Which of the following electron transitions in a hydrogen atom will require the largest amount of energy?

Options

A. (a) from `n=1` to `n=2`
B. (b) `"from" n=2 " to " n=4`
C. (c) `"from" n=5 " to " n=1`
D. (d) `"from" n=3 " to " n=5`

Answer

To determine which electron transition in a hydrogen atom requires the largest amount of energy, we can follow these steps:

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

### Step 2: Calculate the Change in Energy for Each Transition
The change in energy (\( \Delta E \)) when an electron transitions from an initial state \( n_1 \) to a final state \( n_2 \) is given by:
\[ \Delta E = E_{n_1} - E_{n_2} = -13.6 \, \text{eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

### Step 3: Evaluate Each Option
Let’s evaluate the energy change for each transition provided in the options:

1. **Transition from \( n_1 = 1 \) to \( n_2 = 2 \)**:
\[ \Delta E = -13.6 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = -13.6 \left( 1 - \frac{1}{4} \right) = -13.6 \left( \frac{3}{4} \right) = -10.2 \, \text{eV} \]

2. **Transition from \( n_1 = 2 \) to \( n_2 = 4 \)**:
\[ \Delta E = -13.6 \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = -13.6 \left( \frac{1}{4} - \frac{1}{16} \right) = -13.6 \left( \frac{4 - 1}{16} \right) = -13.6 \left( \frac{3}{16} \right) = -2.55 \, \text{eV} \]

3. **Transition from \( n_1 = 5 \) to \( n_2 = 1 \)**:
\[ \Delta E = -13.6 \left( \frac{1}{5^2} - \frac{1}{1^2} \right) = -13.6 \left( \frac{1}{25} - 1 \right) = -13.6 \left( \frac{1 - 25}{25} \right) = -13.6 \left( -\frac{24}{25} \right) = 13.056 \, \text{eV} \]

4. **Transition from \( n_1 = 3 \) to \( n_2 = 5 \)**:
\[ \Delta E = -13.6 \left( \frac{1}{3^2} - \frac{1}{5^2} \right) = -13.6 \left( \frac{1}{9} - \frac{1}{25} \right) = -13.6 \left( \frac{25 - 9}{225} \right) = -13.6 \left( \frac{16}{225} \right) = -0.96 \, \text{eV} \]

### Step 4: Compare the Energies
Now we compare the absolute values of the energy changes:
- Transition 1: \( 10.2 \, \text{eV} \)
- Transition 2: \( 2.55 \, \text{eV} \)
- Transition 3: \( 13.056 \, \text{eV} \)
- Transition 4: \( 0.96 \, \text{eV} \)

### Conclusion
The transition from \( n_1 = 5 \) to \( n_2 = 1 \) requires the largest amount of energy, which is \( 13.056 \, \text{eV} \).

**Final Answer: Transition from \( n = 5 \) to \( n = 1 \) requires the largest amount of energy.**

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