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Energy of Electron in Bohr Orbit: En = -RH/n² Formula

Chemistry · Structure Of Atom · NEET

In Bohr's model, the energy of an electron in the nth orbit of a hydrogen-like atom is En = -RH·Z²/n². For hydrogen (Z=1) this is En = -13.6/n² eV, so the electron is most tightly bound in n=1 (-13.6 eV) and its energy rises toward 0 as n grows. Memory hook: "Big n, small pull" - larger orbit means energy closer to zero (less negative), so the electron is less tightly held.
Energy levels of hydrogen: E n = -13.6 / n² eVEnergy (eV)n = ∞ , E = 0 (free electron)n = 3 , -1.51 eVn = 2 , -3.40 eVn = 1 , -13.6 eV (ground)Higher n → energy closer to 0 (less negative, less bound)
Hydrogen energy levels from En = -13.6/n² eV. The n=1 orbit sits lowest (most negative, most stable); as n rises the levels crowd toward 0, where the electron becomes free.

Your doubts, answered

What exactly is the formula for the energy of an electron in a Bohr orbit?

For a hydrogen-like atom (one electron, nuclear charge Z), En = -RH·Z²/n². Here RH = 2.18 x 10^-18 J per atom (or 13.6 eV, or 1312 kJ/mol). For hydrogen Z=1, so En = -13.6/n² eV. The n is the orbit number (1, 2, 3...). Only the whole numbers are allowed, which is why energy is quantised.

Why is there a minus sign in En = -RH/n²?

The minus sign means the electron is bound to the nucleus. We fix the zero of energy at n = infinity, where the electron is free and just escaped the atom. Any bound electron has LESS energy than a free one, so its energy is below zero (negative). It does not mean the amount of energy is negative in a physical sense - it is a comparison to the free state. NEET loves testing this sign.

Does the energy become more negative or less negative as n increases?

As n increases, n² gets bigger, so RH/n² gets smaller, so En becomes LESS negative (closer to 0). Example: E1 = -13.6 eV, E2 = -3.4 eV, E3 = -1.51 eV. So higher orbits have HIGHER energy (less negative). Students often flip this - remember: closer to nucleus = more negative = more stable.

Where does the Z² come from and how do I use it?

Z is the atomic number (nuclear charge). For H it is 1, He+ it is 2, Li2+ it is 3, Be3+ it is 4. Energy depends on Z²/n². So He+ in n=1 has energy -13.6 x (2²/1²) = -54.4 eV. NEET often gives you H's energy and asks for a hydrogen-like ion - just scale by Z²/n².

What is the value of RH and in which units?

RH (the Rydberg energy) = 2.18 x 10^-18 J/atom = 13.6 eV/atom = 1312 kJ/mol. Use joules when the question gives values in joules, and eV for quick shell energies. 1 eV = 1.6 x 10^-19 J. Note: this energy RH is different from the Rydberg CONSTANT (1.097 x 10^7 m^-1) used in the spectral wavenumber formula - do not mix them up.

Two orbits have the same Z²/n² - do they have the same energy?

Yes. Energy depends only on the ratio Z²/n². For He+ (Z=2, n=1): Z²/n² = 4. For Be3+ (Z=4, n=2): Z²/n² = 16/4 = 4. Same ratio, same energy. NEET 2024 used exactly this trick - the answer was that both energies are equal.

How is kinetic energy, potential energy and total energy related here?

For a Bohr orbit: Total energy E = -RH·Z²/n² (negative). Kinetic energy KE = +RH·Z²/n² = -E (positive). Potential energy PE = -2RH·Z²/n² = 2E (twice as negative). So KE = -E and PE = 2E, giving E = KE + PE = -KE. This is the virial relation and NEET sometimes asks for the ratio.

⚠️ The NEET trap
✗ As n increases the electron energy becomes more negative because it is farther from the nucleus.
✓ As n increases, RH/n² shrinks, so En becomes LESS negative (rises toward 0). The n=1 orbit has the lowest (most negative) energy and is most stable.
🧠 More negative = more tightly bound = smaller n. Bigger orbit always means energy CLOSER to zero, never further below it.

Real NEET questions

NEET 2024

The energy of an electron in the ground state (n=1) for the He+ ion is -x J. Then the energy of an electron in the n=2 state for the Be3+ ion, in J, is:

A · -x/9
B · -4x
C · -4x/9
D · -x ✓
Solution: En is proportional to -Z²/n². For He+ (Z=2, n=1): Z²/n² = 4/1 = 4, and this equals -x J. For Be3+ (Z=4, n=2): Z²/n² = 16/4 = 4 - the SAME value. So the energy is also -x J. Answer (D). This tests that energy depends only on the ratio Z²/n².
NEET 2025

Energy and radius of the ground-state Bohr orbit of He+ and Li2+ are: [Given RH = 2.18 x 10^-18 J, a0 = 52.9 pm]

A · En(Li2+) = -19.62 x 10^-16 J, rn(Li2+) = 17.6 pm; En(He+) = 8.72 x 10^-16 J, rn(He+) = 26.4 pm
B · En(Li2+) = -8.72 x 10^-16 J, rn(Li2+) = 17.6 pm; En(He+) = -19.62 x 10^-16 J, rn(He+) = 17.6 pm
C · En(Li2+) = -19.62 x 10^-18 J, rn(Li2+) = 17.6 pm; En(He+) = -8.72 x 10^-18 J, rn(He+) = 26.4 pm ✓
D · En(Li2+) = -8.72 x 10^-18 J, rn(Li2+) = 26.4 pm; En(He+) = -19.62 x 10^-18 J, rn(He+) = 17.6 pm
Solution: Use En = -RH·Z²/n² with n=1. For Li2+ (Z=3): En = -2.18 x 10^-18 x 9 = -19.62 x 10^-18 J. For He+ (Z=2): En = -2.18 x 10^-18 x 4 = -8.72 x 10^-18 J. Radius rn = a0·n²/Z: Li2+ = 52.9/3 = 17.6 pm; He+ = 52.9/2 = 26.4 pm. This matches option (C). Note the energy stays in 10^-18 J, so options with 10^-16 are wrong.

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Frequently asked

What is the energy of an electron in the first Bohr orbit of hydrogen?

E1 = -13.6 eV = -2.18 x 10^-18 J = -1312 kJ/mol. This is the ground state and the most stable (most negative) level.

Is the -13.6 eV the same as the ionisation energy of hydrogen?

Yes in magnitude. To remove the electron from n=1 (E1 = -13.6 eV) to n=infinity (E = 0) you must add +13.6 eV. So the ionisation energy of hydrogen equals +13.6 eV.

How do I get the energy of a hydrogen-like ion like He+ or Li2+?

Multiply the hydrogen value by Z². En = -13.6 x Z²/n² eV. For He+ (Z=2) ground state: -13.6 x 4 = -54.4 eV.

What is the difference between RH here and the Rydberg constant in spectra?

RH here is the Rydberg ENERGY = 2.18 x 10^-18 J = 13.6 eV, used for orbit energies. The Rydberg CONSTANT in the wavenumber formula is 1.097 x 10^7 m^-1. They are related but have different units - do not confuse them in NEET.

Why does NEET keep asking about this formula?

Because it links energy, radius, ionisation energy and spectral lines. One formula En = -RH·Z²/n² lets NEET test the sign, the Z² scaling and quick numeric ratios - all common single-mark questions in Structure of Atom.

Related concepts:
Why Electron Energy Is Negative in Bohr's ModelBohr's Postulates for the Hydrogen AtomIonisation Energy of Hydrogen from Bohr ModelVelocity of Electron in a Bohr OrbitRydberg Formula for the Hydrogen Spectrum
Practise Chemistry for NEET:
Chemistry MCQs (free, chapter-wise)Structure Of Atom PYQs