Problem 92 - Chemistry The Central Science In Si Units

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

(a) 3214.5 g/mol for penicillin G; (b) 65647 g/mol for hemoglobin.

Step 1: Understanding the Concept of Molar Mass

The molar mass of a substance is the mass of one mole of its entities (atoms, molecules, etc.). For a molecule, the molar mass can be determined using the individual molecular mass and Avogadro's number, which is approximately \(6.022 \times 10^{23}\) molecules/mol.

Step 2: Calculate Molar Mass of Penicillin G

To find the molar mass of penicillin G, multiply the mass of one molecule by Avogadro's number. Thus, the molar mass \(M\) is calculated as follows: \[ M = 5.342 \times 10^{-21} \text{ g/molecule} \times 6.022 \times 10^{23} \text{ molecules/mol} = 3214.5 \text{ g/mol}. \]

Step 3: Identify Hemoglobin's Iron Content by Mass

Given that hemoglobin contains 0.340% iron by mass and has 4 iron atoms per molecule, first calculate the total mass of iron in one mole of hemoglobin molecules. The atomic mass of iron (Fe) is approximately 55.85 g/mol.

Step 4: Determine Mass of Iron in Hemoglobin Per Mole

Calculate the total mass of iron in one mole of hemoglobin using the molecular content: \[ \text{Mass of iron per mole} = 4 \times 55.85 \text{ g/mol} = 223.4 \text{ g/mol}. \]

Step 5: Use Iron Percentage to Determine Hemoglobin's Molar Mass

Knowing that this iron mass constitutes 0.340% of hemoglobin's molar mass, set up the equation: \[ 223.4 \text{ g/mole} = 0.340\% \times M, \] where \(M\) is the molar mass of hemoglobin. Solve for \(M\) by dividing both sides by 0.0034 (from 0.340%):\[ M = \frac{223.4}{0.0034} \approx 65647 \text{ g/mol}. \]

Step 6: Final Step: Interpret Results

Thus, the molar mass of penicillin G is approximately 3214.5 g/mol, and the molar mass of hemoglobin is approximately 65647 g/mol.

Key Concepts

Avogadro's Number

To understand molar mass calculations, getting familiar with Avogadro's number is crucial. This number, approximately equal to \(6.022 \times 10^{23}\), represents the number of atoms, molecules, or particles in one mole of a substance.
Avogadro's number bridges the gap between atomic scale and macroscopic scale by allowing us to quantify a mole of entities.

• This means that if you have a mole of something, you have \(6.022 \times 10^{23}\) of those items, whether they are atoms, molecules, or any other countable particles.
• This makes it possible to convert between the mass of a single molecule and the molar mass, which is the mass of one mole of those molecules.

For instance, in the example of penicillin G, we used Avogadro's number to convert the known mass of a molecule to the molar mass, which is the basis for stoichiometric calculations in chemistry.

Molecular Mass

Molecular mass is the sum of the atomic masses of all the atoms in a molecule. It is typically measured in atomic mass units (amu) or grams per mole (g/mol) when referenced in molar quantities. This mass gives us a stepping stone to determine other properties like molar mass.
Molecular mass involves adding up each atom's contribution to the molecule, using the periodic table to obtain individual atomic masses.

• For example, if a molecule consists of two hydrogen atoms and one oxygen atom (H₂O), its molecular mass is the sum of the mass of two hydrogen atoms and one oxygen atom.
• By knowing this property of molecular structure, chemists can predict how chemicals behave in reactions.

In the case of penicillin G, scientists have calculated the mass of one whole molecule, which we used to find the molar mass by factoring in Avogadro's number.

Chemical Composition

Chemical composition refers to the identity and amount of each element or compound in a chemical substance. Understanding composition is essential for determining the proportions of elements within a molecule and is crucial for calculating molar mass correctly.
This concept helps to understand how much of each element makes up a larger complex molecule.

• For example, hemoglobin is a complex molecule with a specific composition of different elements including iron.
• Knowing the chemical composition allows us to calculate how much of each constituent is present, such as determining the mass contribution of iron in hemoglobin.

In exercises like those you're working on, noting chemical composition details is essential for correctly determining molar mass values based on known or calculable proportions.

Iron Content

Iron content is a specific consideration when evaluating compounds that include iron atoms, such as hemoglobin. The specific percentage or amount of iron can influence both the calculation and understanding of the compound's function.
In chemical terms, the iron content is usually expressed as a percentage of the total molecular mass.

• For hemoglobin, knowing it has a 0.340% iron content means calculating how this contributes to its overall mass.
• This percentage tells us what part of the molar mass is due to iron, allowing us to back-calculate the molar mass of hemoglobin using this percentage.

This exercise utilized the iron content to determine hemoglobin’s molar mass, showcasing how percentage compositions of elements help discern structural and functional aspects of biomolecules.