Problem 40 - Chemistry: The Central Science

Question

\( \mathrm{~A}\) sample of the male sex hormone testosterone, \(\mathrm{C}_{19} \mathrm{H}_{28} \mathrm{O}_{2}\), contains \(7.08 \times 10^{20}\) 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

a) The sample contains \(4.83 \times 10^{20}\) carbon atoms. b) The sample contains \(2.53 \times 10^{19}\) molecules of testosterone. c) The sample contains \(1.175 \times 10^{-5}\) moles of testosterone. d) The mass of the sample is \(3.39 \times 10^{-3}\) grams.

Step 1: Determine the Number of Testosterone Molecules in the Sample

To find the number of testosterone molecules in the sample, we need to know how many of them contain the given hydrogen atoms. Since there are 28 hydrogen atoms in one molecule of testosterone, we can find the number of testosterone molecules in the sample by dividing the total number of hydrogen atoms (7.08 × 10^20) by 28: Number of testosterone molecules = \(\frac{7.08 \times 10^{20}}{28}\)

Step 2: Calculate the Number of Carbon Atoms

Now that we know the number of testosterone molecules in the sample, we can find out the number of carbon atoms in it, as there are 19 carbon atoms in each molecule of testosterone. We can do this by multiplying the number of testosterone molecules (calculated above) by 19: Number of carbon atoms = \(\frac{7.08 \times 10^{20}}{28} \times 19\)

Step 3: Determine the Number of Moles of Testosterone

To calculate the number of moles, we just need to divide the number of testosterone molecules (calculated in step 1) by Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol): Number of moles = \(\frac{\frac{7.08 \times 10^{20}}{28}}{6.022 \times 10^{23}}\)

Step 4: Calculate the Mass of the Sample in Grams

Finally, we can find the mass of the sample in grams using the molar mass of testosterone. The molar mass of testosterone can be calculated as follows: Molar mass of testosterone = (\(19 \times 12.01\)) (for C) \(+ (28 \times 1.01)\) (for H) \(+ (2 \times 16.00)\) (for O) = \(288.43~\text{g/mol}\) Next, we multiply the number of moles (calculated in step 3) by the molar mass of testosterone to find the mass in grams: Mass in grams = \(\frac{\frac{7.08 \times 10^{20}}{28}}{6.022 \times 10^{23}} \times 288.43\) Now we can just plug in the numbers and compute the required values: a) Number of carbon atoms = \(\frac{7.08 \times 10^{20}}{28} \times 19 = 4.83 \times 10^{20}\) carbon atoms b) Number of testosterone molecules = \(\frac{7.08 \times 10^{20}}{28} = 2.53 \times 10^{19}\) molecules c) Number of moles of testosterone = \(\frac{\frac{7.08 \times 10^{20}}{28}}{6.022 \times 10^{23}} = 1.175 \times 10^{-5}\) moles d) Mass of the sample in grams = \(\frac{\frac{7.08 \times 10^{20}}{28}}{6.022 \times 10^{23}} \times 288.43 = 3.39 \times 10^{-3}~\text{g}\)

Key Concepts

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry, crucial for understanding molecular and chemical quantities. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in the amount of substance of one mole.
This number is approximately \(6.022 \times 10^{23}\) particles per mole. Hence, when you have one mole of a substance, it contains \(6.022 \times 10^{23}\) units of that substance—be it atoms, molecules, or ions.
For example, in calculating the number of moles of testosterone in a sample, we used Avogadro's number to determine how many molecules corresponded to the given number of hydrogen atoms.

Molar Mass

The molar mass is an essential concept, representing the mass of one mole of a given substance. It is usually expressed in grams per mole (g/mol).
To calculate the molar mass, you need to sum the atomic masses of all the atoms in a molecule. For testosterone \(\text{C}_{19}\text{H}_{28}\text{O}_{2}\), this involves:

• Carbon: \(19 \times 12.01\ \text{g/mol}\)
• Hydrogen: \(28 \times 1.01\ \text{g/mol}\)
• Oxygen: \(2 \times 16.00\ \text{g/mol}\)

Adding these values gives the molar mass of testosterone as approximately \(288.43\ \text{g/mol}\).
We used this value to calculate the mass of the sample, based on the number of moles calculated in previous steps.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the reactants and products in chemical reactions.
It involves using balanced chemical equations to calculate the masses, moles, and particles of the involved substances.
In our exercise, stoichiometry helped determine the number of carbon atoms from the known quantity of hydrogen atoms. By recognizing that each testosterone molecule contains a set number of hydrogen and carbon atoms, we calculated the respective quantities by using these stoichiometric ratios.

Moles

The concept of a mole is central to chemistry, providing a bridge between the atomic world and the macroscopic world.
One mole equates to \(6.022 \times 10^{23}\) units of anything—be it atoms, molecules, ions, etc.
For example, when calculating the number of moles of testosterone, we used the number of molecules determined by dividing the given hydrogen atoms by the number in each molecule. This allows for converting a tangible measurement (e.g., mass) into a quantifiable chemical amount.

Mass Calculation

Calculating mass from moles or particles is an important application of chemistry principles. It connects the theoretical with practical lab calculations.
To find the mass of a sample, you multiply the number of moles by the molar mass of the substance. This simple formula allows us to convert the abstract idea of a mole into measurable quantities.
In the exercise, once we knew the number of moles of testosterone, multiplying by the molar mass of 288.43 g/mol allowed us to calculate the sample's mass in grams as \(3.39 \times 10^{-3}\) grams. This logical chain from atoms to mass highlights how each concept interrelates.