Ratio of nuclear radii of `""^(135)Cs "to" ""^(40)Ca` is

Options

A. `1.40`
B. `1.50`
C. `2.750`
D. `3.375`

Answer

To find the ratio of the nuclear radii of cesium-135 (Cs) to calcium-40 (Ca), we can use the formula for the radius of a nucleus:

\[ R = R_0 A^{1/3} \]

where:
- \( R \) is the radius of the nucleus,
- \( R_0 \) is a constant (approximately \( 1.2 \, \text{fm} \)),
- \( A \) is the mass number (number of nucleons) of the nucleus.

### Step-by-step Solution:

1. **Identify the mass numbers**:
- For cesium-135, the mass number \( A_{Cs} = 135 \).
- For calcium-40, the mass number \( A_{Ca} = 40 \).

2. **Write the formula for the radii**:
- The radius of the cesium nucleus \( R_{Cs} \) is given by:
\[ R_{Cs} = R_0 (135)^{1/3} \]
- The radius of the calcium nucleus \( R_{Ca} \) is given by:
\[ R_{Ca} = R_0 (40)^{1/3} \]

3. **Calculate the ratio of the nuclear radii**:
- The ratio \( \frac{R_{Cs}}{R_{Ca}} \) can be expressed as:
\[ \frac{R_{Cs}}{R_{Ca}} = \frac{R_0 (135)^{1/3}}{R_0 (40)^{1/3}} \]
- The \( R_0 \) cancels out:
\[ \frac{R_{Cs}}{R_{Ca}} = \frac{(135)^{1/3}}{(40)^{1/3}} = \left(\frac{135}{40}\right)^{1/3} \]

4. **Simplify the fraction**:
- Calculate \( \frac{135}{40} \):
\[ \frac{135}{40} = 3.375 \]

5. **Take the cube root**:
- Now, find the cube root of \( 3.375 \):
\[ \left(3.375\right)^{1/3} \approx 1.50 \]

### Final Answer:
The ratio of the nuclear radii of cesium-135 to calcium-40 is approximately \( 1.50 \).