Problem 90 - Chemistry: The Central Science in SI Units, Global Edition

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) There are 0.00242 moles of ibuprofen in the 500-mg tablet. (b) There are \(1.46 \times 10^{21}\) molecules of ibuprofen in the tablet. (c) There are \(2.92 \times 10^{21}\) oxygen atoms in the tablet.

Step 1: Calculate the molecular weight of ibuprofen

To find the molecular weight of ibuprofen, we need to add the atomic masses of all the atoms in the molecule, i.e. 13 carbon atoms, 18 hydrogen atoms, and 2 oxygen atoms. Using the atomic weights: Carbon (C): 12.01 g/mol Hydrogen (H): 1.01 g/mol Oxygen (O): 16.00 g/mol Molecular weight of ibuprofen = (13 × 12.01 g/mol) + (18 × 1.01 g/mol) + (2 × 16.00 g/mol) = 206.29 g/mol

Step 2: Convert the mass of the tablet to grams

The mass of the tablet is given in milligrams (mg). We need to convert it to grams (g) to match the molecular weight. Mass of the tablet in grams = 500 mg × (1 g / 1000 mg) = 0.5 g

Step 3: Calculate the number of moles of ibuprofen in the tablet

To find the number of moles, we will use the formula: Number of moles = (mass of the tablet) / (molecular weight of ibuprofen) Number of moles = (0.5 g) / (206.29 g/mol) = 0.00242 mol

Step 4: Calculate the number of molecules of ibuprofen in the tablet

To find the number of molecules, we will use Avogadro's number (6.022 × 10^23 molecules/mol): Number of molecules = (number of moles) × (Avogadro's number) Number of molecules = (0.00242 mol) × (6.022 × 10^23 molecules/mol) = 1.46 × 10^21 molecules

Step 5: Calculate the number of oxygen atoms in the tablet

Since there are 2 oxygen atoms in each ibuprofen molecule, we can find the total number of oxygen atoms by multiplying the number of ibuprofen molecules by 2: Number of oxygen atoms = (number of ibuprofen molecules) × 2 Number of oxygen atoms = (1.46 × 10^21 molecules) × 2 = 2.92 × 10^21 oxygen atoms So, (a) there are 0.00242 moles of ibuprofen in the 500-mg tablet, (b) there are 1.46 × 10^21 molecules of ibuprofen in the tablet, and (c) there are 2.92 × 10^21 oxygen atoms in the tablet.

Key Concepts

Molecular Weight Calculation

Calculating molecular weight is a vital first step in stoichiometry. It helps us understand the weight of one mole of a compound. Ibuprofen, our example, consists of elements carbon, hydrogen, and oxygen. Each element contributes to the total molecular weight, requiring you to calculate their combined contributions.
To find the molecular weight of ibuprofen, you must add the atomic weights of each atom type:

• Carbon (C) weighs 12.01 g/mol.
• Hydrogen (H) weighs 1.01 g/mol.
• Oxygen (O) weighs 16.00 g/mol.

The formula for ibuprofen is \(\mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2}\). Thus, each compound contains 13 carbon atoms, 18 hydrogen atoms, and 2 oxygen atoms.
The calculation looks like this:\[(13 \times 12.01\, \text{g/mol}) + (18 \times 1.01\, \text{g/mol}) + (2 \times 16.00\, \text{g/mol}) = 206.29\, \text{g/mol}\]Breaking it down helps determine the molecular weight accurately and lays the groundwork for further calculations in stoichiometry.

Avogadro's Number

Avogadro's number has a special role in chemistry. It is the bridge between the microscopic world of atoms and the macroscopic world we observe. Avogadro's number is 6.022 × 10^23, and it signifies how many molecules or atoms exist in one mole of a substance.
This constant allows us to convert moles into molecules. Thus, if we have calculated the number of moles, like with the ibuprofen tablet, we then can determine the number of actual ibuprofen molecules. For example:

• Given: 0.00242 moles of ibuprofen
• Avogadro's number: 6.022 × 10^23 molecules/mol

This conversion looks like:\[ \text{Number of molecules} = 0.00242\, \text{mol} \times 6.022 \times 10^{23}\, \text{molecules/mol}\]This results in approximately 1.46 × 10^21 molecules, connecting the number of moles to the tangible world.

Mole Concept

The mole concept is a fundamental aspect of chemistry. It allows us to quantify and compare chemical entities. One mole represents an Avogadro's number worth of particles, whether they're atoms, molecules, ions, or other entities.
In stoichiometry problems, the mole acts as the unit for measuring quantity, much like the dozen or the pair in everyday life. To find the moles, the mass of the substance must be divided by its molecular weight.
For instance, in dealing with ibuprofen:

• Mass of ibuprofen tablet: 0.5 g
• Molecular weight of ibuprofen: 206.29 g/mol

The number of moles is given by:\[ \text{Number of moles} = \frac{0.5\, \text{g}}{206.29\, \text{g/mol}}\]This computation leads to 0.00242 moles. Such calculations help clarify how many entities are present in a given mass and form the basis for chemical reactions and equations.