Hydrogen Energy Levels Calculator

Calculate electron energy levels, orbital radii, and spectral transitions in hydrogen-like atoms

Calculate Hydrogen Energy Levels

Calculation Mode

Single Energy Level

Calculate energy, radius, and binding energy for a specific level

Energy Transitions

Calculate photon energy and wavelength for electron transitions

Atomic Number (Z)

Hydrogen (H) - Number of protons in nucleus

Principal Quantum Number (n)

Energy level (n=1 is ground state)

Energy Level Results

Energy Level

-13.606 eV

E₁ = -13.61 eV for hydrogen

Orbital Radius

5.292 × 10^-11 m

a₀ = 0.529 Å for hydrogen

Binding Energy

13.606 eV

Energy to remove electron

Additional Properties

Radius in Bohr radii: 1.00 a₀

Velocity: 2.188 × 10^6 m/s

Formula: Eₙ = -13.61 eV × Z²/n² (Rydberg formula)

Orbital radius: rₙ = n² × a₀/Z, where a₀ = 0.529 Å

Energy Level Diagram

n = 1

E = -13.606 eV

r = 5.29 × 10^-11 m

Selected

n = 2

E = -3.401 eV

r = 2.12 × 10^-10 m

n = 3

E = -1.512 eV

r = 4.76 × 10^-10 m

n = 4

E = -0.850 eV

r = 8.47 × 10^-10 m

n = 5

E = -0.544 eV

r = 1.32 × 10^-9 m

n = 6

E = -0.378 eV

r = 1.91 × 10^-9 m

Energy approaches 0 eV as n → ∞ (ionization threshold)

Example: Hydrogen Lyman Series

Lyman Alpha Transition

Atom: Hydrogen (Z = 1)

Transition: n = 2 → n = 1 (first excited state to ground state)

Energy levels: E₂ = -3.40 eV, E₁ = -13.61 eV

Energy difference: ΔE = E₁ - E₂ = -13.61 - (-3.40) = -10.21 eV

Photon Properties

Photon energy: 10.21 eV (emitted)

Wavelength: λ = hc/E = 121.6 nm

Spectrum region: Ultraviolet (UV-C)

Series: Lyman series (all transitions to n = 1)

Hydrogen Spectral Series

Lyman Series

n → 1 (UV region)

91.2 - 121.6 nm

Balmer Series

n → 2 (Visible region)

364.6 - 656.3 nm

Paschen Series

n → 3 (Near-IR region)

820.4 - 1875 nm

Brackett Series

n → 4 (Mid-IR region)

1.46 - 4.05 μm

Physical Constants

Rydberg Energy

13.606 eV

Bohr Radius (a₀)

0.529 Å = 5.29 × 10⁻¹¹ m

Fine Structure Constant

α = 1/137.036

Electron Mass

9.109 × 10⁻³¹ kg

Atomic Physics Tips

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Lower n values have more negative energies (more bound)

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Energy scales as Z²/n² for hydrogen-like atoms

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Orbital radius increases as n²/Z

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Emission: electron falls to lower energy level

Understanding Hydrogen Energy Levels

The Bohr Model

Niels Bohr proposed that electrons in hydrogen orbit the nucleus in discrete energy levels. Each level is characterized by a principal quantum number n, where n = 1 is the ground state (lowest energy) and higher n values represent excited states.

Energy Quantization

•Electrons can only exist in specific energy levels
•Energy is quantized due to wave nature of electrons
•Transitions between levels emit or absorb photons
•Higher levels are less tightly bound to nucleus

Mathematical Formulation

Eₙ = -13.61 eV × Z²/n²

rₙ = 0.529 Å × n²/Z

E₁: Energy of electron at level n
Z: Atomic number (number of protons)
n: Principal quantum number (1, 2, 3, ...)
rₙ: Orbital radius at level n

Note: The negative energy indicates that the electron is bound to the nucleus. Zero energy corresponds to a free electron (ionization).

Spectral Lines and Transitions

Emission Spectra

When electrons fall from higher to lower energy levels, they emit photons with specific wavelengths, creating characteristic spectral lines.

Absorption Spectra

Electrons can absorb photons of specific energies to jump from lower to higher energy levels, creating dark lines in a continuous spectrum.

Hydrogen-like Ions

The same formulas apply to ions with only one electron (He⁺, Li²⁺, etc.), but with different atomic numbers Z.