Maximum energy is evolved during which of the following transitions ?

Options

A. `n=1" to "n=2`
B. `n=2" to "n=6`
C. `n=2" to "n=1`
D. `n=6" to "n=2`

Answer

To determine which transition releases the maximum energy, we can follow these steps:

### Step 1: Understand the Energy Levels
The energy of an electron in a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

### Step 2: Analyze the Transitions
We need to analyze the transitions given in the options. The transitions typically involve moving from a higher energy level to a lower energy level, which releases energy.

### Step 3: Calculate Energy Differences
For each transition, we will calculate the energy difference using the formula:
\[ \Delta E = E_{\text{final}} - E_{\text{initial}} = -\frac{13.6 \, Z^2}{n_{\text{final}}^2} + \frac{13.6 \, Z^2}{n_{\text{initial}}^2} \]

### Step 4: Evaluate Each Transition
Let’s evaluate the transitions:

1. **Transition from n=2 to n=1**:
\[ \Delta E_{2 \to 1} = -\frac{13.6 \, Z^2}{1^2} + \frac{13.6 \, Z^2}{2^2} = -13.6 Z^2 + \frac{13.6 Z^2}{4} = -13.6 Z^2 \left(1 - \frac{1}{4}\right) = -13.6 Z^2 \cdot \frac{3}{4} = -10.2 Z^2 \, \text{eV} \]

2. **Transition from n=6 to n=2**:
\[ \Delta E_{6 \to 2} = -\frac{13.6 \, Z^2}{2^2} + \frac{13.6 \, Z^2}{6^2} = -\frac{13.6 Z^2}{4} + \frac{13.6 Z^2}{36} = -\frac{13.6 Z^2}{4} + \frac{13.6 Z^2}{36} \]
To combine these, we need a common denominator (36):
\[ = -\frac{13.6 Z^2 \cdot 9}{36} + \frac{13.6 Z^2}{36} = -\frac{13.6 Z^2 (9 - 1)}{36} = -\frac{13.6 Z^2 \cdot 8}{36} = -3.04 Z^2 \, \text{eV} \]

### Step 5: Compare the Energies
Now we can compare the energies released in both transitions:
- For \( n=2 \to n=1 \): \( -10.2 Z^2 \, \text{eV} \)
- For \( n=6 \to n=2 \): \( -3.04 Z^2 \, \text{eV} \)

### Step 6: Conclusion
The transition from \( n=2 \) to \( n=1 \) releases more energy than the transition from \( n=6 \) to \( n=2 \). Therefore, the maximum energy is evolved during the transition from **n=2 to n=1**.

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