Calculate the number of gold atoms in 300 mg of a gold ring of 20 carat gold (atomic mass of gold = 197, pure gold is 24 carat).

Explanation

To calculate the number of gold atoms in 300 mg of a 20 carat gold ring, follow these steps:

### Step 1: Determine the amount of pure gold in the 20 carat gold ring.
- Pure gold is defined as 24 carat. Therefore, the fraction of pure gold in 20 carat gold is:
\[ \text{Fraction of pure gold} = \frac{20}{24} \]
- Calculate the mass of pure gold in the ring:
\[ \text{Mass of pure gold} = \text{Total mass of the ring} \times \text{Fraction of pure gold} \]
\[ \text{Mass of pure gold} = 300 \, \text{mg} \times \frac{20}{24} = 250 \, \text{mg} \]

### Step 2: Convert the mass of pure gold from milligrams to grams.
- Since 1 mg = \(10^{-3}\) g, convert 250 mg to grams:
\[ \text{Mass of pure gold in grams} = 250 \, \text{mg} \times 10^{-3} = 0.250 \, \text{g} \]

### Step 3: Calculate the number of moles of gold.
- Use the formula for moles:
\[ \text{Moles of gold} = \frac{\text{Mass of gold (g)}}{\text{Molar mass of gold (g/mol)}} \]
- The atomic mass of gold is given as 197 g/mol:
\[ \text{Moles of gold} = \frac{0.250 \, \text{g}}{197 \, \text{g/mol}} \approx 0.00127 \, \text{mol} \]

### Step 4: Calculate the number of gold atoms.
- Use Avogadro's number, which is \(6.022 \times 10^{23}\) atoms/mol:
\[ \text{Number of gold atoms} = \text{Moles of gold} \times \text{Avogadro's number} \]
\[ \text{Number of gold atoms} = 0.00127 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 7.64 \times 10^{20} \, \text{atoms} \]

### Final Answer:
The number of gold atoms in 300 mg of a 20 carat gold ring is approximately \(7.64 \times 10^{20}\) atoms.

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