Given mass number of gold `= 197`, Density of gold `= 19.7 g cm^(-1)`. The radious of the gold atom is appoximately:

Options

A. `1.587 xx 10^(-8)m`
B. `1.587 xx 10^(-9)m`
C. `1.587 xx 10^(-10)m`
D. `1.587 xx 10^(-12)m`

Explanation

To find the radius of a gold atom given its mass number and density, we can follow these steps:

### Step 1: Gather the Given Information
- Mass number of gold (A) = 197
- Density of gold (D) = 19.7 g/cm³

### Step 2: Convert Density to kg/m³
To use the density in SI units, we convert it from g/cm³ to kg/m³:
\[ D = 19.7 \, \text{g/cm}^3 = 19.7 \times 10^3 \, \text{kg/m}^3 \]

### Step 3: Calculate the Mass of a Gold Atom
The mass of one gold atom can be calculated using the mass number:
\[ \text{Mass of gold atom} = A \times 1.66 \times 10^{-27} \, \text{kg} \]
\[ \text{Mass of gold atom} = 197 \times 1.66 \times 10^{-27} \, \text{kg} \approx 3.27 \times 10^{-25} \, \text{kg} \]

### Step 4: Use the Density Formula
The density formula is given by:
\[ D = \frac{\text{Mass}}{\text{Volume}} \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, we can rearrange the density formula to find the radius:
\[ D = \frac{m}{\frac{4}{3} \pi r^3} \implies r^3 = \frac{m}{D \cdot \frac{4}{3} \pi} \]

### Step 5: Substitute the Values
Substituting the mass of the gold atom and the density into the equation:
\[ r^3 = \frac{3.27 \times 10^{-25} \, \text{kg}}{19.7 \times 10^3 \, \text{kg/m}^3 \cdot \frac{4}{3} \pi} \]

### Step 6: Calculate \( r^3 \)
Calculating the denominator:
\[ 19.7 \times 10^3 \cdot \frac{4}{3} \pi \approx 8.27 \times 10^4 \, \text{kg/m}^3 \]
Now, substituting this back:
\[ r^3 = \frac{3.27 \times 10^{-25}}{8.27 \times 10^4} \approx 3.95 \times 10^{-30} \, \text{m}^3 \]

### Step 7: Calculate the Radius \( r \)
Taking the cube root of both sides to find \( r \):
\[ r \approx (3.95 \times 10^{-30})^{1/3} \approx 1.587 \times 10^{-10} \, \text{m} \]

### Final Answer
The radius of the gold atom is approximately:
\[ r \approx 1.587 \times 10^{-10} \, \text{m} \]