Question

Which of the following is (are) correct for a hydrogen-like single-electron species?

• A. The energy gap between the consecutive energy levels decreases as the value of "n” increases
• B. The longest wavelength in any spectral series corresponds to α-line in that series.
• C. Each spectral series is bounded by minimum and maximum wavelengths and the range follows a continuous distribution as given by Bohr's theory.
• D. Kinetic energy of the electron decreases, whereas the potential energy increases as the value of "n” increases.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A

The energy gap between the consecutive energy levels decreases as the value of "n” increases

B The longest wavelength in any spectral series corresponds to α-line in that series.
C

Each spectral series is bounded by minimum and maximum wavelengths and the range follows a continuous distribution as given by Bohr's theory.

D Kinetic energy of the electron decreases, whereas the potential energy increases as the value of "n” increases.

Here, we know, from Bohr's postulates

$E_n$ = $-2.18\times {10}^{-18}\left(\frac{Z^2}{n^2}\right)J$

We can see, as value of n increases, the gap of energy between two consecutive energy values decreases.

$\Rightarrow$ More energy gap at lesser values of 'n'.

Option (a) is correct.

$\to$ According to Bohr's theory, each spectral line is bounded by minimum and maximum wavelengths and range follows a continuous distribution.

Option (c) is correct.

$\to$$P.E.\alpha -\frac{1}{n^2}\Rightarrow KE\alpha \frac{1}{n^2}$

Therefore, P.E. with proper sign will increase with increase in value of n whereas KE will decrease.

Option (d) is correct.

The longest wavelength corresponds to $\alpha$ - line in that series.

Option (b) is correct.