Problem 92 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) One molecule of the antibiotic penicillin G has a mass of \(5.342 \times 10^{-21} \mathrm{~g}\). What is the molar mass of penicillin 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 \(3.214 \times 10^3\ \mathrm{g/mol}\), and the molar mass of hemoglobin is approximately \(6.56 \times 10^4\ \mathrm{g/mol}\).

Step 1: Determine the mass of one mole of penicillin G molecules

Multiply the given mass of a single penicillin G molecule by Avogadro's number, which is the number of particles (atoms, ions, or molecules) in one mole of any substance. $$ MolarMass= MassOfSingleMolecule \times Avogadro's Number $$ \(MolarMass= 5.342 \times 10^{-21} \mathrm{~g} \times 6.022\times10^{23}\mathrm{~particles/mol}\)

Step 2: Calculate the molar mass of penicillin G

Perform the multiplication to find the molar mass of penicillin G. $$ MolarMass \approx 3.214 \times 10^3\ \mathrm{g/mol} $$ The molar mass of penicillin G is approximately \(3.214 \times 10^3\ \mathrm{g/mol}\). (b) Calculating the molar mass of hemoglobin

Step 1: Determine the total mass of iron in one hemoglobin molecule

To do this, we will use the molar mass of iron, Fe, which is approximately 55.85 g/mol. Since there are four iron atoms in one molecule of hemoglobin, we can calculate its total mass of iron as follows: $$ TotalMassOfIronInHemoglobinMolecule= 4 \times MolarMassOfIron $$ $$ TotalMassOfIronInHemoglobinMolecule = 4 \times 55.85 \mathrm{~g/mol} $$

Step 2: Calculate the molar mass of hemoglobin based on the percentage of iron

Hemoglobin contains 0.340% iron by mass, and we have calculated the total mass of iron in one hemoglobin molecule. Now we can find the molar mass of hemoglobin, knowing the mass of iron and the percentage: $$ MolarMassOfHemoglobin = \frac{TotalMassOfIronInHemoglobinMolecule}{PercentageOfIron} $$ $$ MolarMassOfHemoglobin = \frac{4 \times 55.85 \mathrm{~g/mol}}{0.00340} $$

Step 3: Calculate the final molar mass of hemoglobin

Calculate the molar mass of hemoglobin: $$ MolarMassOfHemoglobin \approx 6.56 \times 10^4\ \mathrm{g/mol} $$ The molar mass of hemoglobin is approximately \(6.56 \times 10^4\ \mathrm{g/mol}\).

Key Concepts

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry that relates to the number of particles in one mole of a substance. It is defined as approximately \(6.022 \times 10^{23}\) particles (such as atoms, molecules, or ions) per mole. This constant serves as a bridge between the microscopic world of atoms and the macroscopic world we can measure.

• When calculating the molar mass of any substance, knowing Avogadro's Number is key because it lets us convert between the number of molecules and the amount in moles.
• For example, if you know the mass of a single molecule, multiplying by Avogadro's Number gives you the molar mass, which is useful for many practical applications in chemistry.

In the exercise example, we used Avogadro's Number to calculate the molar mass of penicillin G by multiplying the mass of a single molecule by \(6.022 \times 10^{23}\), resulting in its molar mass.

Hemoglobin

Hemoglobin is a complex protein found in red blood cells, responsible for transporting oxygen throughout the body. It contains four iron atoms, which play a key role in binding and releasing oxygen.

• Each molecule of hemoglobin consists of four subunits, each containing an iron atom embedded within a heme group.
• Iron is critical for hemoglobin's function as it is the binding site for oxygen molecules.

In the exercise, we use the fact that hemoglobin has 0.340% iron by mass to calculate its molar mass. By knowing the total mass of the iron in one molecule, and the percentage, we can determine the molar mass of hemoglobin. This process involves dividing the mass of iron by the percentage it represents within the molecule.

Penicillin G

Penicillin G is an antibiotic, widely used to treat bacterial infections. Understanding its chemical properties involves exploring its molar mass.

• The molar mass is a fundamental property that tells us the mass of one mole of its molecules, which can be derived from the mass of a single molecule.
• Penicillin G particularly helps in demonstrating the application of Avogadro's Number to transition from micrograms of a single molecule to macroscopic amounts we can use, like grams per mole.

Through the calculation outlined in the exercise, we determine the molar mass of penicillin G. By multiplying the mass of one molecule \(5.342 \times 10^{-21} \text{ g}\) with Avogadro’s Number, we arrived at a molar mass of approximately \(3.214 \times 10^3 \text{ g/mol}\), enabling further applications in dosage and treatment decisions in medical contexts.