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 difference in orbit radii, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Radius and Principal Quantum Number**:
The radius of the electron 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 and \( n \) is the principal quantum number.

2. **Set Up the Equation for the Difference in Radii**:
We need to find the difference in radii between two states \( n_1 \) and \( n_2 \):
\[ r_{n_2} - r_{n_1} = r_0(n_2^2 - n_1^2) \]
According to the problem, this difference is equal to \( 24 \cdot r_0 \):
\[ r_0(n_2^2 - n_1^2) = 24 \cdot r_0 \]

3. **Simplify the Equation**:
We can divide both sides by \( r_0 \) (since \( r_0 \neq 0 \)):
\[ n_2^2 - n_1^2 = 24 \]

4. **Factor the Difference of Squares**:
The equation \( n_2^2 - n_1^2 = 24 \) can be factored as:
\[ (n_2 - n_1)(n_2 + n_1) = 24 \]

5. **Identify Possible Values for \( n_1 \) and \( n_2 \)**:
We need to find pairs of integers \( (n_1, n_2) \) such that their product equals 24. The factor pairs of 24 are:
- \( (1, 24) \)
- \( (2, 12) \)
- \( (3, 8) \)
- \( (4, 6) \)

We can set \( n_2 - n_1 = a \) and \( n_2 + n_1 = b \), where \( a \cdot b = 24 \).

6. **Solve for Each Pair**:
For each factor pair \( (a, b) \):
- \( n_2 = \frac{a + b}{2} \)
- \( n_1 = \frac{b - a}{2} \)

Let's evaluate the pairs:
- For \( (4, 6) \):
- \( n_2 - n_1 = 4 \)
- \( n_2 + n_1 = 6 \)
- Solving gives \( n_2 = 5 \) and \( n_1 = 1 \).

- For \( (3, 8) \):
- \( n_2 - n_1 = 3 \)
- \( n_2 + n_1 = 8 \)
- Solving gives \( n_2 = 7 \) and \( n_1 = 5 \).

7. **Conclusion**:
The possible transitions are:
- From \( n_1 = 1 \) to \( n_2 = 5 \)
- From \( n_1 = 5 \) to \( n_2 = 7 \)

### Final Answer:
The transitions identified are \( (1 \to 5) \) and \( (5 \to 7) \).