The graph of `1n (R/R_0)` versus `1n A (R = radius` of a nucleus and `A =` its mass number) is

Options

A. a straight line
B. a parabola
C. and ellipse
D. a circle

Answer

To solve the question regarding the graph of \( \ln \left( \frac{R}{R_0} \right) \) versus \( \ln A \) (where \( R \) is the radius of a nucleus and \( A \) is its mass number), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We start with the known relationship between the radius of a nucleus and its mass number. The radius \( R \) of a nucleus is proportional to the cube root of its mass number \( A \):
\[ R \propto A^{1/3} \]

2. **Expressing the Relationship Mathematically**:
We can express this proportionality as:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant.

3. **Finding the Ratio**:
Next, we find the ratio \( \frac{R}{R_0} \):
\[ \frac{R}{R_0} = A^{1/3} \]

4. **Taking the Natural Logarithm**:
We take the natural logarithm of both sides:
\[ \ln \left( \frac{R}{R_0} \right) = \ln \left( A^{1/3} \right) \]

5. **Using Logarithmic Properties**:
Using the property of logarithms, we can simplify the right side:
\[ \ln \left( \frac{R}{R_0} \right) = \frac{1}{3} \ln A \]

6. **Rearranging the Equation**:
We can rearrange this equation to express \( \ln \left( \frac{R}{R_0} \right) \) in terms of \( \ln A \):
\[ \ln \left( \frac{R}{R_0} \right) = \frac{1}{3} \ln A \]

7. **Identifying the Graph**:
This equation shows that \( \ln \left( \frac{R}{R_0} \right) \) is directly proportional to \( \ln A \) with a slope of \( \frac{1}{3} \). Thus, when we plot \( \ln \left( \frac{R}{R_0} \right) \) on the y-axis against \( \ln A \) on the x-axis, we will get a straight line passing through the origin.

### Conclusion:
The graph of \( \ln \left( \frac{R}{R_0} \right) \) versus \( \ln A \) is a straight line passing through the origin.