Question

\(10^{21}\) molecules are removed from \(200 \mathrm{mg}\) of \(\mathrm{CO}_{2}\). The moles of \(\mathrm{CO}_{2}\) left are (a) \(2.88 \times 10^{-3}\) (b) \(28.8 \times 10^{-3}\) (c) \(288 \times 10^{-3}\) (d) \(28.8 \times 10^{3}\)

Short answer

(a) \(2.88 \times 10^{-3}\)

Step-by-step solution

Find the Moles of CO2 in the Original Sample

The molecular weight of CO₂ is 44 g/mol. First, we convert 200 mg to grams: \(200 \, \text{mg} = 0.2 \, \text{g}\). The number of moles in the original 0.2 g of CO₂ is given by \(\frac{\text{mass}}{\text{molar mass}} = \frac{0.2}{44} = 4.5455 \times 10^{-3} \text{ moles}\).

Calculate the Number of Molecules in the Original Sample

To determine how many molecules are in the original moles, use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules/mol. Multiply the number of moles by this constant: \(4.5455 \times 10^{-3} \times 6.022 \times 10^{23} = 2.737 \times 10^{21} \text{ molecules}\).

Calculate the Number of Molecules Left

From the problem, we know that \(10^{21}\) molecules are removed from the original \(2.737 \times 10^{21}\) molecules. This leaves \(2.737 \times 10^{21} - 10^{21} = 1.737 \times 10^{21}\) molecules.

Calculate Moles of CO2 Left

Divide the remaining molecules by Avogadro's number to convert back to moles: \(\frac{1.737 \times 10^{21}}{6.022 \times 10^{23}} = 2.88 \times 10^{-3} \text{ moles}\).

Key concepts

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry representing the number of entities, such as atoms or molecules, in one mole of a substance. The value of Avogadro's number is fixed at \(6.022 \times 10^{23}\), a very large number. This means that one mole of any substance contains \(6.022 \times 10^{23}\) particles, whether they are atoms, molecules, or formula units. Understanding this concept helps in converting moles to molecules and vice versa, making it instrumental in stoichiometry—the part of chemistry that deals with the relative quantities of reactants and products in chemical reactions. Whenever you're asked about the number of molecules or atoms in a certain amount of substance, Avogadro's number is your go-to tool. Using it involves either multiplying or dividing by this number, depending on the direction of the conversion.

Molecular Weight

Molecular weight, sometimes called molecular mass, is the mass of a single molecule of a compound compared to 1/12 of the mass of a carbon-12 atom. Each element has its atomic weight, and the molecular weight is the sum of the atomic weights of all atoms in a molecule. For carbon dioxide (CO₂), the molecular weight is determined by adding the atomic weights of 1 carbon atom (approximately 12) and 2 oxygen atoms (approximately 16 each), resulting in a molecular weight of 44 g/mol. This value plays a critical role when working out conversions in chemistry involving moles, especially when shifting between mass and mole units. Knowing the molecular weight allows for calculating the number of moles from a given mass. For example, in the problem, 200 mg or 0.2 g of CO₂ are converted using its molecular weight to find how many moles that mass contains.

Mole Conversion

Mole conversion is a fundamental process in chemistry that involves converting between different units of measure where moles are involved, such as mass, volume, and particles (atoms, molecules). This process is essential to quantify the substances before and after chemical reactions. In the given problem, we convert mass (grams) to moles using the formula: \[\text{Number of Moles} = \frac{\text{Mass in grams}}{\text{Molar Mass}}\]Starting with 200 mg of CO₂, converted to grams as 0.2 g, we use the molecular weight of CO₂ (44 g/mol) for this conversion. This calculation gives the initial moles of CO₂:\[\frac{0.2}{44} = 4.5455 \times 10^{-3} \text{ moles}\]Understanding mole conversion is significant for any calculations involving a chemical equation, allowing scientists and students alike to comprehend changes in the amount of substances during reactions.

Number of Molecules

Once you know the number of moles of a substance, you can easily determine the number of molecules by employing Avogadro's number. This process involves multiplying the number of moles by \(6.022 \times 10^{23}\) to find out how many molecules are present. In the exercise, initially, 4.5455 \times 10^{-3} moles of CO₂ result in \(2.737 \times 10^{21}\) molecules by applying:\[\text{Number of Molecules} = \text{Number of Moles} \times 6.022 \times 10^{23}\]If some molecules are removed, like the \(10^{21}\) molecules in the problem, simply subtract to find the amount remaining:\[2.737 \times 10^{21} - 10^{21} = 1.737 \times 10^{21} \text{ molecules}\]Finally, to find out how many moles these remaining molecules correspond to, divide by Avogadro's number. Following the given solution, this yields \(2.88 \times 10^{-3}\) moles left in the sample.