Question

The wavelength of spectral line obtained in the spectrum of Li$^{2+$ ion, when the transition takes place between two levels whose sum is 4 and difference is 2, is}

• A. $1.14 \times 10^{-7}$ cm
• B. $2.28 \times 10^{-7}$ cm
• C. $2.28 \times 10^{-6}$ cm
• D. $1.14 \times 10^{-6}$ cm

Hint:

Hydrogen-like species follow the Rydberg equation with $Z^2$ dependence.

Answer 1

The Correct Option is D

Step 1: Finding energy levels.
Let $n_1 + n_2 = 4$ and $n_2 - n_1 = 2$. Solving gives $n_1 = 1$, $n_2 = 3$.
Step 2: Rydberg formula for hydrogen-like ions.
\[ \frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For Li$^{2+}$, $Z = 3$.
Step 3: Substitution.
\[ \frac{1}{\lambda} = 1.097 \times 10^5 \times 9 \left(1 - \frac{1}{9}\right) \]
\[ \lambda = 1.14 \times 10^{-6} \text{ cm} \]