Problem 39 - Chemistry The Central Science

Question

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

Step-by-Step Solution

Answer

(a) The molar mass of allicin is 162.3 g/mol. (b) There are \(3.08 \times 10^{-5}\) moles of allicin in 5.00 mg of the substance. (c) There are \(1.86 \times 10^{19}\) molecules of allicin in 5.00 mg of the substance. (d) There are \(3.72 \times 10^{19}\) sulfur atoms in 5.00 mg of allicin.

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

To find the molar mass of allicin, add the molar masses of all the atoms in the molecular formula. The molecular formula of allicin is given as \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{OS}_{2}\). The molar mass of carbon (C) is 12.01 g/mol, hydrogen (H) is 1.01 g/mol, oxygen (O) is 16.00 g/mol, and sulfur (S) is 32.07 g/mol. Molar mass of allicin = \((6 \times 12.01) + (10 \times 1.01) + (1 \times 16.00) + (2 \times 32.07)\) g/mol = \(72.06 + 10.10 + 16.00 + 64.14\) g/mol = \(162.3\) g/mol The molar mass of allicin is 162.3 g/mol.

Step 2: (b) Calculate the number of moles of allicin

To find the number of moles of allicin in 5.00 mg, convert the mass to grams and use the molar mass we calculated in part (a). 5.00 mg = \(5.00 \times 10^{-3}\) g Number of moles = \(\frac{mass}{molar \, mass}\) Number of moles = \(\frac{5.00 \times 10^{-3}\,g}{162.3\,g/mol}\) = \(3.08 \times 10^{-5}\) mol There are \(3.08 \times 10^{-5}\) moles of allicin in 5.00 mg of the substance.

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

To find the number of molecules of allicin, we multiply the number of moles by Avogadro's number (6.022 x 10^23). Number of molecules = (number of moles) x (Avogadro's number) Number of molecules = \((3.08 \times 10^{-5}\,mol) \times (6.022 \times 10^{23}\,mol^{-1})\) = \(1.86 \times 10^{19}\) molecules There are \(1.86 \times 10^{19}\) molecules of allicin in 5.00 mg of the substance.

Step 4: (d) Calculate the number of sulfur atoms in 5.00 mg of allicin

To determine the number of sulfur atoms present in 5.00 mg of allicin, we must first determine the number of molecules in that sample (which we already calculated in part (c)) and then account for the fact that there are two sulfur atoms in each allicin molecule. Number of S atoms = (number of allicin molecules) x (number of S atoms per molecule) Number of S atoms = \((1.86 \times 10^{19}\,molecules) \times (2\,S\,atoms/molecule)\) = \(3.72 \times 10^{19}\) S atoms There are \(3.72 \times 10^{19}\) sulfur atoms in 5.00 mg of allicin.

Key Concepts

Understanding Molar Mass

Molar mass is a crucial concept in chemistry that refers to the mass of one mole of a given substance. To calculate the molar mass of a compound like allicin, you sum up the molar masses of all the atoms present in its molecular formula. For allicin (\(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{OS}_{2}\)), you add:

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

By adding these together, you find the molar mass of allicin to be 162.3 g/mol.
This tells us how much 6 carbons, 10 hydrogens, 1 oxygen, and 2 sulfurs combined will weigh per mole.

Calculating Moles of a Substance

The calculation of moles is important when relating the mass of a substance to the number of particles it contains. To determine the number of moles, use the formula:\[\text{Number of moles} = \frac{\text{mass}}{\text{molar mass}}\]In the example problem, converting 5.00 mg of allicin into grams gives you 0.00500 g. With a molar mass of 162.3 g/mol, the calculation yields:\[\frac{0.00500 \, g}{162.3 \, g/mol} = 3.08 \times 10^{-5} \, mol\]This means that 5.00 mg of allicin is equivalent to 3.08 x 10^{-5} moles. Calculating moles helps us understand the proportion of a substance available within a given mass.

The Role of Avogadro's Number

Avogadro's number, which is approximately \(6.022 \times 10^{23}\), equals the number of particles (atoms, molecules) in one mole of a substance. This concept makes it possible to link moles with actual numbers of particles.
To find out how many molecules are in 5.00 mg of allicin, you multiply the number of moles by Avogadro's number:\[(3.08 \times 10^{-5} \, mol) \times (6.022 \times 10^{23} \, mol^{-1}) = 1.86 \times 10^{19} \, molecules\]This means that a tiny 5.00 mg sample contains approximately 1.86 x 10^{19} molecules of allicin. Understanding this conversion is essential for grasping the scale at which chemical reactions occur.

Determining Elemental Composition

Elemental composition looks at the number and types of atoms in a compound. It's important for understanding what comprises a substance. In the case of allicin, knowing it has the formula \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{OS}_{2}\) tells us the molecule includes carbon, hydrogen, oxygen, and sulfur.
To find the number of sulfur atoms in 5.00 mg of allicin, knowing the molecular composition helps. Since each molecule of allicin contains 2 sulfur atoms, you multiply the number of molecules by 2:\[(1.86 \times 10^{19} \, molecules) \times (2 \, S/molecule) = 3.72 \times 10^{19} \, S \, atoms\]This calculation reveals that in 5.00 mg of allicin, there are approximately 3.72 x 10^{19} sulfur atoms. Understanding elemental composition helps chemists analyze and make decisions about chemical behavior and properties.