An electron is at ground state of the H atom. The minimum energy required to excite the H atom into the second excited state is

Options

A. 13.6 eV
B. 12.1 eV
C. 10.2 eV
D. 3.4 eV

Answer

To find the minimum energy required to excite a hydrogen atom from its ground state to the second excited state, we can follow these steps:

### Step 1: Identify the energy levels
The energy of an electron in a hydrogen atom at a given energy level \( n \) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Calculate the energy of the ground state (n=1)
For the ground state (where \( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Calculate the energy of the second excited state (n=3)
The second excited state corresponds to \( n = 3 \):
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV} \]

### Step 4: Calculate the energy difference (ΔE)
The energy required to excite the electron from the ground state to the second excited state is the difference in energy between these two states:
\[ \Delta E = E_3 - E_1 \]
Substituting the values we calculated:
\[ \Delta E = -1.51 \, \text{eV} - (-13.6 \, \text{eV}) = -1.51 \, \text{eV} + 13.6 \, \text{eV} = 12.1 \, \text{eV} \]

### Conclusion
The minimum energy required to excite the hydrogen atom from the ground state to the second excited state is:
\[ \Delta E = 12.1 \, \text{eV} \]

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