Problem 37 - Chemistry: The Central Science

Question

The molecular formula of allicin, the compound responsible for the characteristic smell of garlic, is \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{OS}_{2}\). (a) What is the molar mass of allicin? (b) How many moles of allicin are present in \(5.00 \mathrm{mg}\) of this substance? (c) How many molecules of allicin are in \(5.00 \mathrm{mg}\) of this substance? (d) How many \(\mathrm{S}\) atoms are present in 5.00 mg of allicin?

Step-by-Step Solution

Answer

(a) The molar mass of allicin is 162.26 g/mol. (b) There are \(2.72 \times 10^{-5}\) moles of allicin in 5.00 mg of the substance. (c) There are \(1.64 \times 10^{19}\) molecules of allicin in 5.00 mg of the substance. (d) There are \(3.28 \times 10^{19}\) S atoms in 5.00 mg of allicin.

Step 1: (a) Calculating the molar mass of Allicin

First, we need to find the molar mass of Allicin. We are given the molecular formula (\(\mathrm{C}_{6} \mathrm{H}_{10}\mathrm{OS}_{2}\)). The molar mass can be calculated by adding the molar masses of all the individual elements in the molecular formula, multiplied by their respective number of atoms in the formula. Use the atomic masses of C, H, O, and S: C: 12.01 g/mol H: 1.01 g/mol O: 16.00 g/mol S: 32.07 g/mol Molar mass of Allicin = (6 × 12.01) + (10 × 1.01) + (1 × 16.00) + (2 × 32.07) g/mol

Step 2: (b) Finding the number of moles in 5.00 mg of Allicin

To find the number of moles in 5.00 mg of Allicin, use the formula: Number of moles = (mass of substance) / (molar mass of substance) But first, convert the mass of Allicin from mg to g: 5.00 mg × (1 g / 1000 mg) = 0.00500 g Now, divide the mass in grams by the molar mass calculated in part (a): Number of moles = (0.00500 g) / (molar mass from part a)

Step 3: (c) Calculating the number of molecules in 5.00 mg of Allicin

To find the number of molecules in 5.00 mg of Allicin, use Avogadro's constant (6.022 × 10^23 molecules/mole): Number of molecules = (number of moles from part b) × (6.022 × 10^23 molecules/mole)

Step 4: (d) Determining the number of S atoms in 5.00 mg of Allicin

Finally, in order to find the number of S atoms in 5.00 mg of Allicin, we will use the number of molecules (calculated in part c) and the molecular formula. We know that there are 2 S atoms in one molecule of Allicin. Therefore, Number of S atoms = (number of molecules from part c) × 2

Key Concepts

Molar Mass Calculation

Molar mass is a crucial concept when dealing with chemical formulas. It represents the mass of one mole of a substance, typically in grams per mole (g/mol). For the molecule allicin, with the formula \(\mathrm{C}_6 \mathrm{H}_{10} \mathrm{OS}_2\), we calculate its molar mass by adding the products of the atomic masses of each element and their respective number of atoms in the molecule.
For allicin, the calculation is as follows:

• Carbon (C): 6 atoms × 12.01 g/mol = 72.06 g/mol
• Hydrogen (H): 10 atoms × 1.01 g/mol = 10.10 g/mol
• Oxygen (O): 1 atom × 16.00 g/mol = 16.00 g/mol
• Sulfur (S): 2 atoms × 32.07 g/mol = 64.14 g/mol

Adding these together gives the total molar mass for allicin:
\[\text{Molar mass of allicin} = 72.06 + 10.10 + 16.00 + 64.14 = 162.30\, \text{g/mol}\]

Avogadro's Number

Avogadro's number is foundational in stoichiometry, providing a link between the number of moles of a substance and its molecules or atoms. It is defined as \(6.022 \times 10^{23}\). This figure tells us the number of molecules in one mole of any substance.
When you know the number of moles of allicin in 5.00 mg, you can utilize Avogadro's number to calculate the number of molecules.
Remember:

• Calculate moles using the conversion formula: \(\text{Number of moles} = \frac{\text{mass in grams}}{\text{molar mass in g/mol}}\)
• Then, multiply the number of moles by Avogadro's number to find the number of molecules.

This seamless connection between mass, moles, and molecules helps in understanding how chemical reactions proceed quantitatively.

Mole Concept

The mole is a fundamental unit in chemistry used to express amounts of a chemical substance. A mole delineates a specific quantity—much like a dozen denotes 12 units. The mole provides a bridge between the atomic scale and the measurable mass scale.
When working with allicin, knowing its molar mass allows you to convert between grams and moles. For 5.00 mg of allicin, convert mass to moles to further derive the number of molecules and atoms involved in reactions.

• Convert 5.00 mg to grams: \(5.00 \text{ mg} = 0.00500 \text{ g}\)
• Use \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\) to calculate moles.

The mole concept is vital for comprehending reaction stoichiometry and predicting yields, reactants, and products in any chemical reaction.

Molecular Formulas

Molecular formulas like \(\mathrm{C}_6 \mathrm{H}_{10} \mathrm{OS}_2\) in allicin represent the actual number of atoms of each element in a molecule. These formulas provide insight into the composition, proportions, and arrangement within a molecule.
Understanding molecular formulas aids in:

• Determining molecular structure.
• Calculating molar mass.
• Predicting amounts of products in reactions.

For allicin, the composition indicates that each molecule contains 6 carbon atoms, 10 hydrogen atoms, 1 oxygen atom, and 2 sulfur atoms. In solving problems, these details help compute quantities at the molecular level—whether it's determining how much of a compound is present or how many sulfur atoms are in a given mass of allicin.