Q164 / 180 · COMEDK 2024 (Morning Shift) · Physics
The ratio of the radii of the nucleus of two element $\mathrm{X}$ and $\mathrm{Y}$ having the mass numbers 232 and 29 is:
- A1 : 4
- B1 : 2
- C4 : 1
- Correct answerD2 : 1
Solution
The radius of a nucleus is given by the formula $R = R_{0} A^{1/3}$, where $R_{0}$ is a constant and $A$ is the mass number of the nucleus.
For the two elements X and Y, the ratio of their radii is given by $\dfrac{R_{X}}{R_{Y}} = \dfrac{R_{0} A_{X}^{1/3}}{R_{0} A_{Y}^{1/3}} = \left( \dfrac{A_{X}}{A_{Y}} \right)^{1/3}$.
Given $A_{X} = 232$ and $A_{Y} = 29$, we substitute these values into the ratio:
$\dfrac{R_{X}}{R_{Y}} = \left( \dfrac{232}{29} \right)^{1/3}$.
Since $232 = 29 \times 8$, we have $\dfrac{232}{29} = 8$.
Therefore, $\dfrac{R_{X}}{R_{Y}} = (8)^{1/3} = 2$.
The ratio is $2 : 1$.
Answer: 2 : 1
For the two elements X and Y, the ratio of their radii is given by $\dfrac{R_{X}}{R_{Y}} = \dfrac{R_{0} A_{X}^{1/3}}{R_{0} A_{Y}^{1/3}} = \left( \dfrac{A_{X}}{A_{Y}} \right)^{1/3}$.
Given $A_{X} = 232$ and $A_{Y} = 29$, we substitute these values into the ratio:
$\dfrac{R_{X}}{R_{Y}} = \left( \dfrac{232}{29} \right)^{1/3}$.
Since $232 = 29 \times 8$, we have $\dfrac{232}{29} = 8$.
Therefore, $\dfrac{R_{X}}{R_{Y}} = (8)^{1/3} = 2$.
The ratio is $2 : 1$.
Answer: 2 : 1
Hence, the correct answer is option: 'D'.