- Calculate the molar mass of \(\text{CaCO}_3\) by adding the atomic masses of its components: \(40.08 + 12.01 + 3 \times 16.00 = 100.09 \text{ g/mol}\).
- Determine the number of moles by dividing the sample mass by the molar mass: \(98.6 \text{ g} / 100.09 \text{ g/mol} \approx 0.985113 \text{ mol}\).
- Round the result to three significant figures to match the given precision: \(\boxed{0.985 \text{ mol}}\).
Explanation
- Determine the Molar Mass of Calcium Carbonate (\(\text{CaCO}_3\))
To find the number of moles in a given mass of a substance, we need the chemical formula to determine the molar mass. The chemical formula for calcium carbonate is \(\text{CaCO}_3\). This compound consists of one calcium atom (Ca), one carbon atom (C), and three oxygen atoms (O). The atomic masses are approximately:
- Ca: \(40.08 \text{ g/mol}\)
- C: \(12.01 \text{ g/mol}\)
- O: \(16.00 \text{ g/mol}\)
The molar mass of \(\text{CaCO}_3\) is calculated as:
\[M = (1 \times 40.08) + (1 \times 12.01) + (3 \times 16.00) = 40.08 + 12.01 + 48.00 = 100.09 \text{ g/mol}\]
- Calculate the Number of Moles
The number of moles \(n\) can be calculated by dividing the mass of the substance \(m\) by its molar mass \(M\).
Given:
- Mass \(m = 98.6 \text{ g}\)
- Molar mass \(M = 100.09 \text{ g/mol}\)
The calculation is:
\[n = \frac{m}{M} = \frac{98.6 \text{ g}}{100.09 \text{ g/mol}} \approx 0.985113 \text{ mol}\]
Since the given mass \(98.6 \text{ g}\) has three significant figures, we round our final answer to three significant figures, which is \(0.985 \text{ mol}\).
Examples
In chemistry and biology, knowing the number of moles is essential for quantitative analysis. For instance, marine biologists studying coral reef growth calculate the amount of calcium carbonate deposited by corals over time. By knowing the mass of the deposits, they can determine the number of moles to understand the rate of calcification, which helps them assess the health of the reef in response to changing ocean conditions.