Question

Determine the mass, in grams, of (a) \(7.34 \mathrm{mol} \mathrm{N}_{2} \mathrm{O}_{4}\) (b) \(3.16 \times 10^{24} \mathrm{O}_{2}\) molecules; (c) \(18.6 \mathrm{mol} \mathrm{CuSO}_{4} \cdot 5 \mathrm{H}_{2} \mathrm{O}\) (d) \(4.18 \times 10^{24}\) molecules of \(\mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{OH})_{2}\)

Short answer

The masses are (a) 675.47 g \(N_{2}O_{4}\), (b) 168.00 g \(O_{2}\), (c) 4644.25 g \(CuSO_{4} \cdot 5H_{2}O\), and (d) 431.05 g \(C_{2}H_{4}(OH)_{2}\)

Step-by-step solution

Calculate the mass of \(7.34 \mathrm{mol} \mathrm{N}_{2} \mathrm{O}_{4}\)

First, calculate the molar mass of \(N_{2}O_{4}\) by adding the molar masses of its constituent atoms. The molar mass is \(2(14.01 \, g/mol) + 4(16.00 \, g/mol) = 92.02 \, g/mol\). Multiply the given number of moles by the molar mass to find the mass: \(7.34 \, mol \times 92.02 \, g/mol = 675.47 \, g\)

Calculate the mass of \(3.16 \times 10^{24}\) O2 molecules

First, convert the number of molecules to moles by dividing by Avogadro's number, \(6.022 \times 10^{23} mol^{-1}\): \(3.16 \times 10^{24} molecules / 6.022 \times 10^{23}\, mol^{-1} = 5.25 \, mol\). The molar mass of \(O_{2}\) is \(2(16.00 g/mol) = 32.00 g/mol\). Multiply the number of moles by the molar mass to find the mass: \(5.25 \, mol \times 32.00 \, g/mol = 168.00 \,g\)

Calculate the mass of \(18.6 \, mol CuSO_{4} \cdot 5H_{2}O\)

First calculate the molar mass of \(CuSO_{4} \cdot 5H_{2}O\): \((63.55 g/mol) + (32.07 g/mol) + 4(16.00 g/mol) + 5((2.02 g/mol) + (16.00 g/mol)) = 249.68 g/mol\). Then multiply the given number of moles by the molar mass to find the mass: \(18.6 mol \times 249.68 g/mol = 4644.25 g\)

Calculate the mass of \(4.18 \times 10^{24}\) C2H4(OH)2 molecules

First convert the number of molecules to moles: \(4.18 \times 10^{24} molecules / 6.022 \times 10^{23} \, mol^{-1} = 6.94 mol\). Then calculate the molar mass of \(C_{2}H_{4}(OH)_{2}\): \(2(12.01 g/mol) + 4(1.01 g/mol) + 2((16.00 g/mol) + (1.01 g/mol) + (1.01 g/mol)) = 62.07 g/mol\). Multiply the number of moles by the molar mass to find the mass: \(6.94 mol \times 62.07 g/mol = 431.05 g\)

Key concepts

Avogadro's number

Avogadro's number is a fundamental constant used in chemistry, crucial for understanding the relationship between moles and molecules. It is defined as the number of atoms, ions, or molecules in one mole of a substance, which is approximately \(6.022 \times 10^{23}\). This constant offers a bridge from the microscopic scale to the macroscopic quantities we can measure.
To imagine this, think of Avogadro's number as a super-large group of particles. For example, if you have one mole of water (\(H_2O\)), it contains \(6.022 \times 10^{23}\) water molecules!
When converting from molecules to moles, Avogadro's number helps you transition from a countable number of molecules to the substantial mass we weigh out in grams or other measurable amounts. This allows chemists to calculate how much of a reactant is needed or how much of a product will be formed in a chemical reaction.

Moles to Mass Conversion

Converting moles to mass is an essential skill in chemistry that involves using the molar mass of a compound. The molar mass (also known as molecular weight) is the mass of one mole of a substance in grams, typically expressed in \( g/mol \).
The formula you use for this conversion is: \[\text{mass} = \text{moles} \times \text{molar mass}\]
In practice, if you have a certain number of moles of a compound, you multiply by its molar mass to find out how many grams you have. For example, if you have \(7.34\) moles of \(N_2O_4\), and its molar mass is \(92.02 \ g/mol\), multiplying these gives the total mass in grams: \(7.34 \ mol \times 92.02 \ g/mol = 675.47 \ g\).
This method helps in calculating real-world quantities from laboratory scales to industrial levels, ensuring accurate measurement and calculation of chemical substances.

Molecular Weight Determination

Molecular weight determination, often synonymous with molar mass calculation, is the process of summing the atomic weights of all atoms in a molecule. This plays a vital role in solving chemical problems, from simple classroom tasks to complex research projects.
To calculate the molecular weight, follow these steps:

• Identify each element in the compound and note the number of atoms of each.
• Look up the atomic weights of these elements on the periodic table (usually expressed in \( g/mol \)).
• Multiply the number of atoms of each element by their respective atomic weights.
• Add all these values together to get the molar mass (molecular weight) of the compound.

For instance, let's say you have \(CuSO_4 \cdot 5H_2O\). You would find the molar mass as follows: find the atomic weights of \(Cu\), \(S\), and \(O\), and multiply accordingly — adding the weights of water molecules multiplied by 5. Summing these gives \(249.68 \ g/mol\).
This calculated molar mass allows us to easily convert between mass and moles, and understand the proportions and interactions in chemical reactions.