Calculate Bohr model energy levels, photon transitions, wavelength, and frequency for hydrogen-like atoms and ions with step-by-step solutions.
Energy Difference (ΔE)
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Initial Energy Level (E₁)
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Final Energy Level (E₂)
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Photon Wavelength (λ)
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Photon Frequency (ν)
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Spectral Series
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Bohr's atomic theory describes electrons orbiting the nucleus in fixed energy levels. Electrons can only exist in specific quantized orbits, and they emit or absorb photons when transitioning between these energy levels.
The energy of an electron in the nth Bohr orbit is calculated using En = -13.6 × Z²/n² eV, where Z is the atomic number and n is the principal quantum number. The negative sign indicates the electron is bound to the nucleus.
Theoretically, hydrogen has infinite energy levels (n = 1, 2, 3, ...). However, practically observable transitions typically involve the first few levels, with n=1 being the ground state and higher levels representing excited states.
The energy of a photon is directly proportional to its frequency: E = h×ν, where h is Planck's constant. When an electron transitions between energy levels, the energy difference equals the photon energy emitted or absorbed.
Hydrogen-like atoms are ions with only one electron but different nuclear charges (Z > 1). They follow the same Bohr model equations but with Z² scaling, making their energy levels more tightly bound than hydrogen.
The main series are: Lyman (transitions to n=1, UV region), Balmer (transitions to n=2, visible region), Paschen (transitions to n=3, infrared), and Brackett (transitions to n=4, far infrared).
Wavelength is inversely related to energy difference through λ = hc/|ΔE|, where h is Planck's constant and c is the speed of light. Higher energy transitions produce shorter wavelengths (bluer light).
When an electron absorbs a photon, it jumps to a higher energy level (excitation). When it emits a photon, it falls to a lower energy level. The photon energy must exactly match the energy difference between levels.