Problem 42 - Chemistry: The Central Science

Question

A sample of the male sex hormone testosterone, \(\mathrm{C}_{19} \mathrm{H}_{28} \mathrm{O}_{2}\), contains \(3.88 \times 10^{21}\) hydrogen atoms. (a) How many atoms of carbon does it contain? (b) How many molecules of testosterone does it contain? (c) How many moles of testosterone does it contain? (d) What is the mass of this sample in grams?

Step-by-Step Solution

Answer

The sample of testosterone contains approximately (a) \(2.67 \times 10^{21}\) carbon atoms, (b) \(1.39 \times 10^{20}\) testosterone molecules, (c) \(2.31 \times 10^{-4}\) moles of testosterone, and (d) 0.0667 grams of mass.

Step 1: (Determine the ratio of atoms in the testosterone molecule)

The molecular formula for testosterone is C19H28O2. Thus, there are 19 carbon atoms, 28 hydrogen atoms, and 2 oxygen atoms in each testosterone molecule. To find the number of carbon atoms, we will first determine the ratio between carbon and hydrogen atoms in the molecule, which is \(\frac{19}{28}\).

Step 2: (a) Calculate the number of carbon atoms in the sample)

Given the sample contains \(3.88 \times 10^{21}\) hydrogen atoms, we can use the ratio \(\frac{19}{28}\) to find the number of carbon atoms. Carbon atoms = \(\frac{19}{28}\) x \(3.88 \times 10^{21}\) Carbon atoms ≈ \(2.67 \times 10^{21}\) So, the sample contains approximately \(2.67 \times 10^{21}\) carbon atoms.

Step 3: (b) Calculate the number of testosterone molecules)

We know that there are 28 hydrogen atoms in each testosterone molecule. Therefore, we can divide the total number of hydrogen atoms by 28 to calculate the number of testosterone molecules: Testosterone molecules = \(\frac{3.88 \times 10^{21}}{28}\) Testosterone molecules ≈ \(1.39 \times 10^{20}\) So, the sample contains approximately \(1.39 \times 10^{20}\) testosterone molecules.

Step 4: (c) Calculate the number of moles of testosterone)

We will use Avogadro's number (\(6.022 \times 10^{23}\)) to calculate the number of moles of testosterone: Moles of testosterone = \(\frac{1.39 \times 10^{20}}{6.022 \times 10^{23}}\) Moles of testosterone ≈ \(2.31 \times 10^{-4}\) moles So, the sample contains approximately \(2.31 \times 10^{-4}\) moles of testosterone.

Step 5: (d) Calculate the mass of the sample)

To find the mass of the sample, we would need the molar mass of testosterone. The molar mass of testosterone (C19H28O2) can be calculated as: Molar mass = (19 x 12.01 g/mol of C) + (28 x 1.01 g/mol of H) + (2 x 16.00 g/mol of O) Molar mass = 288.43 g/mol Now, we can calculate the mass of the sample by multiplying the number of moles by the molar mass: Mass of sample = \(2.31 \times 10^{-4}\) moles x 288.43 g/mol Mass of sample ≈ 0.0667 g So, the mass of this sample is approximately 0.0667 grams.