Question

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

Short answer

(a) The molar mass of allicin is 162.26 g/mol. (b) There are \(3.08 \times 10^{-5}\) moles of allicin in a 5 mg sample. (c) There are \(1.85 \times 10^{19}\) molecules of allicin in a 5 mg sample. (d) There are \(3.70 \times 10^{19}\) sulfur atoms in a 5 mg sample of allicin.

Step-by-step solution

We need to find the molar mass of allicin, with molecular formula C6H10OS2. To do this, we will first find the molar mass of each element (taking values from the periodic table) and then multiply by the number of atoms of each element: Molar_mass_of_allicin = (6 × molar_mass_of_C) + (10 × molar_mass_of_H)+(1 × molar_mass_of_O) + (2 × molar_mass_of_S) Molar_mass_of_C = 12.01 g/mol Molar_mass_of_H = 1.01 g/mol Molar_mass_of_O = 16.00 g/mol Molar_mass_of_S = 32.07 g/mol Complete the calculation: Molar_mass_of_allicin= (6 × 12.01 g/mol) + (10 × 1.01 g/mol) + (1 × 16.00 g/mol) + (2 × 32.07 g/mol) = 162.26 g/mol (a) The molar mass of allicin is 162.26 g/mol. #Step 2: Calculate the number of moles of allicin in a 5 mg sample#

To find the number of moles of allicin, we will divide the mass of the allicin sample (5 mg) by the molar mass of allicin: Number_of_moles = mass_of_sample / Molar_mass_of_allicin Convert 5.00 mg to grams (1 mg = 0.001 g): 5.00 mg = 5.00 × 0.001 g = 0.005 g Complete the calculation: Number_of_moles = 0.005 g / 162.26 g/mol = 3.08 × 10^(-5) mol (b) There are 3.08 × 10^(-5) moles of allicin in a 5 mg sample. #Step 3: Calculate the number of allicin molecules in a 5 mg sample#

Now, to find the number of allicin molecules, we will multiply the number of moles by Avogadro's number (approximately 6.022 × 10^23/mol): Number_of_molecules = Number_of_moles × Avogadro's_number Complete the calculation: Number_of_molecules = 3.08 × 10^(-5) mol × 6.022 × 10^23/mol = 1.85 × 10^(19) molecules (c) There are 1.85 × 10^(19) molecules of allicin in a 5 mg sample. #Step 4: Calculate the number of sulfur atoms in a 5 mg sample of allicin#

Remember that in the molecular formula of allicin, there are two sulfur atoms. To find the total number of sulfur atoms present in a 5 mg sample of allicin, we can simply multiply the number of allicin molecules by the number of sulfur atoms per molecule: Number_of_S_atoms = Number_of_molecules × 2 Complete the calculation: Number_of_S_atoms = 1.85 × 10^19 × 2 = 3.70 × 10^19 S atoms (d) There are 3.70 × 10^19 sulfur atoms in a 5 mg sample of allicin.

Key concepts

Molecular Formula

Understanding the molecular formula of a compound gives insight into the types and numbers of atoms that make up a molecule. For allicin, the molecular formula is \( \mathrm{C}_6 \mathrm{H}_{10} \mathrm{OS}_2 \). This formula tells us that one molecule of allicin contains six carbon (C) atoms, ten hydrogen (H) atoms, one oxygen (O) atom, and two sulfur (S) atoms.
The molecular formula is not only a representation of the molecule's composition but also essential in determining other properties, such as its molar mass.
The molar mass is calculated by adding the atomic masses of all atoms present in the molecular formula. Here, you can use the atomic masses found in the periodic table, which are measured in grams per mole (g/mol).

• For carbon (C): Mass = 12.01 g/mol.
• For hydrogen (H): Mass = 1.01 g/mol.
• For oxygen (O): Mass = 16.00 g/mol.
• For sulfur (S): Mass = 32.07 g/mol.

To find the molar mass of allicin, you multiply these atomic masses by their respective number of atoms in the formula, and sum them up. Performing this operation gives the molar mass, which, for allicin, calculates to 162.26 g/mol.

Avogadro's Number

Avogadro’s number is a cornerstone concept in chemistry, simplifying the understanding of relationships between atoms, molecules, and moles. The number is defined as \(6.022 \times 10^{23}\), a remarkably large constant that conveys the amount of atoms or molecules in a mole.
Utilizing Avogadro's number allows chemists to bridge the gap between the microscopic scale (atoms and molecules) and reality in lab practice with tangible masses and volumes.
For instance, in calculating the number of molecules in a sample, such as our 5 mg of allicin, once the number of moles is ascertained (let's say it was calculated to be \(3.08 \times 10^{-5} \text{ mol}\)) via formulae shown above, multiplying this by Avogadro's number gives the number of molecules within that sample:

• Total molecules of allicin = \(3.08 \times 10^{-5} \text{ mol} \times 6.022 \times 10^{23}\text{ molecules/mol}\)

This calculation allows us to understand chemical quantities more precisely and aligns with stoichiometry in reaction mechanisms and product yields in laboratory or industrial practices.

Chemical Calculations

Chemical calculations allow us to determine various chemical properties and predict outcomes in chemical processes. These calculations are founded on basic principles such as the conservation of mass and use of a standard unit of measurement called the mole.
In this particular exercise with allicin, chemical calculations were employed to:

• Determine allicin's molar mass using its molecular formula, which is crucial for converting between grams and moles.
• Convert milligrams to grams, since using a consistent unit throughout calculations ensures accuracy (5 mg converts to 0.005 g).
• Calculate the number of moles in a given mass by dividing the mass of the sample by the molar mass. This step unravels the sample's stoichiometric potential.
• Determine molecular population via calculation of the number of molecules using Avogadro's number. This bridges mass-based measurement to particle count.
• Quantify element-specific atom counts in a substance—like sulfur atoms—in terms of molecular formula specifics (as two sulfur atoms per allicin molecule implies two per entire molecular count).

These calculations empower chemists to plan experiments, balance equations, and predict how substances will react, making chemistry a quantifiable and predictive science.