Problem 84 - Chemistry: The Central Science

Question

(a) One molecule of the antibiotic known as penicillin \(\mathrm{G}\) 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

(a) The molar mass of penicillin G is \(5.342\mathrm{~g/mol}\). (b) The molar mass of hemoglobin is approximately \(65647\mathrm{~g/mol}\).

Step 1: Convert the mass of one molecule of penicillin G to moles

To do this, we'll use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules/mol. Divide the mass of one molecule by Avogadro's number to get the mass in moles: \[\frac{5.342 \times 10^{-21}\mathrm{~g}}{6.022 \times 10^{23}\text{ molecules/mol}}\]

Step 2: Calculate the molar mass of penicillin G

To find the molar mass, multiply the mass in moles (obtained in step 1) by Avogadro's number (in grams/mole): \[\text{Molar mass of penicillin G} = \frac{5.342 \times 10^{-21}\mathrm{~g}}{6.022 \times 10^{23}\text{ molecules/mol}} \times 6.022 \times 10^{23}\mathrm{~g/mol}\] The numbers \(6.022 \times 10^{23}\) molecules/mol and \(6.022 \times 10^{23}\) g/mol cancel each other out, and you are left with: \[\text{Molar mass of penicillin G} = 5.342 \times 10^{-21}\mathrm{~g} \times \frac{1\mathrm{~mol}}{6.022 \times 10^{23}\text{ molecules}} \times 6.022 \times 10^{23}\mathrm{~g/mol} = 5.342\mathrm{~g/mol}\] # For part (b) # Given that hemoglobin has four iron atoms per molecule, and it contains \(0.340\%\) iron by mass, let's proceed with the following steps to calculate the molar mass of hemoglobin:

Step 1: Calculate the mass of iron in hemoglobin

To do this, we'll use the percentage of iron by mass: \[0.340\% = \frac{\text{Mass of iron}}{\text{Molar mass of hemoglobin}}\]

Step 2: Use the molar mass of iron to find the mass of all four iron atoms in a hemoglobin molecule

The molar mass of iron is \(55.8\mathrm{~g/mol}\). Given that there are four iron atoms in each hemoglobin molecule, the total mass of iron per hemoglobin molecule is: \(4 \times 55.8 = 223.2\mathrm{~g/mol}\)

Step 3: Calculate the molar mass of hemoglobin

Now, we can plug in the mass of iron into the equation from step 1 to find the molar mass of hemoglobin: \[0.340\% = \frac{223.2\mathrm{~g/mol}}{\text{Molar mass of hemoglobin}}\] Rearrange the equation and solve for the molar mass of hemoglobin: \[\text{Molar mass of hemoglobin} = \frac{223.2\mathrm{~g/mol}}{0.00340}\]

Step 4: Compute the numerical value for the molar mass of hemoglobin

Divide the mass of iron by the percentage of iron by mass to get the molar mass of hemoglobin: \[\text{Molar mass of hemoglobin} = \frac{223.2\mathrm{~g/mol}}{0.00340} = 65647\mathrm{~g/mol}\] So, the molar mass of hemoglobin is approximately \(65647\mathrm{~g/mol}\).

Key Concepts

Penicillin G

Penicillin G is a type of antibiotic that fights bacterial infections. It works by killing bacteria or stopping them from growing. Understanding its molar mass helps chemists and pharmacists estimate how much of the compound is needed for medicinal use.

• To calculate the molar mass of penicillin G, we need the mass of one molecule and Avogadro's number.
• The mass of a single penicillin G molecule is given as \(5.342 \times 10^{-21} \mathrm{~g}\).
• Avogadro's number, \(6.022 \times 10^{23}\) molecules/mol, bridges single molecules to bulk substances.

By dividing the mass of one molecule by Avogadro's number, and then using this quotient to adjust into terms of molar mass, we confirm the conversion to be straightforward.

Molecular Mass

Molecular mass is a fundamental concept in chemistry, referring to the weight of a molecule based on the sum of its atoms. This value is expressed in grams per mole (g/mol).

• You calculate it by summing the atomic masses of all atoms in a molecule.
• It is a critical factor in determining stoichiometry and reaction yields.

In the case of penicillin G and hemoglobin, understanding the molecular masses helps in comprehending their chemical properties and potential uses in biological systems. The calculation steps involve understanding the structure of the molecule, which can inform dosing in medical applications.

Avogadro's Number

A fundamental constant in chemistry, Avogadro's number \(6.022 \times 10^{23}\) represents the number of constituent particles, usually atoms or molecules, in one mole of a substance.

• It facilitates the conversion between atomic mass units and grams.
• Allows chemists to scale experiences at the molecular level to quantities measurable in the laboratory.

In calculations like those for penicillin G, Avogadro's number provides the essential link to determine a molecule's weight on a macroscopic scale, which is crucial for practical applications in chemistry and pharmacology.

Hemoglobin

Hemoglobin is an essential protein in red blood cells, responsible for transporting oxygen throughout the body.

• It consists of four iron atoms per molecule, enabling it to bind oxygen effectively.
• Understanding its molar mass involves calculating the mass of the iron content as a percentage of the total mass.

In the exercise, the problem illustrates the relationship between the mass of iron and the total molar mass of hemoglobin. The given \(0.340\%\) iron by mass helps in calculating the total molar mass, revealing insights into its complex structure.

Percentage Composition

Percentage composition provides insights into the proportion of each element within a compound. It is calculated by determining the mass percentage of each constituent element relative to the total molar mass.

• Helps in determining the formula of compounds.
• Essential for understanding properties and reactions of compounds.

In the context of hemoglobin, the percentage composition of iron becomes a crucial factor in estimating its overall molar mass, providing a simple method for chemists to quantify and analyze substances. By using the formula mass ratio, the percentage composition sheds light on the concentration of elements within a compound, essential for reactions and applied chemistry.