Question

(a) Diamond is a natural form of pure carbon. How many moles of carbon are in a \(1.25\) -carat diamond (1 carat \(=0.200 \mathrm{~g}\) )? How many atoms are in this diamond? (b) 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 \(0.500-\mathrm{g}\) tablet of aspirin? How many molecules of \(\mathrm{C}_{9} \mathrm{H}_{6} \mathrm{O}_{4}\) are in this tablet?

Short answer

(a) In a 1.25-carat diamond, there are approximately 0.0208 moles of carbon which corresponds to approximately \(1.25 \times 10^{22}\) carbon atoms. (b) In a 0.500 g tablet of aspirin, there are approximately 0.00277 moles of acetylsalicylic acid (\(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\)), which corresponds to approximately \(1.67 \times 10^{21}\) molecules.

Step-by-step solution

Convert carats to grams

To find the mass of the diamond in grams, we will multiply the given carats (1.25) by the conversion factor (1 carat = 0.200 g): Mass (g) = 1.25 carats × 0.200 g/carat = 0.250 g

Calculate the moles of carbon

To calculate the number of moles, we will divide the mass of the diamond (0.250 g) by the molar mass of carbon (12.01 g/mol): Moles C = Mass (g) / Molar mass of C (g/mol) Moles C = 0.250 g / 12.01 g/mol ≈ 0.0208 moles

Calculate the number of atoms of carbon

To find the number of carbon atoms, we will multiply the number of moles by Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol): Atoms C = Moles C × Avogadro's number Atoms C = 0.0208 moles × \(6.022 \times 10^{23}\) atoms/mol ≈ \(1.25 \times 10^{22}\) atoms (b)

Find the molar mass of acetylsalicylic acid \(\mathrm{C}_{9} \mathrm{H}_{8} \mathrm{O}_{4}\)

First, we need to look up the molar masses of C, H, and O, and calculate the molar mass of the formula: Molar mass = (9 × 12.01 g/mol) + (8 × 1.008 g/mol) + (4 × 16.00 g/mol) ≈ 180.2 g/mol

Calculate the moles of acetylsalicylic acid

To find the number of moles, we will divide the mass of the aspirin tablet (0.500 g) by the molar mass of acetylsalicylic acid (180.2 g/mol): Moles \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = Mass (g) / Molar mass of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) (g/mol) Moles \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = 0.500 g / 180.2 g/mol ≈ 0.00277 moles

Calculate the number of molecules of acetylsalicylic acid

To find the number of \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) molecules, we will multiply the number of moles by Avogadro's number (\(6.022 \times 10^{23}\) molecules/mol): Molecules \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = Moles \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) × Avogadro's number Molecules \(\mathrm{C}_{9}\mathrm{H}_{8}\mathrm{O}_{4}\) = 0.00277 moles × \(6.022 \times 10^{23}\) molecules/mol ≈ \(1.67 \times 10^{21}\) molecules

Key concepts

Conversion Factors

Conversion factors are incredibly useful in chemistry, especially when you need to switch units. These factors help convert measurements from one unit to another without changing the value.
For example, in the given exercise, the diamond's weight is measured in carats, but to find the moles of carbon, it needs to be converted to grams.

• The conversion factor is given: 1 carat equals 0.200 grams.
• To apply this factor, multiply the carat weight (1.25 carats) by the conversion factor (0.200 g/carat).
• Thus, you find that 1.25 carats equal 0.250 grams.

This factor ensures all measurements are in compatible units when performing calculations related to mass, moles, or atoms.

Molar Mass Calculation

Molar mass holds the key to unlocking the number of moles from a given mass. The molar mass of an element or compound represents the mass of one mole of its particles, expressed in grams per mole.
To find the molar mass of a compound like acetylsalicylic acid (C₉H₈O₄):

• You need to know the molar masses of individual elements: C (12.01 g/mol), H (1.008 g/mol), and O (16.00 g/mol).
• Then, compute using the formula of the compound: \[(9 \times 12.01) + (8 \times 1.008) + (4 \times 16.00) \approx 180.2 \, \text{g/mol}.\]

This calculated molar mass is necessary to convert the mass of the sample to the number of moles.

Avogadro's Number

Avogadro's number is a central concept when dealing with the number of particles in chemistry. It is defined as the number of atoms or molecules in one mole of a substance, which is approximately \(6.022 \times 10^{23}\).
This huge number makes it possible to count particles on a relatable scale. In the exercise:

• For diamond, the number of moles of carbon is calculated first.
• The total number of atoms is found by multiplying the moles by Avogadro's number: \[\text{Atoms of Carbon} = 0.0208\, \text{moles} \times 6.022 \times 10^{23}\, \text{atoms/mole},\] resulting in approximately \(1.25 \times 10^{22}\) atoms.

This approach gives a tangible understanding of molecules and atoms participating in chemical processes.

Chemical Formulas

Chemical formulas express the types and numbers of atoms present in a compound. They play a vital role in stoichiometry and calculations involving moles.
For a compound like acetylsalicylic acid, its formula is C₉H₈O₄, indicating:

• 9 carbon atoms,
• 8 hydrogen atoms,
• and 4 oxygen atoms.

With this formula, chemists can compute the compound's molar mass and predict the behavior of the substance in reactions.
In the exercise, understanding these numbers allowed for calculating moles and, subsequently, the number of molecules in a sample. Each element’s contribution to the compound is considered in the total mass calculation.

Mass-to-Mole Conversion

Converting mass to moles links a measurable quantity to a fundamental chemistry unit. Understanding this conversion is pivotal for determining how substances interact. To convert the mass of a material into moles:

• Divide the mass by its molar mass.
• When you have the mass of a substance, like the 0.500 g of aspirin:
• Use the molar mass calculated earlier:
\[\text{Moles} = \frac{0.500 \, \text{g}}{180.2 \, \text{g/mol}} \approx 0.00277\, \text{moles}.\]

This conversion is generic in chemistry to link the tangible and the particle world, allowing the calculation of further quantities like molecules or atoms using Avogadro’s number.