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To calculate the mass of gold(III) chloride (Au₂Cl₆) containing 1 billion gold atoms (1.000 × 10⁹ atoms), follow these steps:

### Step 1: Determine the number of moles of gold atoms
To do this, use Avogadro's number, which is \( 6.022 \times 10^{23} \) atoms/mol.

\[
\text{Moles of gold atoms} = \frac{\text{Number of atoms}}{\text{Avogadro's number}} = \frac{1.000 \times 10^9}{6.022 \times 10^{23}} 
\]

\[
\text{Moles of gold atoms} = 1.6606 \times 10^{-15} \, \text{mol}
\]

### Step 2: Relate the moles of gold atoms to moles of Au₂Cl₆
Each molecule of Au₂Cl₆ contains 2 gold atoms. Therefore, the number of moles of Au₂Cl₆ is half the number of moles of gold atoms:

\[
\text{Moles of Au₂Cl₆} = \frac{\text{Moles of gold atoms}}{2} = \frac{1.6606 \times 10^{-15}}{2} = 8.303 \times 10^{-16} \, \text{mol}
\]

### Step 3: Calculate the molar mass of Au₂Cl₆
The molar masses of the elements are:
- Gold (Au): \( 196.97 \, \text{g/mol} \)
- Chlorine (Cl): \( 35.45 \, \text{g/mol} \)

The molar mass of Au₂Cl₆ is:
\[
\text{Molar mass of Au₂Cl₆} = (2 \times 196.97) + (6 \times 35.45) = 393.94 + 212.7 = 606.64 \, \text{g/mol}
\]

### Step 4: Calculate the mass of Au₂Cl₆
Finally, use the moles of Au₂Cl₆ and the molar mass to calculate the mass:

\[
\text{Mass of Au₂Cl₆} = \text{Moles of Au₂Cl₆} \times \text{Molar mass of Au₂Cl₆}
\]
\[
\text{Mass of Au₂Cl₆} = (8.303 \times 10^{-16}) \times 606.64 \, \text{g/mol} = 5.034 \times 10^{-13} \, \text{g}
\]

### Final Answer:
The mass of Au₂Cl₆ that contains \( 1.000 \times 10^9 \) gold atoms is approximately \( 5.034 \times 10^{-13} \, \text{g} \), rounded to four significant digits.

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Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the mass change if there were 2 billion gold atoms?
2. What if the compound was gold(I) chloride (AuCl) instead of Au₂Cl₆?
3. How is the molar mass of a compound calculated from individual atoms?
4. Why do chemists use Avogadro's number in such calculations?
5. What is the significance of gold being in its +3 oxidation state in Au₂Cl₆?

**Tip**: Always double-check the units in calculations to ensure accuracy!

Calculate the mass of gold(III) chloride (Au₂Cl₆) that contains a billion (1.000 × 10⁹) gold atoms.

Learn how to calculate the mass of gold(III) chloride (Au₂Cl₆) containing 1 billion gold atoms. This solution covers stoichiometry, the use of Avogadro's number, and molar mass calculations. Suitable for high school students in Grades 10-12.

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