Which electronic transition in a hydrogen atom releases the greatest amount of energy?

Options

A. n = 1 to n = 2
B. n = 2 to n = 4
C. n = 3 to n = 6
D. `n = oo` to n = 1

Explanation

To determine which electronic transition in a hydrogen atom releases the greatest amount of energy, we can follow these steps:

### Step 1: Understand the Energy Levels
In a hydrogen atom, the energy levels are quantized and can be described by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( E_n \) is the energy of the level, and \( n \) is the principal quantum number (n = 1, 2, 3,...).

### Step 2: Energy and Quantum Number Relationship
From the formula, we can see that energy is inversely proportional to the square of the principal quantum number \( n \). This means:
- As \( n \) increases, the energy becomes less negative (increases).
- The lower the value of \( n \), the more negative the energy, indicating a higher energy state when transitioning to a lower level.

### Step 3: Identify the Transition with Maximum Energy Release
To find the transition that releases the greatest amount of energy, we need to look for the transition from a higher energy level (higher \( n \)) to a lower energy level (lower \( n \)). The transition that releases the most energy occurs when an electron falls from the highest energy level (infinity) to the lowest energy level (n = 1).

### Step 4: Calculate the Energy for the Transition
The transition from \( n = \infty \) to \( n = 1 \) can be calculated as follows:
- Energy at \( n = \infty \):
\[ E_{\infty} = 0 \, \text{eV} \]
- Energy at \( n = 1 \):
\[ E_1 = -13.6 \, \text{eV} \]

The energy released during the transition from \( n = \infty \) to \( n = 1 \) is:
\[ \Delta E = E_1 - E_{\infty} = -13.6 \, \text{eV} - 0 \, \text{eV} = -13.6 \, \text{eV} \]
This indicates that 13.6 eV of energy is released.

### Conclusion
The electronic transition in a hydrogen atom that releases the greatest amount of energy is from \( n = \infty \) to \( n = 1 \).

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