Problem 39 - Chemistry: The Central Science

Question

The molecular formula of allicin, the compound responsible for the characteristic smell of garlic, is \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{OS} \mathrm{S}_{2}\). (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 \(5.00 \mathrm{mg}\) of allicin?

Step-by-Step Solution

Answer

(a) The molar mass of allicin is \(162.26 \mathrm{g}\cdot\mathrm{mol}^{-1}\). (b) There are \(3.08 \times 10^{-5}\) moles of allicin in \(5.00 \mathrm{mg}\) of the substance. (c) There are \(1.85 \times 10^{19}\) molecules of allicin in \(5.00 \mathrm{mg}\) of the substance. (d) There are \(3.70 \times 10^{19}\) S atoms in \(5.00 \mathrm{mg}\) of allicin.

Step 1: (a) Finding the molar mass of allicin

To calculate the molar mass of allicin, we need to determine the number of each type of atom in the given molecular formula and then sum up their molar masses. The molar mass of an atom is roughly the atomic number, taken from the periodic table. The molar mass of allicin in atomic mass units (\(\mathrm{amu}\)) can be calculated as follows: Molar mass of Allicin = (Number of C atoms x Molar mass of C) + (Number of H atoms x Molar mass of H) + (Number of O atoms x Molar mass of O) + (Number of S atoms x Molar mass of S)

Step 2: (b) Finding the moles of allicin

To find the moles of allicin, we'll use the equation: Moles = Mass / Molar Mass Given the mass of the allicin is \(5.00 \mathrm{mg}\), we'll convert it to grams before plugging in the values.

Step 3: (c) Finding the number of allicin molecules

To find the number of allicin molecules in \(5.00 \mathrm{mg}\) of the substance, we'll use the equation: Number of molecules = Moles * Avogadro's constant = Moles * \(6.022 \times 10^{23}\)

Step 4: (d) Finding the number of S atoms

To find the number of S atoms in \(5.00 \mathrm{mg}\) of allicin, we can multiply the number of allicin molecules by the number of sulfur atoms in one molecule: Number of S atoms = Number of allicin molecules * Number of S atoms per allicin molecule

Key Concepts

Molecular Formula

Understanding the molecular formula of a substance is crucial in chemistry as it provides the exact number of each type of atom present in a molecule. In the case of allicin, the compound that gives garlic its characteristic smell, the molecular formula is represented as \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{O}\mathrm{S} \mathrm{S}_{2}\).

This formula tells us that each molecule of allicin contains 6 carbon atoms (C), 10 hydrogen atoms (H), 1 oxygen atom (O), and 2 sulfur atoms (S). Knowing the molecular formula is the first step in the molar mass calculation, which allows us to quantify the amounts of a substance involved in chemical reactions.

Avogadro's Constant

Avogadro's constant is a fundamental figure in chemistry, symbolizing the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. It is numerically equivalent to \(6.022 \times 10^{23}\) particles per mole. This vast number enables chemists to work with the submicroscopic particles on a macroscopic scale.

When calculating the number of molecules in a given mass of a substance, like allicin, Avogadro's constant provides the bridge between the mole concept and the actual number of molecules or atoms in that mass. This allows for precise calculations in chemical formulations and stoichiometry.

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It is based on the conservation of mass where the mass of the reactants equals the mass of the products. Stoichiometry involves calculations of the masses and moles in a chemical equation, leveraging the balanced reaction to determine the amount of each substance required or produced.

In exercises like our allicin example, understanding stoichiometry allows students to solve for various quantities, such as the number of moles of allicin in a given mass, by applying the molar mass of the substance, thus stretching the concept beyond basic molar mass calculations.

Atomic Mass Units

The atomic mass unit (amu) is a standard unit of mass that quantifies mass at an atomic or molecular level. It is defined as one twelfth of the mass of a carbon-12 atom, which makes it extremely convenient for expressing the weights of atoms and molecules on an understandable scale.

When calculating the molar mass of a molecule, such as allicin, each atom's mass is considered in atomic mass units. The sum of the atomic masses of all the atoms in the molecular formula gives us the molar mass of the molecule. This is vital for various chemical calculations including those involving stoichiometry and determining the number of moles in a certain mass of a compound.