, \text{O} = 16.00\, \text{g/mol}\). \[\text{Molar mass} = 9(12.01) + 8(1.008) + 4(16.00) = 180.16\, \text{g/mol}.\]

Calculate Mass of Aspirin in Grams

To find the mass, multiply the number of moles by the molar mass: \(\text{mass} = \text{moles} \times \text{molar mass}\). Substitute the values found: \[\text{mass} = 8.30 \times 10^{-4}\, \text{mol} \times 180.16\, \text{g/mol} \approx 0.149\, \text{g}.\]

Key concepts

Avogadro's number

Avogadro's number is a fundamental constant in chemistry that connects the macroscopic scale of substances we can see and touch with the microscopic scale of atoms and molecules. It's a large number: approximately \(6.022 \times 10^{23}\). This number tells us how many particles (like atoms or molecules) are in one mole of a substance.
When we discuss chemical reactions and compositions, it's crucial to have a common unit to count these tiny particles. Avogadro's number serves this purpose by allowing chemists to work in moles, which relate directly to grams for practical measurement.
To convert the number of molecules into moles, you simply divide the number of molecules by Avogadro's number. For example, if you have \(5.0 \times 10^{20}\) molecules of aspirin, you divide this by \(6.022 \times 10^{23}\) to find how many moles you have. This conversion simplifies calculations for further understanding and usage in chemical reactions.

Molar mass

Molar mass is another cornerstone concept in chemistry that you need to grasp when performing mole calculations. It describes the mass of one mole of a substance, measured in grams per mole (g/mol).
To determine the molar mass of a compound, you sum the masses of each element as indicated by the chemical formula. For aspirin, with the formula \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\), you calculate it by adding together the average atomic masses of carbon, hydrogen, and oxygen based on their count in the formula.
For instance, in aspirin: you'll multiply the atomic mass of carbon (12.01 g/mol) by the number of carbon atoms (9), the atomic mass of hydrogen (1.008 g/mol) by the number of hydrogen atoms (8), and the atomic mass of oxygen (16.00 g/mol) by the number of oxygen atoms (4). Adding these values gives you the molar mass, \(180.16 \, \text{g/mol}\). This measure gives you the weight of one mole of aspirin and is crucial for calculating how many grams appear from a given mole count.

Chemical formulas

Chemical formulas are a symbolic representation of a compound's composition. They show which elements are present and in what numerical proportion. In the chemical formula for aspirin, \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\), the subscripts indicate there are 9 carbon atoms, 8 hydrogen atoms, and 4 oxygen atoms.
Understanding chemical formulas is essential for mole calculations because they directly indicate the number of atoms of each element needed to calculate molar masses. Moreover, these formulas help predict the outcome of chemical reactions by illustrating which elements and how much of each are involved.
When calculating the mass of a compound in grams, the formula helps determine the molar mass by providing accurate atomic ratios needed to sum up the atomic masses. This is why mastering chemical formulas lets you convert between moles and grams and aids in solving complex chemical equations.