Derive the energy expression for the hydrogen atom using the Bohr atom model.

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Q. Derive the energy expression for the hydrogen atom using the Bohr atom model.

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Concept understanding — Bohr Model Energy Levels

The Intuition: Why Can't an Electron Just Sit Anywhere?

Imagine you're rolling a marble on a staircase. The marble can rest on any step — step 1, step 2, step 3 — but it can never float halfway between two steps. The staircase forces the marble into specific, fixed positions.

That's the core idea of the Bohr model. Before Bohr, physicists thought electrons orbited the nucleus like planets around the sun — they could be at any distance, any energy. But experiments showed something strange: atoms only emit or absorb light at very specific colours (wavelengths), not a continuous rainbow. That meant electrons could only have certain, fixed energies — like the steps of a staircase.

Bohr's genius was to say: an electron in an atom cannot have any arbitrary energy. It can only occupy certain "allowed" energy levels. When it jumps from one level to another, it either absorbs or emits a photon of light whose energy exactly matches the difference between those levels.

The Precise Statement

In the Bohr model of the hydrogen atom (and hydrogen-like ions with one electron), the electron moves in circular orbits around the nucleus. But only those orbits are allowed where the electron's angular momentum is an integer multiple of $\frac{h}{2\pi}$ (where $h$ is Planck's constant).

This quantisation condition leads to a simple formula for the energy of the electron in the $n$ -th orbit:

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

Here:

$n$ is the principal quantum number — a positive integer ( $n = 1, 2, 3, \dots$ )
$E_n$ is the energy of the electron in that level (in electronvolts)
The negative sign means the electron is bound to the nucleus — you need to add energy to free it

The lowest energy state ( $n=1$ ) is called the ground state. Its energy is $-13.6 \text{ eV}$ . The higher states ( $n=2, 3, 4, \dots$ ) are excited states — they have less negative (higher) energies.

Note

As $n$ increases, the energy levels get closer together. At $n = \infty$ , the energy becomes $0 \text{ eV}$ — the electron is completely free from the atom (ionisation).

What This Explains

When an electron jumps from a higher level ( $n_i$ ) to a lower level ( $n_f$ ), it emits a photon of energy:

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

This single formula predicts all the spectral lines of hydrogen — the Lyman series (jumps to $n=1$ ), Balmer series (to $n=2$ ), Paschen series (to $n=3$ ), and so on. Each series corresponds to a different "final step" on the staircase.

Watch out

The Bohr model works perfectly only for hydrogen and one-electron ions (like $\text{He}^+$ , $\text{Li}^{2+}$ ). For multi-electron atoms, it fails — electron-electron repulsion changes the energy levels in ways Bohr's simple picture cannot capture. That's where quantum mechanics takes over.

Key Takeaways for Exams

Energy levels are quantised — only specific values allowed, given by $E_n = -13.6/n^2$ eV for hydrogen
The ground state ( $n=1$ ) is the most stable, lowest energy
Excited states ( $n>1$ ) are higher in energy (less negative)
Transitions between levels produce line spectra — not continuous
The ionisation energy of hydrogen (energy to remove the electron from ground state) is $+13.6$ eV

Important

The negative sign in $E_n$ is not optional — it tells you the electron is bound. A positive energy would mean a free electron (ionised atom).

Searches like "Bohr model energy levels formula" and "hydrogen spectrum series class 12 physics" are extremely common, since this concept anchors the Atoms chapter of the NCERT/CBSE Class 12 Physics curriculum. Energy-level transition and spectral-series questions built on this model are a staple of JEE Main and NEET.

Why this formula?

Why the Bohr Model Gives Those Energy Levels

The Bohr model is a beautiful piece of physics because it takes a simple, almost desperate idea — "electrons only exist in certain orbits" — and derives the entire hydrogen spectrum from it. The key is that Bohr didn't just assume the energy levels; he forced them to be consistent with classical physics in one specific way, then broke with it in another.

The Two Non-Negotiable Pieces

First, the electron moves in a circle around the proton. That's pure classical mechanics: the Coulomb attraction provides the centripetal force.

\frac{1}{4\pi\varepsilon_0} \frac{e^2}{r^2} = \frac{mv^2}{r}

This gives you a relation between speed $v$ and radius $r$ :

v^2 = \frac{e^2}{4\pi\varepsilon_0 m r}

Second, the total energy of the electron is the sum of its kinetic and potential energies. Potential energy for a Coulomb force is $-\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}$ (negative because the force is attractive, and we set zero at infinity).

E = \frac12 mv^2 - \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}

Substitute $v^2$ from above:

E = \frac12 \left( \frac{e^2}{4\pi\varepsilon_0 r} \right) - \frac{e^2}{4\pi\varepsilon_0 r} = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0 r}

So far, nothing is quantised. Any radius $r$ gives a valid classical orbit, and the energy just follows from that radius. The problem is that a classical electron in a curved path radiates energy and spirals into the nucleus — atoms should collapse. Bohr needed a rule to pick out stable orbits.

The Quantisation Condition

Bohr's revolutionary step was to postulate that the angular momentum of the electron is quantised in units of $\hbar = h/2\pi$ :

m v r = n \hbar, \quad n = 1, 2, 3, \dots

Why this particular rule? Bohr later said it was the simplest way to get the right answer. But there's a deeper physical motivation: if you think of the electron as a wave (de Broglie's idea, which came a decade later), the condition that a standing wave fits exactly around the circumference $2\pi r = n \lambda$ gives $m v r = n \hbar$ directly. So the quantisation condition is really a wave condition imposed on a particle picture.

Note

The angular momentum quantisation is the only non-classical assumption in the Bohr model. Everything else follows from classical mechanics and electromagnetism.

Deriving the Allowed Radii and Energies

From $m v r = n \hbar$ , we get $v = n\hbar / (mr)$ . Substitute this into the centripetal force equation:

\frac{1}{4\pi\varepsilon_0} \frac{e^2}{r^2} = \frac{m}{r} \left( \frac{n\hbar}{mr} \right)^2 = \frac{n^2 \hbar^2}{m r^3}

Solve for $r$ :

r_n = \frac{4\pi\varepsilon_0 \hbar^2}{m e^2} \, n^2

The constant in front is the Bohr radius $a_0 \approx 0.529 \, \text{Å}$ . So the radii are $r_n = a_0 n^2$ .

Now plug $r_n$ back into the energy expression $E = -\frac12 \frac{e^2}{4\pi\varepsilon_0 r}$ :

E_n = -\frac12 \frac{e^2}{4\pi\varepsilon_0} \cdot \frac{1}{a_0 n^2} = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0 a_0} \cdot \frac{1}{n^2}

Substitute $a_0 = \frac{4\pi\varepsilon_0 \hbar^2}{m e^2}$ :

E_n = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0} \cdot \frac{m e^2}{4\pi\varepsilon_0 \hbar^2} \cdot \frac{1}{n^2} = -\frac{m e^4}{8 \varepsilon_0^2 h^2} \cdot \frac{1}{n^2}

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

That's the famous result. The $1/n^2$ dependence comes directly from the $n^2$ dependence of the radius, which came from the angular momentum quantisation.

Why the Negative Sign Matters

The energy is negative because the electron is bound. To remove the electron from the atom (ionise it), you need to add $+13.6 \text{ eV}$ to get it to $E=0$ (free electron at rest). The ground state ( $n=1$ ) is the most tightly bound; higher $n$ states are less negative, meaning they're closer to being free.

Watch out

A common mistake is to think the energy levels are equally spaced. They're not — the gap between $n=1$ and $n=2$ is about 10.2 eV, while between $n=2$ and $n=3$ is only 1.9 eV. The spacing shrinks as $1/n^3$ for large $n$ .

The Physical Picture

The Bohr model gives you a ladder of energies because the electron can only exist in orbits whose angular momentum is an integer multiple of $\hbar$ . Each orbit has a specific radius, and therefore a specific energy. When the electron jumps from a higher orbit to a lower one, the energy difference is emitted as a photon of frequency $f = (E_i - E_f)/h$ — which exactly matches the hydrogen spectral lines.

The model fails for multi-electron atoms and doesn't explain why angular momentum is quantised in the first place. But for hydrogen, it's remarkably accurate — and the derivation shows that the $1/n^2$ energy law is a direct consequence of combining classical circular motion with a single quantisation postulate.

Step 1. Force balance. For an electron of mass $m$ in a circular orbit of radius $r_n$ around a nucleus of charge $+Ze$ , the Coulomb force supplies the centripetal force:

\frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r_n^2}=\frac{m\upsilon_n^2}{r_n}\ \Rightarrow\ r_n=\frac{4\pi\varepsilon_0 m\upsilon_n^2 r_n^2}{Ze^2}\quad\text{(rearranged for use below)}

Step 2. Quantisation. Bohr's postulate quantises angular momentum: $m\upsilon_n r_n=n\hbar$ , i.e. $\upsilon_n=n\hbar/(mr_n)$ .

Step 3. Radius. Substituting Step 2's $\upsilon_n$ into the force-balance equation and solving for $r_n$ gives

r_n=\frac{4\pi\varepsilon_0 n^2\hbar^2}{Zme^2}=a_0\frac{n^2}{Z},\qquad a_0=0.529\ \text{Å}

Step 4. Potential and kinetic energy. The potential energy is $U_n=-\dfrac{1}{4\pi\varepsilon_0}\dfrac{Ze^2}{r_n}$ ; substituting $r_n$ from Step 3 gives $U_n=-\dfrac{Z^2me^4}{4\varepsilon_0^2h^2n^2}$ . Working through the mechanics shows $KE_n=-\tfrac12U_n$ (the virial-theorem-like relation for an inverse-square force).

Step 5. Total energy. Adding kinetic and potential energy, $E_n=KE_n+U_n=\tfrac12U_n$ :

E_n=-\frac{me^4Z^2}{8\varepsilon_0^2h^2n^2}

Step 6. Numerical form. Substituting the known values of $m$ , $e$ , $\varepsilon_0$ and $h$ , and expressing in electron-volts, this simplifies to the standard result

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

For hydrogen ( $Z=1$ ): $E_1=-13.6$ eV, $E_2=-3.4$ eV, $E_3=-1.51$ eV, matching the observed hydrogen energy-level spacings exactly.

✓Final answer

E_n=-\dfrac{me^4Z^2}{8\varepsilon_0^2h^2n^2}=-\dfrac{13.6Z^2}{n^2}

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Q.Derive the energy expression for the hydrogen atom using the Bohr atom model.

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