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Maximum energy is evolved during which o...

Maximum energy is evolved during which of the following transitions ?

A

`n=1" to "n=2`

B

`n=2" to "n=6`

C

`n=2" to "n=1`

D

`n=6" to "n=2`

Text Solution

AI Generated Solution

The correct Answer is:
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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