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

Options

A. From n= 1 to n = 2
B. From n = 2 to n = 3
C. From `n =oo` to n =1
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 use the formula for the energy of an electron transition:

\[ E = -13.6 \, \text{eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

Where:
- \( E \) is the energy of the transition,
- \( n_1 \) is the principal quantum number of the initial state,
- \( n_2 \) is the principal quantum number of the final state (with \( n_2 > n_1 \)),
- \( 13.6 \, \text{eV} \) is a constant for hydrogen.

### Step-by-Step Solution:

1. **Identify the transitions**: We need to evaluate the energy for each of the given transitions based on the principal quantum numbers \( n_1 \) and \( n_2 \).

2. **Calculate the energy for each transition**:
- **Option 1**: \( n_1 = 1, n_2 = 2 \)
\[ E_1 = -13.6 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = -13.6 \left( 1 - \frac{1}{4} \right) = -13.6 \left( \frac{3}{4} \right) = -10.2 \, \text{eV} \]

- **Option 2**: \( n_1 = 2, n_2 = 3 \)
\[ E_2 = -13.6 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = -13.6 \left( \frac{1}{4} - \frac{1}{9} \right) = -13.6 \left( \frac{9 - 4}{36} \right) = -13.6 \left( \frac{5}{36} \right) \approx -1.89 \, \text{eV} \]

- **Option 3**: \( n_1 = 1, n_2 = \infty \)
\[ E_3 = -13.6 \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = -13.6 \left( 1 - 0 \right) = -13.6 \, \text{eV} \]

- **Option 4**: \( n_1 = 3, n_2 = 5 \)
\[ E_4 = -13.6 \left( \frac{1}{3^2} - \frac{1}{5^2} \right) = -13.6 \left( \frac{1}{9} - \frac{1}{25} \right) = -13.6 \left( \frac{25 - 9}{225} \right) = -13.6 \left( \frac{16}{225} \right) \approx -0.96 \, \text{eV} \]

3. **Compare the energies**: Now we compare the absolute values of the energies calculated:
- \( |E_1| = 10.2 \, \text{eV} \)
- \( |E_2| \approx 1.89 \, \text{eV} \)
- \( |E_3| = 13.6 \, \text{eV} \)
- \( |E_4| \approx 0.96 \, \text{eV} \)

4. **Determine the largest energy transition**: The transition with the largest energy change (absolute value) is from \( n=1 \) to \( n=\infty \) (Option 3), which requires 13.6 eV.

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