The electron energy in hydrogen atom is given by $E_n = (-2.18 \times 10^{-18})/n^2\ J$. Calculate the energy required to remove an electron completely from the n = 2 orbit. What is the longest wavelength of light in cm that can be used to cause this transition?

The electron energy in hydrogen atom is given by Eₙ = (-2.18 × 10⁻¹⁸)/n²J. Calculate the energy required to re

Exercises · 2.19

E_n = (-2.18 \times 10^{-18})/n^2\ J

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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 energy required to remove an electron from the $n=2$ orbit of hydrogen is $5.45 \times 10^{-19}\ \text{J}$ , and the longest wavelength of light that can cause this transition is $3.65 \times 10^{-5}\ \text{cm}$ .

The key idea here is that "removing an electron completely" means taking it from a bound state ( $n=2$ ) to the ionization limit ( $n = \infty$ ), where the electron has zero total energy. The energy needed is simply the difference between the final energy ( $E_\infty = 0$ ) and the initial energy ( $E_2$ ). The longest wavelength corresponds to the smallest energy photon that can still cause this transition — that is, a photon with energy exactly equal to the ionization energy from $n=2$ .

Energy level quantization in hydrogen means the electron can only occupy specific orbits, each with a well-defined energy given by $E_n = -\frac{2.18 \times 10^{-18}}{n^2}\ \text{J}$ . The negative sign indicates that the electron is bound to the nucleus — you must add energy to free it. At $n = \infty$ , the electron is free and at rest, so $E_\infty = 0$ . The energy required to remove the electron from any level $n$ is therefore $0 - E_n = -E_n$ , which is positive.

Let's work through the calculation.

Find the energy of the electron in the $n=2$ orbit. Using the given formula:

E_2 = -\frac{2.18 \times 10^{-18}}{2^2} = -\frac{2.18 \times 10^{-18}}{4} = -5.45 \times 10^{-19}\ \text{J}

Calculate the ionization energy from $n=2$ . The energy required to remove the electron is the difference:

\Delta E = E_\infty - E_2 = 0 - (-5.45 \times 10^{-19}) = 5.45 \times 10^{-19}\ \text{J}

This is the minimum energy a photon must have to eject the electron from the $n=2$ level.

Relate photon energy to wavelength. The energy of a photon is given by $E = \frac{hc}{\lambda}$ , where $h = 6.626 \times 10^{-34}\ \text{J·s}$ is Planck's constant and $c = 3.00 \times 10^8\ \text{m/s}$ is the speed of light. Solving for wavelength:

\lambda = \frac{hc}{\Delta E}

Plug in the numbers.

\lambda = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{5.45 \times 10^{-19}}

First compute the numerator:

6.626 \times 10^{-34} \times 3.00 \times 10^8 = 1.9878 \times 10^{-25}\ \text{J·m}

Then divide:

$\lambda = \frac{1.9878 \times 10^{-25}}{5.45 \times 10^{-19}} = 3.647 \times 10^{-7}\ \text{m}$ …

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