The radius of the nucleus with nucleon number 2 is `1.5 xx 10^(-15)`m, then the radius of nucleus with nucleon number 54 will be

Options

A. `3xx 10^(-15) m`
B. `4.5 xx 10^(-15)m`
C. `6 xx 10^(-15) m`
D. `9.5 xx 10^(-15)m`

Answer

To find the radius of a nucleus with nucleon number 54, given that the radius of a nucleus with nucleon number 2 is \(1.5 \times 10^{-15}\) m, we can use the formula that relates the radius of a nucleus to its nucleon number:

### Step 1: Understand the formula for nuclear radius
The radius \(R\) of a nucleus can be expressed as:
\[ R = R_0 A^{1/3} \]
where \(R_0\) is a constant (the Fermi constant) and \(A\) is the nucleon number.

### Step 2: Set up the equation for nucleon number 2
For the nucleus with nucleon number 2:
\[ R = 1.5 \times 10^{-15} \text{ m} \]
\[ A = 2 \]
Substituting into the formula:
\[ 1.5 \times 10^{-15} = R_0 \cdot 2^{1/3} \]

### Step 3: Solve for \(R_0\)
To find \(R_0\), we rearrange the equation:
\[ R_0 = \frac{1.5 \times 10^{-15}}{2^{1/3}} \]

### Step 4: Set up the equation for nucleon number 54
Now, we need to find the radius \(R\) for the nucleus with nucleon number 54:
\[ A = 54 \]
Using the formula:
\[ R = R_0 \cdot 54^{1/3} \]

### Step 5: Substitute \(R_0\) into the equation for nucleon number 54
Substituting \(R_0\) from step 3 into this equation:
\[ R = \left(\frac{1.5 \times 10^{-15}}{2^{1/3}}\right) \cdot 54^{1/3} \]

### Step 6: Simplify the expression
Now we can simplify:
\[ R = 1.5 \times 10^{-15} \cdot \frac{54^{1/3}}{2^{1/3}} = 1.5 \times 10^{-15} \cdot \left(\frac{54}{2}\right)^{1/3} = 1.5 \times 10^{-15} \cdot 27^{1/3} \]
Since \(27^{1/3} = 3\):
\[ R = 1.5 \times 10^{-15} \cdot 3 \]

### Step 7: Calculate the final radius
Calculating this gives:
\[ R = 4.5 \times 10^{-15} \text{ m} \]

### Final Answer
The radius of the nucleus with nucleon number 54 is:
\[ R = 4.5 \times 10^{-15} \text{ m} \]

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