In a certain electronic transition in the hydrogen atoms from an initial state `(1)` to a final state `(2)`, the difference in the orbit radius `((r_(1)-r_(2))` is 24 times the first Bohr radius. Identify the transition-

Options

A. `5rarr1`
B. `25rarr1`
C. `8rarr3`
D. `7rarr5`

Explanation

To solve the problem of identifying the electronic transition in a hydrogen atom based on the given difference in orbit radius, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to identify the transition from an initial state (n1) to a final state (n2) in a hydrogen atom, given that the difference in orbit radius (r1 - r2) is 24 times the first Bohr radius (r0).

2. **Recall the Formula for Bohr Radius**: The radius of the nth orbit in a hydrogen atom is given by the formula:
\[ r_n = n^2 \cdot r_0 \]
where \( r_0 \) is the first Bohr radius.

3. **Express the Difference in Radii**: The difference in the radii can be expressed as:
\[ r_1 - r_2 = r_{n1} - r_{n2} = (n1^2 \cdot r_0) - (n2^2 \cdot r_0) = r_0(n1^2 - n2^2) \]

4. **Set Up the Equation**: From the problem, we know:
\[ r_1 - r_2 = 24 \cdot r_0 \]
Therefore, we can set up the equation:
\[ r_0(n1^2 - n2^2) = 24 \cdot r_0 \]
Dividing both sides by \( r_0 \) (assuming \( r_0 \neq 0 \)):
\[ n1^2 - n2^2 = 24 \]

5. **Factor the Difference of Squares**: The equation \( n1^2 - n2^2 = 24 \) can be factored as:
\[ (n1 - n2)(n1 + n2) = 24 \]

6. **Identify Possible Integer Solutions**: We need to find pairs of integers (n1, n2) such that their product equals 24. The pairs of factors of 24 are:
- (1, 24)
- (2, 12)
- (3, 8)
- (4, 6)

We can use these pairs to find possible values for \( n1 \) and \( n2 \).

7. **Test Possible Values**: We will test pairs to see if they yield valid quantum numbers:
- For \( n1 - n2 = 4 \) and \( n1 + n2 = 6 \):
- Solving these gives \( n1 = 5 \) and \( n2 = 1 \).
- For \( n1 - n2 = 6 \) and \( n1 + n2 = 4 \):
- This does not yield valid quantum numbers since \( n1 \) cannot be less than \( n2 \).

- For \( n1 - n2 = 8 \) and \( n1 + n2 = 3 \):
- This also does not yield valid quantum numbers.

- For \( n1 - n2 = 12 \) and \( n1 + n2 = 2 \):
- This does not yield valid quantum numbers.

- For \( n1 - n2 = 3 \) and \( n1 + n2 = 8 \):
- Solving these gives \( n1 = 5 \) and \( n2 = 7 \).

8. **Conclusion**: The possible transitions are:
- From \( n1 = 5 \) to \( n2 = 1 \)
- From \( n1 = 7 \) to \( n2 = 5 \)

Thus, the identified transitions are \( n = 5 \to n = 1 \) and \( n = 7 \to n = 5 \).