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?

Short answer

\(= 2.62 \times 10^{21}\) carbon atoms

Step-by-step solution

Identify Components in the Molecule

The testosterone molecule has a molecular formula of \(\mathrm{C}_{19}\mathrm{H}_{28}\mathrm{O}_{2}\). This means that the molecule consists of 19 carbon atoms, 28 hydrogen atoms, and 2 oxygen atoms. In this problem, we are given information about the number of hydrogen atoms in the sample.

Calculate Number of Carbon Atoms

We can find the number of carbon atoms by using the ratio of carbon to hydrogen atoms in the molecule. Since there are 19 carbon atoms for every 28 hydrogen atoms, we can write the proportion as: $$\frac{\text{Number of Carbon Atoms}}{\text{Number of Hydrogen Atoms}} = \frac{19}{28}$$ Given that there are \(3.88 \times 10^{21}\) hydrogen atoms, let's solve for the number of carbon atoms: $$\text{Number of Carbon Atoms} = \frac{19}{28} \times (3.88 \times 10^{21})$$

Key concepts

Molecular Formula

Understanding the molecular formula of a compound is crucial in stoichiometry. A molecular formula describes the exact number and type of atoms present in a single molecule of a compound. In the case of testosterone, the molecular formula is \(\mathrm{C}_{19}\mathrm{H}_{28}\mathrm{O}_{2}\). This notation informs us that each molecule of testosterone consists of: - 19 carbon atoms (\(\mathrm{C}\)) - 28 hydrogen atoms (\(\mathrm{H}\)) - 2 oxygen atoms (\(\mathrm{O}\))The molecular formula is your starting point for all calculations related to the substance. By knowing this formula, you can proceed to calculate other parameters such as the number of certain atoms, the total number of molecules, or even the mass of the sample.The molecular formula plays a crucial role because it not only identifies the makeup of each molecule but also serves as a roadmap for the relationships and proportions of each type of atom in the compound.

Atoms Calculation

To find the number of atoms of a specific element in a substance, use the answer you get from the molecular formula and apply it to the given data.Suppose you know that you have \(3.88 \times 10^{21}\) hydrogen atoms. The molecular formula of testosterone, \(\mathrm{C}_{19}\mathrm{H}_{28}\mathrm{O}_{2}\), indicates that for every 28 hydrogen atoms, there are 19 carbon atoms. Therefore, the ratio is \(\frac{19}{28}\).By multiplying this ratio by the known number of hydrogen atoms, you calculate the number of carbon atoms:\[ \text{Number of Carbon Atoms} = \frac{19}{28} \times (3.88 \times 10^{21}) \]This step is essential for predicting the quantity of another type of atom based on the known quantity of one type of atom. Always use the relative amounts provided in the molecular formula to find your answer.

Moles Calculation

A mole is a fundamental unit in chemistry used to express amounts of chemical substances. One mole equals approximately \(6.022 \times 10^{23}\) entities (Avogadro's number), such as atoms or molecules. When calculating moles from a known number of molecules, simply divide the number of molecules by Avogadro’s number. To find the number of moles of testosterone molecules given the number of molecules, use the formula:\[\text{Moles of Testosterone} = \frac{\text{Number of Testosterone Molecules}}{6.022 \times 10^{23}}\]If you had, for instance, \(1.39 \times 10^{20}\) testosterone molecules, you'd divide this by \(6.022 \times 10^{23}\) to convert this quantity into moles. Understanding moles is vital as it helps transition from the microscopic world of atoms to the macroscopic world we can measure.

Mass Calculation

Mass calculation involves converting moles of a substance into grams, which is often the most familiar unit of measurement for students. To calculate mass, you need to know the molar mass of the substance, which is the mass of one mole of that substance.For testosterone, you’d calculate the molar mass by adding up the mass of all the atoms in the molecular formula:- Carbon: 19 atoms \(\times 12.01 \text{ g/mol} = 228.19 \text{ g/mol}\)- Hydrogen: 28 atoms \(\times 1.008 \text{ g/mol} = 28.224 \text{ g/mol}\)- Oxygen: 2 atoms \(\times 16.00 \text{ g/mol} = 32.00 \text{ g/mol}\)Adding these together gives you the molar mass of testosterone: \[ \text{Molar Mass} = 228.19 + 28.224 + 32.00 = 288.414 \text{ g/mol}\]To find the mass of a given number of moles in grams:\[ \text{Mass (g)} = \text{Moles} \times \text{Molar Mass (g/mol)} \]This conversion allows the quantification of the sample and facilitates further experiments and reactions by providing the substance’s amount in grams.