Question

When gaseous hydrogen at room temperature is bombarded with \(12.75\,\text{eV}\) electrons, the maximum possible number of spectral lines emitted is

• A. \(3\)
• B. \(4\)
• C. \(1\)
• D. \(6\)

Hint:

If an atom is excited up to the level \(n\), the maximum possible number of emission lines is \[ \boxed{ N=\frac{n(n-1)}{2}. } \] For hydrogen, \[ E_n=13.6\left(1-\frac1{n^2}\right)\text{ eV}. \]

Answer 1

The Correct Option is D

Step 1: Determine the highest excited energy level. The excitation energy of hydrogen is \[ E_n=13.6\left(1-\frac1{n^2}\right)\text{ eV}. \] For \[ n=4, \] \[ E_4 = 13.6\left(1-\frac1{16}\right) = 12.75\,\text{eV}. \] Thus, the incident electrons can excite hydrogen atoms up to the \[ \boxed{n=4} \] state.

Step 2: Find the maximum number of spectral lines. If the atom is excited up to the \(n^{\text{th}}\) level, the maximum number of emission lines is \[ N=\frac{n(n-1)}{2}. \] For \[ n=4, \] \[ N=\frac{4\times3}{2}=6. \] Hence, \[ \boxed{6.} \] Therefore, the correct option is \(\boxed{(D)}\).