Physics - Modern Physics Question with Solution | TestHub

The longest wavelength corresponds to the minimum energy transition.

For the Balmer series, this is the transition from $n=3$ to $n=2$.

For the Lyman series, it's from $n=2$ to $n=1$.

Since the energy difference between $n=3$ and $n=2$ is smaller than that between $n=2$ and $n=1$, the longest wavelength in the Balmer series is longer than the longest wavelength in the Lyman series.

$${\lambda }_{longest}\propto \frac{1}{E_{min}}$$

Balmer: $n=3\to n=2$

Lyman: $n=2\to n=1$

$E_{3\to 2}<E_{2\to 1}$

therefore ${\lambda }_{3\to 2}>{\lambda }_{2\to 1}$