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) \(2.88 \times 10^{-3}\) moles remain.

Step-by-step solution

Calculate Moles in 200 mg of CO2

Find the number of moles of \( \mathrm{CO}_2 \) in \( 200 \mathrm{mg} \). First, convert grams to moles using the molar mass of \( \mathrm{CO}_2 \), which is approximately \( 44 \mathrm{g/mol} \).Convert grams to moles:\[200 \mathrm{mg} = 0.2 \mathrm{g}\]\[\text{Moles of } \mathrm{CO}_2 = \frac{0.2 \mathrm{g}}{44 \mathrm{g/mol}} \approx 4.545 \times 10^{-3} \text{ moles}\]

Determine Total Molecules in Original Sample

Use Avogadro's number \( 6.022 \times 10^{23} \) to find the total molecules in the original \( 0.2 \mathrm{g} \) of \( \mathrm{CO}_2 \).\[\text{Total molecules} = 4.545 \times 10^{-3} \text{ moles} \times 6.022 \times 10^{23} \frac{\text{molecules}}{\text{mole}}= 2.74 \times 10^{21} \text{ molecules}\]

Subtract Removed Molecules

Remove \( 10^{21} \) molecules from the total calculated in the previous step.Subtract:\[2.74 \times 10^{21} - 10^{21} = 1.74 \times 10^{21} \text{ molecules remaining}\]

Convert Remaining Molecules Back to Moles

Convert the remaining molecules back into moles using Avogadro's number.\[\text{Moles of remaining } \mathrm{CO}_2 = \frac{1.74 \times 10^{21} \text{ molecules}}{6.022 \times 10^{23} \frac{\text{molecules}}{\text{mole}}} \approx 2.88 \times 10^{-3} \text{ moles}\]

Match the Answer

Compare with the given choices:(a) \(2.88 \times 10^{-3}\) — Correct answer(b) \(28.8 \times 10^{-3}\)(c) \(288 \times 10^{-3}\)(d) \(28.8 \times 10^{3}\)

Key concepts

Molar Mass Calculation

To calculate the number of moles in a given substance, you'll first need to know its molar mass. Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). For carbon dioxide (CO₂), the molar mass is approximately 44 g/mol.

This value is derived from the sum of the atomic masses of carbon (≈12 g/mol) and two oxygen atoms (≈16 g/mol each), giving us 44 g/mol. When you have the mass of a sample, like 200 mg of CO₂, you'll convert that mass into grams (0.2 g) and divide it by the molar mass of CO₂ to calculate the moles.

This is a crucial step whenever you want to convert a known mass into moles, and you'll use this frequently in chemistry calculations.

Avogadro's Number

Avogadro's number, named after the scientist Amedeo Avogadro, is a fundamental constant in chemistry. It is defined as approximately \(6.022 \times 10^{23}\) molecules per mole. This incredibly large number represents the number of constituent particles, usually atoms or molecules, that are contained in the mole of a substance.

When you're given a number of moles of a substance, this constant helps you determine how many molecules or atoms you have. For example, when you know there are \(4.545 \times 10^{-3}\) moles of CO₂, you can use Avogadro's number to find the total number of molecules: \(4.545 \times 10^{-3} \text{ moles} \times 6.022 \times 10^{23} \approx 2.74 \times 10^{21} \text{ molecules}\).

Understanding this relationship is key when you're working with mole-to-molecule conversions and vice versa.

Molecule Subtraction

Molecule subtraction refers to the process of calculating the number of molecules remaining after a certain quantity has been removed from a sample. In the context of our problem, if you start with a known number of molecules and then remove a specific number, you perform subtraction to find out how many molecules are left.

In the exercise, you were initially given \(2.74 \times 10^{21}\) molecules. If you subtract \(10^{21}\) molecules, you're left with \(1.74 \times 10^{21}\) molecules. It's a simple arithmetic operation but ensures understanding of the flow and changes in the molecules during reactions or other processes.

Performing this operation correctly is essential to accurately calculate further changes or conversions in mole-related chemistry tasks.

Mole Conversion

Mole conversion ties together various concepts like molar mass and Avogadro's number, allowing you to interchangeably convert between masses, moles, and molecules. In chemical calculations, after determining how many molecules are left using the molecule subtraction process, you convert those molecules back into moles using Avogadro's number.

In the problem discussed, you had \(1.74 \times 10^{21}\) remaining molecules. Dividing this by Avogadro's number \(6.022 \times 10^{23} \text{ molecules/mole}\), gives you the moles of the substance, calculated as approximately \(2.88 \times 10^{-3}\) moles.

This step ensures you align your final answer in moles, which is often required in many chemistry problems, especially when assessing the quantity of reactants or products in a chemical reaction.