Compare the radii of two nuclei with mass number 27 and 125 respectively.

Options

A. 2/3
B. 3/2
C. 3/5
D. 5/3

Explanation

To compare the radii of two nuclei with mass numbers 27 and 125, we can use the formula for the radius of a nucleus:

### Step 1: Understand the formula for nuclear radius
The radius \( R \) of a nucleus is given by the formula:
\[ R = R_0 A^{1/3} \]
where:
- \( R_0 \) is a constant (approximately \( 1.2 \, \text{fm} \) or \( 1.2 \times 10^{-15} \, \text{m} \)),
- \( A \) is the mass number of the nucleus.

### Step 2: Set up the comparison of the two nuclei
Let \( A_1 = 27 \) (mass number of the first nucleus) and \( A_2 = 125 \) (mass number of the second nucleus). We want to find the ratio of their radii \( R_1 \) and \( R_2 \).

### Step 3: Write the expressions for the radii
Using the formula for nuclear radius:
\[ R_1 = R_0 A_1^{1/3} = R_0 (27)^{1/3} \]
\[ R_2 = R_0 A_2^{1/3} = R_0 (125)^{1/3} \]

### Step 4: Find the ratio of the radii
Now, we can find the ratio of the two radii:
\[ \frac{R_1}{R_2} = \frac{R_0 (27)^{1/3}}{R_0 (125)^{1/3}} \]
The \( R_0 \) cancels out:
\[ \frac{R_1}{R_2} = \frac{(27)^{1/3}}{(125)^{1/3}} = \left(\frac{27}{125}\right)^{1/3} \]

### Step 5: Simplify the ratio
Now we can simplify \( \frac{27}{125} \):
\[ \frac{27}{125} = \frac{3^3}{5^3} = \left(\frac{3}{5}\right)^3 \]
Thus,
\[ \frac{R_1}{R_2} = \left(\frac{3}{5}\right)^{1} = \frac{3}{5} \]

### Step 6: Conclusion
From the ratio \( \frac{R_1}{R_2} = \frac{3}{5} \), we can conclude that the radius of the nucleus with mass number 27 is smaller than that of the nucleus with mass number 125.