Question

Which of the following figure represents the variation of \( \ln \left( \frac{R}{R_0} \right) \) with \( \ln A \) (If \( R \) is the radius of a nucleus and \( A \) is its mass number)?

• A.
• B.
• C.
• D.

Hint:

The radius of a nucleus is related to the mass number by the formula \( R = R_0 A^{1/3} \). The logarithmic relationship results in a straight line when plotted as \( \ln \left( \frac{R}{R_0} \right) \) versus \( \ln A \).

Answer 1

The Correct Option is B

The relationship between the radius \( R \) of a nucleus and its mass number \( A \) is given by the empirical formula: \[ R = R_0 A^{1/3}, \] where \( R_0 \) is a constant. Taking the natural logarithm of both sides, we get: \[ \ln \left( \frac{R}{R_0} \right) = \frac{1}{3} \ln A. \] This shows that the graph of \( \ln \left( \frac{R}{R_0} \right) \) versus \( \ln A \) is a straight line with a slope of \( \frac{1}{3} \). Thus, the correct option is the one that shows a straight line.