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

### Step 2: Calculate the energy of the ground state (n=1)
For the ground state (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 \( n=3 \):
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} = -1.51 \, \text{eV} \]

### Step 4: Determine the energy difference
The minimum energy required to excite the electron from the ground state to the second excited state is the difference between the energies of these two states:
\[ \Delta E = E_3 - E_1 \]
Substituting the values we calculated:
\[ \Delta E = (-1.51 \, \text{eV}) - (-13.6 \, \text{eV}) = -1.51 \, \text{eV} + 13.6 \, \text{eV} = 12.09 \, \text{eV} \]

### Step 5: Conclusion
Thus, the minimum energy required to excite the hydrogen atom from the ground state to the second excited state is:
\[ \Delta E = 12.09 \, \text{eV} \]

### Final Answer:
The minimum energy required to excite the hydrogen atom into the second excited state is **12.09 eV**.

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