Problem 90 - Chemistry The Central Science In Si Units

Question

(a) Ibuprofen is a common over-the-counter analgesic with the formula \(\mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2}\). How many moles of \(\mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2}\) are in a 500-mg tablet of ibuprofen? Assume the tablet is composed entirely of ibuprofen. (b) How many molecules of \(\mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2}\) are in this tablet? (c) How many oxygen atoms are in the tablet?

Step-by-Step Solution

Answer

(a) 0.002425 moles; (b) \(1.46 \times 10^{21}\) molecules; (c) \(2.92 \times 10^{21}\) oxygen atoms.

Step 1: Determine the molar mass of ibuprofen

To find the molar mass of ibuprofen \( \mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2} \), add the molar masses of all atoms in the chemical formula. Carbon (\( \mathrm{C} \)) has a molar mass of 12.01 g/mol, hydrogen (\( \mathrm{H} \)) is 1.01 g/mol, and oxygen (\( \mathrm{O} \)) is 16.00 g/mol. So, the molar mass is calculated as: \( (13 \times 12.01) + (18 \times 1.01) + (2 \times 16.00) = 206.29 \text{ g/mol}. \)

Step 2: Convert mass to moles

To find the number of moles of ibuprofen in a 500 mg tablet, convert mass from milligrams to grams, then use the molar mass. First, convert 500 mg to grams: \( 500 \text{ mg} = 0.500 \text{ g}.\) Then divide by the molar mass: \( \frac{0.500 \text{ g}}{206.29 \text{ g/mol}} = 0.002425 \text{ moles.} \)

Step 3: Calculate the number of molecules

To find how many molecules are in 0.002425 moles of ibuprofen, use Avogadro's number \(6.022 \times 10^{23} \text{ molecules/mol}.\) Multiply the number of moles by Avogadro's number: \( 0.002425 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} \approx 1.46 \times 10^{21} \text{ molecules.}\)

Step 4: Determine the number of oxygen atoms

Each molecule of ibuprofen \( \mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2} \) contains 2 oxygen atoms. Therefore, multiply the total number of molecules by 2 to find the number of oxygen atoms: \( 1.46 \times 10^{21} \text{ molecules} \times 2 = 2.92 \times 10^{21} \text{ oxygen atoms.} \)

Key Concepts

Molar Mass Calculation

Understanding molar mass is crucial when dealing with chemical compounds. Molar mass, also known as molecular weight, is the sum of the atomic masses of each element in a compound. To calculate it, follow these steps:

• Identify the chemical formula of the compound. For ibuprofen, the formula is \( \mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2} \).
• Find the atomic mass of each element from the periodic table. Typically, these are given in grams per mole: Carbon (\( \mathrm{C} \)) is 12.01 \( \text{g/mol} \), Hydrogen (\( \mathrm{H} \)) is 1.01 \( \text{g/mol} \), and Oxygen (\( \mathrm{O} \)) is 16.00 \( \text{g/mol} \).
• Multiply the atomic mass of each element by the number of times that element appears in the molecule. For ibuprofen:
  • Carbon: \( 13 \times 12.01 \text{ g/mol} \)
  • Hydrogen: \( 18 \times 1.01 \text{ g/mol} \)
  • Oxygen: \( 2 \times 16.00 \text{ g/mol} \)
• Add these values together to get the molar mass of ibuprofen: \( 206.29 \text{ g/mol} \).

Avogadro's Number

Avogadro's number is a fundamental constant used in chemistry to count particles in a mole, making it a key concept when working with moles and atoms. This number is a massive 6.022 \( \times 10^{23} \), meaning that one mole of any substance contains exactly that many particles, be they atoms, molecules, ions, or other entities.
When converting moles of a substance to the number of molecules, Avogadro's number is crucial. For instance, when we calculated the moles of ibuprofen to find how many molecules it represents, we multiplied the number of moles (0.002425) by Avogadro's number to determine there are approximately \( 1.46 \times 10^{21} \) molecules in the tablet.
This principle applies irrespective of the substance in question and simplifies comparisons between substances on the molecular level.

Converting Mass to Moles

Converting mass to moles is an integral part of stoichiometry, which is the calculation of reactants and products in chemical reactions. This process helps bridge the gap between the macroscopic world of grams and the microscopic world of molecules and atoms.
To convert mass to moles:

• Ensure the mass is in grams. For instance, we converted the 500 mg of ibuprofen to 0.500 g.
• Divide the mass of the substance by its molar mass. This is done using the formula: \( \text{moles} = \frac{\text{mass in grams}}{\text{molar mass}} \).

Using our example, we calculated the number of moles of ibuprofen by dividing 0.500 g by its molar mass, 206.29 \( \text{g/mol} \), resulting in 0.002425 moles. This method ensures accurate calculations in chemical reactions and formulas.

Oxygen Atoms in Molecules

Understanding the number of specific atoms, like oxygen, in a compound provides insights into its chemical behavior and properties. Ibuprofen's formula \( \mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2} \) tells us it contains two oxygen atoms per molecule.
To find the total number of oxygen atoms in any given amount of substance, follow these steps:

• First, determine the total number of molecules present. In our exercise, this was calculated using the number of moles and Avogadro's number, resulting in \( 1.46 \times 10^{21} \) molecules.
• Multiply the number of these molecules by the number of oxygen atoms in each molecule. Here, each ibuprofen molecule has two oxygen atoms.

Thus, in the ibuprofen tablet, there are \( 2.92 \times 10^{21} \) oxygen atoms. This calculation helps in comprehending reaction mechanisms and stoichiometry in chemical equations.