The ratio of the radius difference between `4^(th)` and `3^(rd)` orbit of H-atom and that of `Li^(2+)` ion is :

Options

A. (a) `1:1`
B. (b) `3:1`
C. (c) `3:4`
D. (d) `9:1`

Answer

To solve the problem of finding the ratio of the radius difference between the 4th and 3rd orbits of the hydrogen atom and that of the Li²⁺ ion, we can follow these steps:

### Step 1: Understand the formula for the radius of an orbit
According to Bohr's theory, the radius of an orbit (R) is given by the formula:
\[ R_n = k \frac{n^2}{Z} \]
where:
- \( R_n \) is the radius of the nth orbit,
- \( k \) is a constant (0.529 Å),
- \( n \) is the principal quantum number (orbit number),
- \( Z \) is the atomic number.

### Step 2: Calculate the radius for the hydrogen atom (Z = 1)
For the hydrogen atom (Z = 1):
- The radius of the 4th orbit (R₄) is:
\[ R_4 = k \frac{4^2}{1} = 16k \]
- The radius of the 3rd orbit (R₃) is:
\[ R_3 = k \frac{3^2}{1} = 9k \]

### Step 3: Find the difference in radius for hydrogen
The difference in radius between the 4th and 3rd orbits for hydrogen is:
\[ R_4 - R_3 = 16k - 9k = 7k \]

### Step 4: Calculate the radius for the Li²⁺ ion (Z = 3)
For the Li²⁺ ion (Z = 3):
- The radius of the 4th orbit (R₄) is:
\[ R_4 = k \frac{4^2}{3} = \frac{16k}{3} \]
- The radius of the 3rd orbit (R₃) is:
\[ R_3 = k \frac{3^2}{3} = 3k \]

### Step 5: Find the difference in radius for Li²⁺
The difference in radius between the 4th and 3rd orbits for Li²⁺ is:
\[ R_4 - R_3 = \frac{16k}{3} - 3k \]
To subtract these, convert 3k into a fraction:
\[ 3k = \frac{9k}{3} \]
So,
\[ R_4 - R_3 = \frac{16k}{3} - \frac{9k}{3} = \frac{7k}{3} \]

### Step 6: Calculate the ratio of the radius differences
Now we can find the ratio of the radius difference for hydrogen to that for Li²⁺:
\[ \text{Ratio} = \frac{R_4 - R_3 \text{ (H)}}{R_4 - R_3 \text{ (Li²⁺)}} = \frac{7k}{\frac{7k}{3}} \]
This simplifies to:
\[ \text{Ratio} = 7k \times \frac{3}{7k} = 3 \]

### Final Answer
The ratio of the radius difference between the 4th and 3rd orbits of the hydrogen atom to that of the Li²⁺ ion is:
\[ \text{Ratio} = 3 : 1 \]