The Balmer series in the hydrogen spectrum corresponds to the transition from n1 = 2 to n2 = 3, 4, ......... .

NCERT Exemplar · Q32

Q. The Balmer series in the hydrogen spectrum corresponds to the transition from n1 = 2 to n2 = 3, 4, ......... . This series lies in the visible region. Calculate the wave number of line associated with the transition in Balmer series when the electron moves to n = 4 orbit. (R_H = 109677 cm^-1)

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Concept understanding — Energy Level Quantization

Energy Level Quantization: From Intuition to Precision

Imagine you're climbing a smooth ramp. You can stop at any height — 1 metre, 1.5 metres, 2.1 metres — anywhere you like. That's how we intuitively think about energy in everyday life: continuous, like a slide.

Now imagine a staircase. You can stand on step 1, step 2, or step 3 — but you cannot stand halfway between step 2 and step 3. There's no such place. The steps are discrete, not continuous.

Energy level quantization is the idea that in the microscopic world of atoms and molecules, energy behaves like a staircase, not a ramp. Electrons in an atom cannot have just any energy — they can only occupy specific, allowed energy levels. Everything else is forbidden.

Why does this happen? The core intuition

In the classical world, an electron orbiting a nucleus would continuously radiate energy, spiral inward, and crash — atoms would be unstable. But atoms are stable. Nature solved this problem by imposing a rule: the electron's angular momentum (and therefore its energy) can only take certain discrete values.

Think of a guitar string. It can only vibrate at specific frequencies — its fundamental and harmonics. You can't pluck it to produce a frequency halfway between two harmonics. The string's vibration is quantized by its boundaries. Similarly, an electron bound to a nucleus is confined in space, and that confinement forces its energy to be quantized.

Note

Quantization is not a mysterious extra rule — it emerges naturally whenever a wave (like an electron's matter wave) is confined. Confinement creates standing waves, and standing waves only exist at specific frequencies.

The precise statement

For a bound system (like an electron in an atom), the total energy $E$ of the system can only take certain discrete values:

E = E_1, E_2, E_3, \dots

where each $E_n$ is a specific, fixed number. The integer $n$ (1, 2, 3, …) is called the principal quantum number. The lowest energy level ( $n=1$ ) is the ground state; higher levels ( $n>1$ ) are excited states.

For the hydrogen atom, the allowed energies are given by:

E_n = -\frac{13.6 \text{ eV}}{n^2}

So:

$n=1$ : $E_1 = -13.6$ eV (ground state)
$n=2$ : $E_2 = -3.4$ eV
$n=3$ : $E_3 = -1.51$ eV
and so on, approaching 0 eV as $n \to \infty$ (the ionization limit)

Important

The negative sign means the electron is bound to the nucleus. Zero energy corresponds to the electron being free (ionized). The more negative the energy, the more tightly bound the electron.

How do we know this is real?

The most direct evidence comes from atomic spectra. When an electron jumps from a higher energy level to a lower one, it emits a photon of light with energy exactly equal to the difference:

\Delta E = E_{\text{higher}} - E_{\text{lower}} = hf

where $h$ is Planck's constant and $f$ is the frequency of the emitted light. …

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The Balmer series arises when electrons fall to the $n=2$ level from higher orbits. For the transition $n=4 \to n=2$ , the wave number is $\boxed{20564.4 \text{ cm}^{-1}}$ .

Why the Balmer series produces visible light

When an electron in hydrogen drops from a higher energy level to a lower one, it releases energy as a photon. The Balmer series specifically describes all transitions that end at $n=2$ . Because the energy gaps to $n=2$ happen to correspond to photon energies in the visible range (roughly 400–700 nm), these lines appear as the characteristic red, blue-green, and violet lines you see in a hydrogen discharge tube.

The wave number $\bar{\nu}$ (the reciprocal of wavelength, measured in cm $^{-1}$ ) tells us how many wave cycles fit in one centimeter. It is directly proportional to the photon's energy, making it a convenient unit in spectroscopy.

The Rydberg formula

The wave number for any hydrogen transition is given by:

\bar{\nu} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

where $n_1$ is the lower (final) level, $n_2$ is the upper (initial) level, and $R_H = 109677 \text{ cm}^{-1}$ is the Rydberg constant for hydrogen.

This formula emerges from the quantized energy levels $E_n = -\frac{13.6 \text{ eV}}{n^2}$ . The energy difference between two levels determines the photon energy, and hence its wave number.

Step-by-step calculation

1. Identify the transition

The problem asks for the line in the Balmer series when the electron moves to $n=4$ orbit. This phrasing is slightly ambiguous, but in the context of the Balmer series (which is defined by transitions ending at $n=2$ ), the electron must be moving from $n=4$ to $n=2$ . So:

$n_1 = 2, \quad n_2 = 4$ …

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Q.The Balmer series in the hydrogen spectrum corresponds to the transition from n1 = 2 to n2 = 3, 4, ......... . This series lies in the visible region. Calculate the wave number of line associated with the transition in Balmer series when the electron moves to n = 4 orbit. (R_H = 109677 cm^-1)

Why the Balmer series produces visible light

The Rydberg formula

Step-by-step calculation

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