(i) The energy associated with the first orbit in the hydrogen atom is $-2.18 \times 10^{-18}\ J\ atom^{-1}$. What is the energy associated with the fifth orbit? (ii) Calculate the radius of Bohr's fifth orbit for hydrogen atom.

(i) The energy associated with the first orbit in the hydrogen atom is -2.18 × 10⁻¹⁸Jatom⁻¹. What is the energ

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-2.18 \times 10^{-18}\ J\ atom^{-1}

(ii) Calculate the radius of Bohr's fifth orbit for hydrogen atom.

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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 of an electron in a hydrogen atom scales as $1/n^2$ , so the fifth orbit has energy $E_5 = E_1 / 25 = -8.72 \times 10^{-20}\ \text{J atom}^{-1}$ . The radius scales as $n^2$ , so $r_5 = 25 \times r_1 = 25 \times 0.529\ \text{Å} = 13.225\ \text{Å}$ .

The Bohr model gives us two beautiful, simple scaling laws for hydrogen: energy goes as $-\frac{1}{n^2}$ , and radius goes as $n^2$ . These aren't arbitrary — they come from balancing the Coulomb attraction with the centripetal force and quantizing angular momentum. Once you know the value for $n=1$ , every other orbit follows by just multiplying by the right factor.

Let's work through both parts.

Part (i): Energy of the fifth orbit

1. Recall the energy quantization rule.

In the Bohr model, the total energy of the electron in the $n$ th orbit is:

E_n = -\frac{13.6\ \text{eV}}{n^2} = -\frac{2.18 \times 10^{-18}\ \text{J}}{n^2}

The negative sign means the electron is bound — you'd need to add energy to free it. The key point: energy is inversely proportional to $n^2$ .

2. Use the given $E_1$ to find $E_5$ .

You're told $E_1 = -2.18 \times 10^{-18}\ \text{J atom}^{-1}$ . For $n=5$ :

E_5 = \frac{E_1}{5^2} = \frac{-2.18 \times 10^{-18}}{25}

3. Do the arithmetic.

E_5 = -8.72 \times 10^{-20}\ \text{J atom}^{-1}

Watch out

A common mistake is to divide by 5 instead of $5^2 = 25$ . Energy scales as $1/n^2$ , not $1/n$ . Always square the principal quantum number.

Part (ii): Radius of the fifth orbit

1. Recall the radius quantization rule.

The Bohr radius (first orbit radius) is:

r_n = n^2 \times a_0

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Q.(i) The energy associated with the first orbit in the hydrogen atom is −2.18×10−18 J atom−1-2.18 \times 10^{-18}\ J\ atom^{-1}−2.18×10−18 J atom−1. What is the energy associated with the fifth orbit?

Part (i): Energy of the fifth orbit

Part (ii): Radius of the fifth orbit

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Q.(i) The energy associated with the first orbit in the hydrogen atom is $-2.18 \times 10^{-18}\ J\ atom^{-1}$ . What is the energy associated with the fifth orbit?

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