What is the energy in eV requried to excite the electron from n=1 to n=2 state in hydrogen atom? (n=principal quantum number)

Options

A. 13.6
B. 3.4
C. 17
D. 10.2

Answer

To determine the energy required to excite an electron in a hydrogen atom from the n=1 state to the n=2 state, we can use the formula for the energy levels of the hydrogen atom:

### Step 1: Understand the energy level formula
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 \( E_n \) is the energy of the electron at level n, and 13.6 eV is the energy of the electron in the ground state (n=1).

### Step 2: Calculate the energy for n=1
For \( n_1 = 1 \):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Calculate the energy for n=2
For \( n_2 = 2 \):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

### Step 4: Calculate the energy difference
The energy required to excite the electron from n=1 to n=2 is the difference between the energies of these two states:
\[ \Delta E = E_2 - E_1 \]
Substituting the values we calculated:
\[ \Delta E = (-3.4 \, \text{eV}) - (-13.6 \, \text{eV}) \]
\[ \Delta E = -3.4 + 13.6 \]
\[ \Delta E = 10.2 \, \text{eV} \]

### Final Answer
The energy required to excite the electron from n=1 to n=2 state in a hydrogen atom is **10.2 eV**.

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