Electrons having kinetic energy 30 eV are made to collide with atomic hydrogen gas (in ground state) and 42.5% of electron energy is used to excite the hydrogen wavelength in emission spectra are

Options

A. 3
B. 6
C. 12
D. 18

Answer

To solve the problem, we need to determine how many unique wavelengths can be emitted when electrons with a kinetic energy of 30 eV collide with atomic hydrogen gas, where 42.5% of the electron energy is used to excite the hydrogen atoms.

### Step-by-Step Solution:

1. **Calculate the Energy Used for Excitation**:
- Given that the kinetic energy of the electrons is 30 eV and 42.5% of this energy is used for excitation, we can calculate the energy used for excitation as follows:
\[ \text{Energy used for excitation} = 0.425 \times 30 \, \text{eV} = 12.75 \, \text{eV} \]

2. **Determine the Excited State**:
- The energy levels of hydrogen can be calculated using the formula:
\[ E_n = -\frac{13.6}{n^2} \, \text{eV} \]
- For the ground state (n=1):
\[ E_1 = -13.6 \, \text{eV} \]
- For the first excited state (n=2):
\[ E_2 = -3.4 \, \text{eV} \]
- For the second excited state (n=3):
\[ E_3 = -1.51 \, \text{eV} \]
- For the third excited state (n=4):
\[ E_4 = -0.85 \, \text{eV} \]

3. **Identify the Maximum Excited State**:
- The energy gained by the hydrogen atom is 12.75 eV. To find the maximum excited state, we need to find the difference in energy between the ground state and the excited state:
\[ E_4 - E_1 = (-0.85) - (-13.6) = 12.75 \, \text{eV} \]
- This indicates that the hydrogen atom can be excited to the fourth excited state (n=4).

4. **Calculate the Number of Unique Wavelengths**:
- The number of unique wavelengths emitted when an electron transitions between energy levels can be calculated using the formula:
\[ \text{Number of unique wavelengths} = \frac{n_2(n_2 - 1)}{2} \]
- Here, \(n_2\) is the uppermost excited state, which is 4:
\[ \text{Number of unique wavelengths} = \frac{4(4 - 1)}{2} = \frac{4 \times 3}{2} = 6 \]

5. **Conclusion**:
- Therefore, the total number of unique wavelengths emitted when the hydrogen atom is excited to the fourth state is **6**.