Question

When an electron beam of energy 10.2 eV is used to excite hydrogen gas, then the possible spectral line is:

• A. First Balmer line
• B. First Lyman line
• C. Second Balmer line
• D. Second Lyman line

Hint:

Energy 10.2 eV corresponds to transition from \(n=2\) to \(n=1\) (first Lyman line) in hydrogen spectrum.

Answer 1

The Correct Option is B

The energy levels of hydrogen atom are given by:
\[ E_n = -13.6 \, \text{eV} \times \frac{1}{n^2} \]

The energy difference between levels corresponds to the energy of emitted or absorbed photons.

- The first excited state energy (from \( n=1 \) to \( n=2 \)) is:
\[ \Delta E = E_2 - E_1 = -13.6 \times \frac{1}{2^2} - \left(-13.6 \times 1\right) = -3.4 + 13.6 = 10.2 \, \text{eV} \]

This energy corresponds exactly to 10.2 eV, which is the energy of the first Lyman line (transition from \( n=2 \to n=1 \)) in the ultraviolet region.

- Balmer lines correspond to transitions ending at \( n=2 \) level and have lower energies (< 10.2 eV).

Hence, excitation by 10.2 eV electrons causes the hydrogen atom to emit the first Lyman line.