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 \(^{135}\text{Cs}\) (Cesium) to \(^{40}\text{Ca}\) (Calcium), we can use the formula for the radius of a nucleus, which is given by:

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

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

### Step 1: Write the formula for the nuclear radii of both isotopes

For \(^{135}\text{Cs}\):
\[ R_{Cs} = R_0 \cdot (135)^{1/3} \]

For \(^{40}\text{Ca}\):
\[ R_{Ca} = R_0 \cdot (40)^{1/3} \]

### Step 2: Set up the ratio of the nuclear radii

Now, we can set up the ratio of the nuclear radii \(R_{Cs}\) to \(R_{Ca}\):

\[ \frac{R_{Cs}}{R_{Ca}} = \frac{R_0 \cdot (135)^{1/3}}{R_0 \cdot (40)^{1/3}} \]

### Step 3: Simplify the ratio

The \(R_0\) cancels out:

\[ \frac{R_{Cs}}{R_{Ca}} = \frac{(135)^{1/3}}{(40)^{1/3}} = \left(\frac{135}{40}\right)^{1/3} \]

### Step 4: Calculate the ratio

Now we need to calculate \(\frac{135}{40}\):

\[ \frac{135}{40} = 3.375 \]

Now take the cube root:

\[ \frac{R_{Cs}}{R_{Ca}} = (3.375)^{1/3} \]

Calculating the cube root:

\[ (3.375)^{1/3} \approx 1.5 \]

### Final Answer

Thus, the ratio of the nuclear radii of \(^{135}\text{Cs}\) to \(^{40}\text{Ca}\) is:

\[ \frac{R_{Cs}}{R_{Ca}} \approx 1.5 \]