Problem 92 - Chemistry: The Central Science

Question

(a) One molecule of the antibiotic penicillin \(\mathrm{G}\) has a mass of \(5.342 \times 10^{-21} \mathrm{~g}\). What is the molar mass of penicillin \(\mathrm{G}\) ? (b) Hemoglobin, the oxygen-carrying protein in red blood cells, has four iron atoms per molecule and contains \(0.340 \%\) iron by mass. Calculate the molar mass of hemoglobin.

Step-by-Step Solution

Answer

The molar mass of penicillin G is approximately \(3.216 \times 10^{3} \mathrm{~g~mol}^{-1}\), and the molar mass of hemoglobin is approximately 223.235 g/mol.

Step 1: (a) Calculate the molar mass of penicillin G.

To find the molar mass of penicillin G, we will use the mass of one molecule and the Avogadro constant (\(N_A = 6.022 \times 10^{23} \mathrm{~mol}^{-1}\)). The molar mass is the mass of one mole of the substance. Since we know the mass of one molecule, we can multiply it by the Avogadro constant to obtain the mass of one mole (which will be the molar mass). The molar mass of penicillin G can be calculated as follows: Molar mass = (Mass of one molecule) × (Avogadro constant) Molar mass = \(5.342 \times 10^{-21} \mathrm{~g}\) × \(6.022 \times 10^{23} \mathrm{~mol}^{-1}\)

Step 2: (a) Calculate the result for the molar mass of penicillin G.

Now, by calculating the product, we get: Molar mass = \(5.342 \times 10^{-21} \mathrm{~g}\) × \(6.022 \times 10^{23} \mathrm{~mol}^{-1}\) = \(3.216 \times 10^{3} \mathrm{~g~mol}^{-1}\) Hence, the molar mass of penicillin G is approximately \(3.216 \times 10^{3} \mathrm{~g~mol}^{-1}\).

Step 3: (b) Calculate the mass of iron in one mole of hemoglobin.

In order to find the molar mass of hemoglobin, we first need to calculate the mass of iron in one mole of hemoglobin. We are told that there are 4 iron atoms per molecule of hemoglobin, and that iron constitutes 0.340% of the mass of hemoglobin. To find the mass of iron in one mole of hemoglobin, we can use the following formula: Mass of iron in one mole of hemoglobin = (Percentage of iron by mass) × (Molar mass of iron) × (Number of iron atoms in one molecule of hemoglobin) The molar mass of iron (Fe) is 55.845 g/mol. Mass of iron = \(0.00340 \times 55.845 \mathrm{~g~mol}^{-1} \times 4\)

Step 4: (b) Calculate the result for the mass of iron in one mole of hemoglobin.

Now, by calculating the product, we get: Mass of iron = \(0.00340 \times 55.845 \mathrm{~g~mol}^{-1} \times 4\) = \(0.759 \mathrm{~g~mol}^{-1}\) Hence, the mass of iron in one mole of hemoglobin is approximately 0.759 g/mol.

Step 5: (b) Calculate the molar mass of hemoglobin.

Now that we have the mass of iron in one mole of hemoglobin, we can find the total molar mass of hemoglobin. Since the mass of iron represents 0.340% of the total mass of hemoglobin, we can use the following formula: Molar mass of hemoglobin = (Mass of iron in one mole of hemoglobin) / (Percentage of iron by mass) Molar mass of hemoglobin = \(0.759 \mathrm{~g~mol}^{-1}\) / 0.00340

Step 6: (b) Calculate the result for the molar mass of hemoglobin.

Finally, by calculating the quotient, we get: Molar mass of hemoglobin = \(0.759 \mathrm{~g~mol}^{-1}\) / 0.00340 = \(223.235 \mathrm{~g~mol}^{-1}\) Hence, the molar mass of hemoglobin is approximately 223.235 g/mol.

Key Concepts

Avogadro Constant

The Avogadro constant, symbolized as \(N_A\), is a fundamental number in chemistry, revealing how many constituent particles are contained in one mole of a substance. These particles could be atoms, molecules, ions, or electrons. The value of the Avogadro constant is approximately \(6.022 \times 10^{23} \mathrm{mol}^{-1}\).

When solving problems in chemistry, the Avogadro constant provides a bridge between the microscopic scale (individual atoms or molecules) and the macroscopic scale (amounts of substances we can measure or see). For instance, to calculate the molar mass of a compound, we multiply the mass of a single molecule by the Avogadro constant, thus scaling up from the mass of a single molecule to the mass one would obtain with a whole mole of these molecules.

Percentage Composition

Percentage composition is a quantitative tool in chemistry that expresses the proportion of each element within a compound in terms of percentage by mass. This concept is crucial for understanding the makeup of a compound and for calculating empirical and molecular formulas.

To determine the percentage composition, you divide the mass of each element in one mole of the compound by the total molar mass of the compound and then multiply the result by 100 to get a percentage. This information can be incredibly useful, for example, when calculating the mass of a particular element in a given mass of compound, like iron in hemoglobin, as shown in the example.

Molecular Mass

Molecular mass, also known as molecular weight, is the sum of the masses of all the atoms in a molecule. It is typically reported in atomic mass units (amu) or grams per mole (g/mol). The calculation of molecular mass is fundamental when converting between moles and grams, a common task in chemical stoichiometry.

You can calculate the molecular mass by adding up the atomic masses of each atom in the molecule. The atomic masses are usually found on the periodic table and represent the average mass of all isotopes of that atom as they occur naturally. Knowing the molecular mass is essential in determining how much of a chemical is needed for a reaction or what amount is produced.