The exact number of moles of aluminum in 1.24 × 10^26 atoms is 62,000 / 301 moles, which is approximately 205.98 moles.
Understanding Moles and Avogadro's Number
In chemistry, a mole is a fundamental unit used to measure the amount of a substance. It provides a convenient way to count extremely large numbers of atoms, molecules, or other particles. Just as a "dozen" means 12 items, a "mole" represents a specific number of particles, which is defined by Avogadro's number.
Avogadro's Number
Avogadro's number is a constant that specifies the number of constituent particles (usually atoms or molecules) found in one mole of a substance. For the purpose of this calculation, we use the value commonly applied in such problems:
- 1 mole = 6.02 × 10^23 atoms
This constant serves as a crucial conversion factor, enabling us to bridge the gap between the macroscopic quantity (moles) and the microscopic quantity (number of individual atoms).
Calculating Moles of Aluminum
To determine the number of moles of aluminum from a given quantity of atoms, we apply a straightforward conversion using Avogadro's number. The method involves dividing the total number of atoms by the value of Avogadro's number.
Here's a step-by-step breakdown of how to calculate the moles of aluminum:
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Identify the Given Number of Atoms: We are provided with 1.24 × 10^26 atoms of aluminum.
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Recall Avogadro's Number: As established, 1 mole contains 6.02 × 10^23 atoms.
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Set Up the Conversion: To convert atoms into moles, we arrange the values as a division:
$$ \text{Moles of Aluminum} = \frac{\text{Number of Aluminum Atoms}}{\text{Avogadro's Number}} $$
$$ \text{Moles of Aluminum} = \frac{1.24 \times 10^{26} \text{ atoms}}{6.02 \times 10^{23} \text{ atoms/mole}} $$
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Perform the Division:
- Divide the numerical coefficients:
$1.24 \div 6.02 = \frac{1.24}{6.02} = \frac{124}{602} = \frac{62}{301}$ - Divide the powers of ten:
$10^{26} \div 10^{23} = 10^{(26-23)} = 10^3$
Combining these results, we get:
$$ \text{Moles of Aluminum} = \left(\frac{62}{301}\right) \times 10^3 \text{ moles} $$
$$ \text{Moles of Aluminum} = \frac{62,000}{301} \text{ moles} $$
- Divide the numerical coefficients:
This fractional value represents the precise, exact answer based on the provided numbers.
Summary of Conversion
For quick reference, here's a summary of the values used in the conversion:
| Quantity Given | Value |
|---|---|
| Number of Aluminum Atoms | 1.24 × 10^26 atoms |
| Avogadro's Number | 6.02 × 10^23 atoms/mole |
| Moles of Aluminum | 62,000 / 301 moles |
Precision and "Exact" Answers in Chemistry
While the exact answer is best expressed as a fraction, it's common in practical chemistry calculations to round results to an appropriate number of significant figures, reflecting the precision of the input measurements. Both 1.24 and 6.02 are given with three significant figures.
- Decimal Approximation: Converting the exact fraction to a decimal yields approximately 205.9800664...
- Rounded Value: If rounded to three significant figures, the result would be approximately 206 moles of aluminum, aligning with typical general chemistry calculations.
However, since the question specifically requested the "exact answer," the fractional form $\frac{62,000}{301}$ is the most precise representation using the given constants.
