Question

Calculate the number of molecules present in each of the following samples. a. \(3.45 \mathrm{~g}\) of \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) b. 3.45 moles of \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) c. \(25.0 \mathrm{~g}\) of \(\mathrm{ICl}_{5}\) d. \(1.00 \mathrm{~g}\) of \(\mathrm{B}_{2} \mathrm{H}_{6}\) e. \(1.05 \mathrm{mmol}\) of \(\mathrm{Al}\left(\mathrm{NO}_{3}\right)_{3}\)

Short answer

a. \(1.153 \times 10^{22}\) molecules b. \(2.077 \times 10^{24}\) molecules c. \(4.941 \times 10^{22}\) molecules d. \(2.175 \times 10^{22}\) molecules e. \(6.323 \times 10^{20}\) molecules

Step-by-step solution

Equation for Number of Molecules Calculation

To calculate the number of molecules, we can use the following equation: Number of molecules = NumberOfMoles × Avogadro's Number where: NumberOfMoles = Sample Mass / Molar Mass (for mass given in grams) Avogadro's Number = 6.022 x 10^23 Now, we will find the number of molecules for each sample given in the exercise.

Calculating the Number of Molecules in Sample (a) of C6H12O6

First, find the molar mass of \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), which is: Molar Mass = (6 × 12.01) + (12 × 1.01) + (6 × 16.00) = 72.06 + 12.12 + 96.00 = 180.18 g/mol Next, find the number of moles in 3.45 g of C6H12O6: NumberOfMoles = Sample Mass / Molar Mass = (3.45 g) / (180.18 g/mol) = 0.01916 moles Finally, use the above equation to find the number of molecules: Number of molecules = NumberOfMoles × Avogadro's Number = 0.01916 moles × 6.022 × 10^23 = 1.153 × 10^22 molecules

Calculating the Number of Molecules in Sample (b) 3.45 moles of C6H12O6

In this case, number of moles is already given. We directly calculate the number of molecules by using the above equation: Number of molecules = NumberOfMoles × Avogadro's Number = 3.45 moles × 6.022 × 10^23 = 2.077 × 10^24 molecules

Calculating the Number of Molecules in Sample (c) of ICl5

First, find the molar mass of \(\mathrm{ICl}_{5}\): Molar Mass = 126.90 (I) + 5 × 35.45 (Cl) = 126.90 + 177.25 = 304.15 g/mol Next, find the number of moles in 25.0 g of ICl5: NumberOfMoles = Sample Mass / Molar Mass = (25.0 g) / (304.15 g/mol) = 0.08213 moles Finally, calculate the number of molecules: Number of molecules = NumberOfMoles × Avogadro's Number = 0.08213 moles × 6.022 × 10^23 = 4.941 × 10^22 molecules

Calculating the Number of Molecules in Sample (d) of B2H6

First, find the molar mass of \(\mathrm{B}_{2} \mathrm{H}_{6}\): Molar Mass = (2 × 10.81) + (6 × 1.01) = 21.62 + 6.06 = 27.68 g/mol Next, find the number of moles in 1.00 g of B2H6: NumberOfMoles = Sample Mass / Molar Mass = (1.00 g) / (27.68 g/mol) = 0.03615 moles Finally, calculate the number of molecules: Number of molecules = NumberOfMoles × Avogadro's Number = 0.03615 moles × 6.022 × 10^23 = 2.175 × 10^22 molecules

Calculating the Number of Molecules in Sample (e) of Al(NO3)3

Convert mmol to moles: NumberOfMoles = 1.05 mmol × (1 mol / 1000 mmol) = 0.00105 moles Find the molar mass of \(\mathrm{Al}\left(\mathrm{NO}_{3}\right)_{3}\): Molar Mass = 26.98 (Al) + 3 × (1 × 14.01 (N) + 3 × 16.00 (O)) = 26.98 + 3 × (14.01 + 48.00) = 26.98 + 186.03 = 213.01 g/mol Calculate the number of molecules: Number of molecules = NumberOfMoles × Avogadro's Number = 0.00105 moles × 6.022 × 10^23 = 6.323 × 10^20 molecules

Key concepts

Avogadro's Number

Understanding Avogadro's number is key to calculating the number of particles in a chemical sample. It is defined as the number of atoms contained in one mole of a substance, which is a standard unit in chemistry to express amounts of a chemical substance. The numerical value of Avogadro's number is approximately \(6.022 \times 10^{23}\), which means that one mole of any substance contains this many constituent particles, be they atoms, ions, or molecules.

When facing problems like calculating the number of molecules in a given sample, Avogadro's number allows us to make the conversion from moles to actual number of molecules. This concept is vital because it provides a bridge between the macroscopic quantities we measure in the lab (like grams of a substance) and the microscopic amount of molecules or atoms those grams represent.

Molar Mass

The molar mass of a substance is the mass of one mole of that substance. It is a key concept in chemistry because it relates the mass of substances to the number of particles they contain using Avogadro's number. Molar mass is usually expressed in grams per mole (g/mol), and it's calculated by summing the masses of the individual atoms in a molecule as given by the periodic table.

For instance, the molar mass of a water molecule (H₂O) would be calculated using the atomic masses of hydrogen (1.01 g/mol) and oxygen (16.00 g/mol), resulting in approximately 18.02 g/mol. When given a certain mass of a substance, one can determine the number of moles by dividing this mass by the molar mass, which is a crucial step before using Avogadro's number to find the number of molecules in the sample.

Chemical Formulas

Chemical formulas such as \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) for glucose or \(\mathrm{ICl}_{5}\) for iodine pentachloride represent the composition of a chemical compound using element symbols and numerical subscripts. These formulas provide not only the types of atoms in the molecule but also how many of each type.

Understanding Chemical Formulas

Each element symbol corresponds to one atom of that element; for example, \(\mathrm{H}\) represents hydrogen. The subscript after an element tells you how many of those atoms are present in one molecule. If there's no subscript, it means there's only one atom of that element in the molecule. In chemical calculations, these subscripts are fundamental in determining the molar mass and, consequently, the number of molecules when given a certain amount of a substance.