If the radii of nuclei of `._(13)Al^(27) and ._(30)Zn^(64)` are `R_(1) and R_(2)` respectively, then `(R_(1))/(R_(2))=`

Options

A. `(64)/(27)`
B. `(3)/(4)`
C. `(27)/(64)`
D. `(4)/(3)`

Answer

To solve the problem of finding the ratio of the radii of the nuclei of Aluminum and Zinc, we can follow these steps:

### Step 1: Understand the relationship between volume and mass number
The volume of a nucleus is directly proportional to its mass number (A). This means:
\[ V \propto A \]

### Step 2: Express the volume of a spherical nucleus
The volume (V) of a spherical nucleus can be expressed as:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the nucleus.

### Step 3: Relate volume to mass number
From the above, we can say:
\[ \frac{4}{3} \pi R^3 \propto A \]
Since \( \frac{4}{3} \pi \) is a constant, we can simplify this to:
\[ R^3 \propto A \]

### Step 4: Establish the relationship between radius and mass number
From the proportionality, we can express the radius in terms of mass number:
\[ R \propto A^{1/3} \]

### Step 5: Write the expressions for the radii of Aluminum and Zinc
Let \( R_1 \) be the radius of Aluminum-27 and \( R_2 \) be the radius of Zinc-64. We can write:
\[ R_1 \propto A_1^{1/3} \]
\[ R_2 \propto A_2^{1/3} \]
where \( A_1 = 27 \) (mass number of Aluminum) and \( A_2 = 64 \) (mass number of Zinc).

### Step 6: Formulate the ratio of the radii
Now, we can write the ratio of the radii:
\[ \frac{R_1}{R_2} = \frac{A_1^{1/3}}{A_2^{1/3}} \]

### Step 7: Substitute the mass numbers
Substituting the values of \( A_1 \) and \( A_2 \):
\[ \frac{R_1}{R_2} = \frac{27^{1/3}}{64^{1/3}} \]

### Step 8: Simplify the expression
We can simplify this further:
\[ \frac{R_1}{R_2} = \left(\frac{27}{64}\right)^{1/3} \]

### Step 9: Calculate the cube root
Calculating the cube root:
\[ \frac{27}{64} = \left(\frac{3}{4}\right)^3 \]
Thus:
\[ \left(\frac{27}{64}\right)^{1/3} = \frac{3}{4} \]

### Final Answer
Therefore, the ratio of the radii of the nuclei of Aluminum and Zinc is:
\[ \frac{R_1}{R_2} = \frac{3}{4} \]