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? \((\mathrm{c})\) How many molecules of allicin are in \(5.00 \mathrm{mg}\) of this substance? (d) How many S atoms are present in \(5.00 \mathrm{mg}\) of allicin?

Short answer

(a) The molar mass of allicin is 162.27 g/mol. (b) There are 3.08 x 10^-5 moles of allicin in 5.00 mg of the substance. (c) There are 1.85 x 10^19 molecules of allicin in 5.00 mg of the substance. (d) There are 3.70 x 10^19 S atoms in 5.00 mg of allicin.

Step-by-step solution

Find the atomic masses of the elements in allicin

Let's start by locating the atomic masses of the constituent elements. Using the periodic table, we find: Carbon (C): 12.01 g/mol Hydrogen (H): 1.01 g/mol Oxygen (O): 16.00 g/mol Sulfur (S): 32.07 g/mol

Calculate the molar mass of allicin

Now, we can calculate the molar mass of allicin using the atomic masses and the given molecular formula C_6H_10OS_2: Molar mass of allicin = (6 * 12.01 g/mol) + (10 * 1.01 g/mol) + (1 * 16.00 g/mol) + (2 * 32.07 g/mol) = 162.27 g/mol. (a) The molar mass of allicin is 162.27 g/mol.

Convert mass to moles

For part (b), we need to convert 5.00 mg of allicin to moles using the molar mass: 1. Convert the mass to grams: 5.00 mg * (1 g / 1000 mg) = 0.00500 g 2. Divide the mass by the molar mass: 0.00500 g / 162.27 g/mol = 3.08 x 10^-5 mol (b) There are 3.08 x 10^-5 moles of allicin in 5.00 mg of the substance.

Calculate the number of molecules

For part (c), we need to convert moles to the number of molecules using Avogadro's number (6.022 x 10^23 molecules/mol): Molecules in 5.00 mg allicin = (3.08 x 10^-5 mol) * (6.022 x 10^23 molecules/mol) = 1.85 x 10^19 molecules (c) There are 1.85 x 10^19 molecules of allicin in 5.00 mg of the substance.

Find the number of S atoms in 5.00 mg of allicin

For part (d), we need to find the number of S atoms. Since there are two S atoms per molecule of allicin, we can multiply the number of allicin molecules by 2: S atoms in 5.00 mg allicin = (1.85 x 10^19 molecules) * (2 S atoms/molecule) = 3.70 x 10^19 S atoms (d) There are 3.70 x 10^19 S atoms in 5.00 mg of allicin.

Key concepts

Molecular Formula

A molecular formula represents the composition of a molecule, indicating the number and type of atoms present. In the case of allicin, the molecular formula is \(\mathrm{C}_{6}\mathrm{H}_{10}\mathrm{OS}_{2} \), which tells us that each molecule is comprised of six carbon atoms, ten hydrogen atoms, one oxygen atom, and two sulfur atoms.
This formula helps to determine the molar mass, which is the mass of one mole of a substance. By understanding the molecular formula, we can move on to essential calculations like the total molecular weight. This ultimately allows us to perform conversions between mass, moles, and the number of molecules.

Allicin Calculation

Allicin, the compound responsible for garlic's scent, requires calculation steps based on its molecular formula. First, we figure out the molar mass by adding up the atomic masses of all atoms in its formula:

• Carbon \((C): 6 \times 12.01 = 72.06 \text{ g/mol}\)
• Hydrogen \((H): 10 \times 1.01 = 10.10 \text{ g/mol}\)
• Oxygen \((O): 1 \times 16.00 = 16.00 \text{ g/mol}\)
• Sulfur \((S): 2 \times 32.07 = 64.14 \text{ g/mol}\)

Adding these gives allicin a molar mass of \(162.27 \text{ g/mol}\).
Next, to find the moles of allicin in a 5.00 mg sample, convert the mg to grams (5.00 mg = 0.00500 g) and divide by the molar mass. Thus, \[0.00500 \text{ g} / 162.27 \text{ g/mol} = 3.08 \times 10^{-5} \text{ mol}\].
This allows further calculations, such as the number of molecules and atoms using Avogadro's number.

Avogadro's Number

Avogadro's number is an essential constant in chemistry, providing a link between moles and individual entities like atoms and molecules. The number is \(6.022 \times 10^{23} \text{ entities/mol}\).
It allows us to calculate how many molecules are present in a given amount of substance. In the case of our allicin sample, multiplying the number of moles \((3.08 \times 10^{-5} \text{ mol})\) by Avogadro's number gives the number of molecules:\[3.08 \times 10^{-5} \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol} = 1.85 \times 10^{19} \text{ molecules}\].
This concept is significant for making precise calculations about chemical quantities and reactions.

Atomic Mass

The atomic mass is the mass of an individual atom, typically measured in atomic mass units \(\text{(amu)}\) or grams per mole \(\text{(g/mol)}\). For allicin, we consider the atomic masses of each element involved: carbon, hydrogen, oxygen, and sulfur. These are:

• Carbon: 12.01 g/mol
• Hydrogen: 1.01 g/mol
• Oxygen: 16.00 g/mol
• Sulfur: 32.07 g/mol

These atomic masses, sourced from the periodic table, allow the calculation of a compound's molar mass by adding the weighted contributions of each atom.
The importance of knowing atomic mass is highlighted when determining molecular formula mass, reacting masses, and deriving important stoichiometric relationships for chemical reactions.