A hydrogen atom (ionisation potential `13.6 eV`) makes a transition from third excited state to first excied state. The enegry of the photon emitted in the process is

Options

A. `1.89 eV`
B. `2.55 eV`
C. `12.09eV`
D. `12.75 eV`

Explanation

To solve the problem of finding the energy of the photon emitted when a hydrogen atom transitions from the third excited state to the first excited state, we can follow these steps:

### Step 1: Identify the energy levels
The ionization potential of hydrogen is given as 13.6 eV. The energy levels of a hydrogen atom can be calculated using the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Determine the principal quantum numbers
- The third excited state corresponds to \( n = 4 \) (since the ground state is \( n = 1 \), the first excited state is \( n = 2 \), the second excited state is \( n = 3 \), and the third excited state is \( n = 4 \)).
- The first excited state corresponds to \( n = 2 \).

### Step 3: Calculate the energies of the two states
Using the formula for the energy levels:
- For \( n = 4 \):
\[ E_4 = -\frac{13.6 \, \text{eV}}{4^2} = -\frac{13.6 \, \text{eV}}{16} = -0.85 \, \text{eV} \]
- For \( n = 2 \):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

### Step 4: Calculate the energy difference
The energy of the photon emitted during the transition from \( n = 4 \) to \( n = 2 \) is given by the difference in energy between these two states:
\[ E_{\text{photon}} = E_2 - E_4 \]
Substituting the values we calculated:
\[ E_{\text{photon}} = (-3.4 \, \text{eV}) - (-0.85 \, \text{eV}) = -3.4 \, \text{eV} + 0.85 \, \text{eV} = -2.55 \, \text{eV} \]
Since we are interested in the magnitude of the energy emitted, we take the absolute value:
\[ E_{\text{photon}} = 2.55 \, \text{eV} \]

### Step 5: Conclusion
The energy of the photon emitted when the hydrogen atom transitions from the third excited state to the first excited state is:
\[ \boxed{2.55 \, \text{eV}} \]