NEET (UG) 2023 — Question 30

Use Rydberg formula for hydrogen: $\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$, where $n_1$ is lower energy level, $n_2$ is higher. For shortest wavelength in a series, $n_2 \to \infty$. For Balmer series, $n_1 = 2$. Then $\frac{1}{\lambda_B} = R\left(\frac{1}{2^2} - 0\right) = \frac{R}{4}$, so $\lambda_B = \frac{4}{R}$. For Brackett series, $n_1 = 4$. Then $\frac{1}{\lambda_{Br}} = R\left(\frac{1}{4^2} - 0\right) = \frac{R}{16}$, so $\lambda_{Br} = \frac{16}{R}$. Taking ratio: $\lambda_{Br} / \lambda_B = (16/R) / (4/R) = 4$, thus $\lambda_{Br} = 4\lambda$.