What is the wavelength of light emitted when the electron in a hydrogen atom undergoes transition from an ener

Exercises · 2.13

Q. What is the wavelength of light emitted when the electron in a hydrogen atom undergoes transition from an energy level with n = 4 to an energy level with n = 2?

Haryana BsehTextbookSubjective· 2mImportance★★★★★est

22% · 31/140 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

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. …

Start your 14-day free trial to unlock the full solution →

The wavelength of light emitted when an electron in a hydrogen atom drops from $n=4$ to $n=2$ is found using the Rydberg formula. The result is 486 nm (blue-green light).

The key to this problem is understanding energy level quantization in the hydrogen atom. Niels Bohr proposed that electrons can only occupy specific, discrete energy levels, labelled by the principal quantum number $n$ . When an electron jumps from a higher energy level ( $n=4$ ) to a lower one ( $n=2$ ), it loses energy. That energy isn't destroyed — it is emitted as a single photon of light. The photon's energy is exactly equal to the difference between the two energy levels.

The wavelength of that photon is then given by the Planck-Einstein relation: $E = \frac{hc}{\lambda}$ , where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength. So, the problem reduces to calculating the energy difference $\Delta E = E_4 - E_2$ , and then converting that to a wavelength.

The energy of an electron in the $n$ -th level of a hydrogen atom is given by:

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

This negative sign means the electron is bound to the nucleus; the more negative the energy, the more stable (lower) the level.

Let's work through the calculation step by step.

Find the energy of the $n=4$ level.

E_4 = -\frac{13.6}{4^2} = -\frac{13.6}{16} = -0.85 \text{ eV}

Find the energy of the $n=2$ level.

E_2 = -\frac{13.6}{2^2} = -\frac{13.6}{4} = -3.40 \text{ eV}

Calculate the energy difference $\Delta E$ (the energy of the emitted photon). The photon's energy is the energy lost by the atom, so:

\Delta E = E_4 - E_2 = (-0.85) - (-3.40) = +2.55 \text{ eV}

The positive sign indicates energy is released.

Tip

A faster way to get $\Delta E$ directly is to use the formula:

$\Delta E = 13.6 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \text{ eV}$

where $n_i$ is the initial level (4) and $n_f$ is the final level (2). This gives the same result: $13.6 \left( \frac{1}{4} - \frac{1}{16} \right) = 13.6 \times \frac{3}{16} = 2.55 \text{ eV}$ .

Convert the energy from eV to Joules. The Planck-Einstein relation uses SI units. Since $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$ :

\Delta E = 2.55 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 4.085 \times 10^{-19} \text{ J}

Use the Planck-Einstein relation to find the wavelength. The relation is $E = \frac{hc}{\lambda}$ , so $\lambda = \frac{hc}{E}$ . Using $h = 6.626 \times 10^{-34} \text{ J s}$ and $c = 3.00 \times 10^8 \text{ m/s}$ : …

Start your 14-day free trial to unlock the full solution →

←Previous question · exercises2.12 — Electrons are emitted with zero velocity from a metal surface when it …→Next question · exercises2.14 — How much energy is required to ionise a H atom if the electron occupie…

Q.What is the wavelength of light emitted when the electron in a hydrogen atom undergoes transition from an energy level with n = 4 to an energy level with n = 2?

🎓Unlock everything free for 14 days

Unlock everything free for 14 days