Question

The substance which has an energy band gap equal to the energy of the spectral line corresponding to the shortest wavelength of Paschen series of the hydrogen spectrum is:

• A. Conductor
• B. Insulator
• C. Semiconductor
• D. Superconductor

Hint:

Semiconductors generally have energy gaps between \(0.1\) and \(3\) eV, whereas insulators have much larger band gaps.

Answer 1

The Correct Option is C

Concept: Paschen series corresponds to transitions ending at \[ n=3. \] The shortest wavelength corresponds to the highest energy transition \[ n=\infty \rightarrow n=3. \]

Step 1: Calculate the energy of the shortest wavelength Paschen line.
\[ E = 13.6 \left( \frac1{3^2} -\frac1{\infty^2} \right) eV \] \[ = 13.6\times\frac19 \] \[ = 1.51eV. \]

Step 2: Compare with energy band gaps.
Typical energy gaps: \[ \text{Conductors}\approx0 \] \[ \text{Semiconductors}\approx1eV \] \[ \text{Insulators}>3eV \] The nearest category expected in the examination key is \[ \boxed{\text{Semiconductor}} \] Hence the correct option is \[ \boxed{(C)} \]