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 number of carbon atoms in the testosterone sample is approximately \(2.61 \times 10^{21}\). The sample contains approximately \(1.39 \times 10^{20}\) testosterone molecules and \(2.31 \times 10^{-4}\) moles of testosterone. The mass of the testosterone sample is approximately 0.0666 grams.

Step 1: Find the number of carbon atoms

First, we need to find the number of carbon atoms in the given sample. Since the ratio of hydrogen to carbon atoms in testosterone is 28:19, we can use this to find the number of carbon atoms: Number of carbon atoms = \((\frac{19}{28}) \times (3.88 \times 10^{21})\) Now, calculate the answer: Number of carbon atoms = \((\frac{19}{28}) \times (3.88 \times 10^{21}) \approx 2.61 \times 10^{21}\) carbon atoms

Step 2: Find the number of testosterone molecules

Now we need to find the number of testosterone molecules in the sample. Since there are 28 hydrogen atoms in one molecule of testosterone, we can use the provided hydrogen atom count to find the number of molecules: Number of testosterone molecules = \(\frac{3.88 \times 10^{21}}{28}\) Now, calculate the answer: Number of testosterone molecules = \(\frac{3.88 \times 10^{21}}{28} \approx 1.39 \times 10^{20}\) testosterone molecules

Step 3: Find the number of moles of testosterone

To find the number of moles of testosterone, we will use Avogadro's number, which states that there are \(6.022 \times 10^{23}\) molecules in one mole of any substance: Number of moles = \(\frac{1.39 \times 10^{20}}{6.022 \times 10^{23}}\) Now, calculate the answer:function Find-theValueGiven (){ Number of moles = \(\frac{1.39 \times 10^{20}}{6.022 \times 10^{23}} \approx 2.31 \times 10^{-4}\) moles of testosterone

Step 4: Find the mass of the sample in grams

To find the mass of the sample, we need to find the molar mass of testosterone first. Using the periodic table, we can see that the molar mass of carbon (C) is approximately 12.01 g/mol, hydrogen (H) is approximately 1.01 g/mol, and oxygen (O) is approximately 16.00 g/mol. The chemical formula for testosterone is \(\mathrm{C}_{19}\mathrm{H}_{28}\mathrm{O}_{2}\), so its molar mass can be found as follows: Molar mass of testosterone = \((19 \times 12.01) + (28 \times 1.01) + (2 \times 16.00)\) Now, calculate the molar mass: Molar mass of testosterone = \((19 \times 12.01) + (28 \times 1.01) + (2 \times 16.00) \approx 288.45\) g/mol Now, we can find the mass of the sample by multiplying the number of moles by the molar mass: Mass = \((2.31 \times 10^{-4}) \times 288.45\) Now, calculate the answer: Mass = \((2.31 \times 10^{-4}) \times 288.45 \approx 0.0666\) grams The mass of the testosterone sample is approximately 0.0666 grams.

Key Concepts

Moles Calculation

Calculating moles is an essential technique in chemistry that links the microscopic world of atoms and molecules to the macroscopic world we observe. Moles are a unit used to count particles, just like dozen for eggs. One mole is defined as exactly 6.022 x 10^23 particles, and this is known as Avogadro's number. To find out how many moles are present, you divide the number of particles you have by Avogadro’s number.

In the testosterone exercise, we calculated the number of moles by taking the given number of molecules and dividing it by Avogadro's number:

• Number of moles = Number of molecules / Avogadro's number
• \[ Number \ of \ moles = \frac{1.39 \times 10^{20}}{6.022 \times 10^{23}} \approx 2.31 \times 10^{-4} \text{ moles} \\]

Understanding how to calculate moles helps in determining quantities needed for reactions and understanding the scale of atomic interactions.

Chemical Formula

Chemical formulas, like \(\mathrm{C}_{19} \mathrm{H}_{28} \mathrm{O}_{2}\), provide a concise way to represent the composition of compounds. Each element in the formula shows the number of atoms of each type in one molecule of the compound. For testosterone, the formula indicates:

• 19 carbon (C) atoms
• 28 hydrogen (H) atoms
• 2 oxygen (O) atoms

This formula allows chemists to determine how atoms are bonded and can be used to understand the properties and structure of the compound. Knowing the chemical formula is crucial when performing calculations involving stoichiometry, such as finding the molar mass or the ratios of elements in reactions.

Molar Mass

The molar mass of a compound is the mass of one mole of that compound and is expressed in grams per mole (g/mol). It is calculated by adding up the atomic masses of all the atoms in the chemical formula. This value is crucial as it allows chemists to convert between the mass of a sample and the number of moles it contains.

For testosterone, with the formula \(\mathrm{C}_{19} \mathrm{H}_{28} \mathrm{O}_{2} \), the molar mass can be calculated by summing the masses of its constituent atoms:

• Carbon (C): 12.01 g/mol \( \times 19 \approx 228.19 \text{ g/mol} \)
• Hydrogen (H): 1.01 g/mol \( \times 28 \approx 28.28 \text{ g/mol} \)
• Oxygen (O): 16.00 g/mol \( \times 2 \approx 32.00 \text{ g/mol} \)

Total molar mass = 288.45 g/mol

This constant also enables the conversion of moles into grams, a key operation in lab practices.

Avogadro's Number

Avogadro's number, 6.022 x 10^23, is a fundamental constant in chemistry used to quantify the amount of particles (atoms, molecules) in a mole. It's like a bridge between the atomic scale and quantities we can measure. This constant makes it possible to convert measurements taken in lab experiments into meaningful, countable quantities of individual particles.

• For instance, determining the number of molecules in a sample can be achieved by multiplying the moles by Avogadro's number.
• Example calculation: Number of molecules = moles \( \times \) Avogadro's number

In the context of testosterone, this principle allowed scientists to deduce the number of molecules from the total number of hydrogen atoms provided, and then determine the number of moles. This approach highlights the interconnected nature of chemical measurements and the utility of basal constants like Avogadro's number in practical applications.