The energy (in eV) required to excite an electron from `n = 2` to `n = 4` state in hydrogen atom is

Options

A. `-0.85`
B. `+4.25`
C. `-3.4`
D. `+2.55`

Answer

To find the energy required to excite an electron from the \( n = 2 \) state to the \( n = 4 \) state in a hydrogen atom, we can follow these steps:

### Step 1: Understand the Energy Levels
The energy of an electron in the \( n \)th energy level of a hydrogen atom is given by the formula:
\[ E_n = -\frac{E_0 Z^2}{n^2} \]
where:
- \( E_0 = 13.6 \, \text{eV} \) (the energy of the ground state for hydrogen),
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( n \) is the principal quantum number.

### Step 2: Calculate Energy at \( n = 2 \)
For \( n = 2 \):
\[ E_2 = -\frac{13.6 \times 1^2}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]

### Step 3: Calculate Energy at \( n = 4 \)
For \( n = 4 \):
\[ E_4 = -\frac{13.6 \times 1^2}{4^2} = -\frac{13.6}{16} = -0.85 \, \text{eV} \]

### Step 4: Calculate the Energy Difference
The energy required to excite the electron from \( n = 2 \) to \( n = 4 \) is the difference in energy between these two states:
\[ \Delta E = E_4 - E_2 \]
Substituting the values we calculated:
\[ \Delta E = (-0.85) - (-3.4) = -0.85 + 3.4 = 2.55 \, \text{eV} \]

### Conclusion
The energy required to excite the electron from \( n = 2 \) to \( n = 4 \) is \( 2.55 \, \text{eV} \).

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