Compare the radii of two nuclei with mass number 1 and 125 respectively.

Options

A. 5/1
B. 1/5
C. 2/5
D. 3/5

Answer

To compare the radii of two nuclei with mass numbers 1 and 125, we can use the formula for the radius of a nucleus:

### Step 1: Understand the formula for nuclear radius
The radius of a nucleus (r) is given by the formula:
\[ r = r_0 A^{1/3} \]
where:
- \( r_0 \) is a constant (approximately \( 1.2 \, \text{fm} \) for nuclear radii),
- \( A \) is the mass number of the nucleus.

### Step 2: Define the mass numbers
Let:
- \( A_1 = 1 \) (for the first nucleus),
- \( A_2 = 125 \) (for the second nucleus).

### Step 3: Write the radii for both nuclei
Using the formula:
- For the first nucleus:
\[ r_1 = r_0 A_1^{1/3} = r_0 (1)^{1/3} = r_0 \]
- For the second nucleus:
\[ r_2 = r_0 A_2^{1/3} = r_0 (125)^{1/3} \]

### Step 4: Calculate \( (125)^{1/3} \)
To simplify \( (125)^{1/3} \):
\[ (125)^{1/3} = 5 \]
Thus,
\[ r_2 = r_0 \times 5 \]

### Step 5: Compare the two radii
Now we can compare \( r_1 \) and \( r_2 \):
- \( r_1 = r_0 \)
- \( r_2 = 5r_0 \)

### Step 6: Ratio of the radii
To find the ratio of the two radii:
\[ \frac{r_1}{r_2} = \frac{r_0}{5r_0} = \frac{1}{5} \]

### Conclusion
The radius of the nucleus with mass number 1 is \( \frac{1}{5} \) the radius of the nucleus with mass number 125.

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