If radius of second shell of `Li^(+2)` ion is `R` , then the radius of its
third shell will be `:-`

Options

A. `(9R)/(4)`
B. `(4R)/(9)`
C. `(3R)/(2)`
D. `(2R)/(3)`

Explanation

To find the radius of the third shell of the `Li^(+2)` ion given that the radius of the second shell is `R`, we can follow these steps:

### Step 1: Understand the Formula for Shell Radius
The radius of the nth shell in a hydrogen-like atom is given by the formula:
\[ R_n = \frac{0.529 \, n^2}{Z} \]
where:
- \( R_n \) is the radius of the nth shell,
- \( n \) is the principal quantum number (shell number),
- \( Z \) is the atomic number of the ion.

### Step 2: Determine the Atomic Number
For the `Li^(+2)` ion, lithium (Li) has an atomic number \( Z = 3 \).

### Step 3: Calculate the Radius of the Second Shell
Using the formula for the second shell (\( n = 2 \)):
\[ R_2 = \frac{0.529 \, (2^2)}{Z} = \frac{0.529 \, \cdot 4}{3} \]
Given that \( R_2 = R \), we can express this as:
\[ R = \frac{2.116}{3} \]

### Step 4: Calculate the Radius of the Third Shell
Now, we calculate the radius of the third shell (\( n = 3 \)):
\[ R_3 = \frac{0.529 \, (3^2)}{Z} = \frac{0.529 \cdot 9}{3} \]
\[ R_3 = \frac{4.761}{3} \]

### Step 5: Relate \( R_3 \) to \( R_2 \)
To find the relationship between \( R_3 \) and \( R_2 \):
\[ \frac{R_3}{R_2} = \frac{0.529 \cdot 9 / 3}{0.529 \cdot 4 / 3} \]
The \( 0.529 \) and \( 3 \) cancel out:
\[ \frac{R_3}{R_2} = \frac{9}{4} \]
Thus, we can express \( R_3 \) in terms of \( R \):
\[ R_3 = \frac{9}{4} R \]

### Final Answer
The radius of the third shell of the `Li^(+2)` ion is:
\[ R_3 = \frac{9}{4} R \]

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