Which of the followin g orbits of hydrogen atom should have the value of their radius in the radius `1:4`?

Options

A. K and L
B. L and N
C. M and N
D. A and B are correct

Answer

To solve the question of which orbits of the hydrogen atom have a radius ratio of 1:4, we can use Bohr's model of the hydrogen atom. Here's a step-by-step solution:

### Step 1: Understand the Bohr Model
According to Bohr's model, the radius of the nth orbit of a hydrogen atom is given by the formula:
\[ r_n \propto n^2 \]
This means that the radius of the orbit is directly proportional to the square of the principal quantum number (n).

### Step 2: Set Up the Ratio
We need to find two orbits (n1 and n2) such that:
\[ \frac{r_{n1}}{r_{n2}} = \frac{1}{4} \]
Using the relationship from Bohr's model, we can express this as:
\[ \frac{r_{n1}}{r_{n2}} = \frac{n_1^2}{n_2^2} \]
Thus, we have:
\[ \frac{n_1^2}{n_2^2} = \frac{1}{4} \]

### Step 3: Solve for the Ratio of n
Taking the square root of both sides gives us:
\[ \frac{n_1}{n_2} = \frac{1}{2} \]
This means that n1 is half of n2:
\[ n_1 = \frac{1}{2} n_2 \]

### Step 4: Identify Possible Values
Now, we can assign values to n1 and n2 based on the ratio we found. If we let:
- \( n_2 = 2 \), then \( n_1 = 1 \)
- \( n_2 = 4 \), then \( n_1 = 2 \)

### Step 5: Check the Options
Now we can check the options given in the question:
- **Option A:** (K, L) corresponds to (1, 2) → Ratio 1:2 (valid)
- **Option B:** (L, N) corresponds to (2, 4) → Ratio 1:2 (valid)
- **Option C:** (M, N) corresponds to (3, 4) → Ratio 3:4 (invalid)
- **Option D:** (K, N) corresponds to (1, 4) → Ratio 1:4 (valid)

### Conclusion
The orbits of the hydrogen atom that have the value of their radius in the ratio 1:4 are:
- Option D: (K, N)