If the diameter of two different nuclei are in the ratio 2: 1, then calculate the ratio of their mass number

Answer

To solve the problem of finding the ratio of the mass numbers of two different nuclei given that their diameters are in the ratio of 2:1, we can follow these steps:

### Step 1: Understand the Relationship Between Diameter and Radius
The diameter of a nucleus is related to its radius. The relationship is given by:
\[ \text{Diameter} = 2 \times \text{Radius} \]
Thus, if the diameters of two nuclei are in the ratio of 2:1, we can express this as:
\[ \frac{D_1}{D_2} = \frac{2}{1} \]

### Step 2: Express the Diameter in Terms of Radius
Using the relationship between diameter and radius, we can rewrite the ratio of diameters in terms of radii:
\[ \frac{D_1}{D_2} = \frac{2R_1}{2R_2} = \frac{R_1}{R_2} \]
This simplifies to:
\[ \frac{R_1}{R_2} = \frac{2}{1} \]

### Step 3: Relate Radius to Mass Number
The radius of a nucleus can be expressed in terms of its mass number \( A \) using the formula:
\[ R = r_0 A^{1/3} \]
where \( r_0 \) is a constant (approximately \( 1.33 \times 10^{-15} \) m). Therefore, we can write:
\[ R_1 = r_0 A_1^{1/3} \quad \text{and} \quad R_2 = r_0 A_2^{1/3} \]

### Step 4: Substitute the Radius Expressions into the Ratio
Substituting these expressions into the ratio we derived earlier:
\[ \frac{R_1}{R_2} = \frac{r_0 A_1^{1/3}}{r_0 A_2^{1/3}} = \frac{A_1^{1/3}}{A_2^{1/3}} \]
Since \( r_0 \) cancels out, we have:
\[ \frac{A_1^{1/3}}{A_2^{1/3}} = \frac{2}{1} \]

### Step 5: Cube Both Sides to Find the Mass Number Ratio
Cubing both sides gives us:
\[ \left(\frac{A_1^{1/3}}{A_2^{1/3}}\right)^3 = \left(\frac{2}{1}\right)^3 \]
This simplifies to:
\[ \frac{A_1}{A_2} = 2^3 = 8 \]

### Conclusion
Thus, the ratio of the mass numbers of the two nuclei is:
\[ \frac{A_1}{A_2} = 8 \]

### Final Answer
The ratio of their mass numbers is \( 8:1 \).

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