Bohr Model
Calculate electron orbit radius, energy levels, and velocity for hydrogen-like atoms using the Bohr model. Free quantum mechanics calculator with charts.
About This Calculator
The Bohr Model Calculator computes three fundamental properties of electrons in hydrogen-like atoms: the orbit radius, the energy level, and the orbital velocity. Based on Niels Bohr's 1913 atomic model, this tool is ideal for physics students learning atomic structure, educators demonstrating quantised energy levels, and anyone studying introductory quantum mechanics. Simply enter the atomic number (Z) and the principal quantum number (n) to get instant results.
The calculations use the standard Bohr model formulas derived from the quantisation of angular momentum. The orbit radius follows r = a₀ · n² / Z where a₀ = 5.292 × 10⁻¹¹ m is the Bohr radius. The energy is given by E = -13.606 · Z² / n² eV, with negative values indicating a bound electron state. The orbital velocity follows v = αc · Z / n where α is the fine-structure constant and c is the speed of light, giving v ≈ 2.188 × 10⁶ · Z / n m/s.
About the Bohr Model
The Bohr model revolutionised atomic physics by introducing the concept of quantised energy levels. Unlike classical physics which predicted that orbiting electrons would continuously radiate energy and spiral into the nucleus, Bohr postulated that electrons could only occupy specific stable orbits with quantised angular momentum. When an electron jumps between orbits, it emits or absorbs a photon whose energy exactly matches the difference between the two energy levels. This model successfully explained the hydrogen emission spectrum, including the Lyman, Balmer, and Paschen series.
Applications
This calculator is useful for solving textbook problems in atomic physics, estimating the size and energy of single-electron ions, and understanding the relationship between atomic number and orbit properties. The charts visualise how radius increases with n², energy becomes less negative with larger n, and velocity decreases as the electron moves farther from the nucleus.
Frequently Asked Questions
What is the Bohr model of the atom?
The Bohr model describes the atom as a small positively charged nucleus surrounded by electrons orbiting in discrete circular paths called shells or energy levels. It was proposed by Niels Bohr in 1913 for the hydrogen atom and explains atomic emission spectra through quantised energy transitions.
How do you calculate the radius of an electron orbit in the Bohr model?
The orbit radius is calculated using r = a₀ × n² / Z, where a₀ = 5.292 × 10⁻¹¹ m is the Bohr radius, n is the principal quantum number, and Z is the atomic number. For hydrogen (Z=1) in the ground state (n=1), the radius equals the Bohr radius of 5.29 × 10⁻¹¹ m.
What is the energy of an electron in the Bohr model?
The energy level is given by E = -13.606 × Z² / n² eV, where 13.606 eV is the Rydberg constant. For hydrogen in the ground state, the energy is -13.606 eV. Higher orbits have less negative (higher) energies, approaching zero as n approaches infinity.
Why are energy levels negative in the Bohr model?
Negative energy values indicate that the electron is bound to the nucleus. Zero energy corresponds to the electron being completely free (ionised). The more negative the energy, the more tightly the electron is bound to the nucleus and the lower the orbit.
How do you calculate electron velocity in the Bohr model?
The electron orbital velocity is calculated using v = 2.188 × 10⁶ × Z / n m/s. For hydrogen ground state (Z=1, n=1), the velocity is approximately 2.19 × 10⁶ m/s, which is about 0.73% of the speed of light.
What atoms does the Bohr model work for?
The Bohr model accurately describes hydrogen-like atoms which have only one electron orbiting the nucleus. This includes hydrogen (Z=1), singly ionised helium He⁺ (Z=2), doubly ionised lithium Li²⁺ (Z=3), and other ions with a single electron.
What is the difference between initial and final energy in the Bohr model?
When an electron transitions from a higher orbit to a lower orbit, it emits a photon with energy equal to the difference between the two energy levels. This explains atomic emission spectra. Conversely, an electron absorbs a photon to jump from a lower to a higher orbit.
How accurate is the Bohr model compared to modern quantum mechanics?
The Bohr model gives correct energy levels for hydrogen-like atoms but fails for multi-electron atoms, fine structure, and magnetic effects. Modern quantum mechanics using the Schrödinger equation provides a more complete description. The Bohr model remains an excellent pedagogical tool for introducing quantised energy levels.