Using a quick and easy formula, you can calculate the Bohr model equation to find the frequency of the photons emitted or absorbed by an atom during an electronic transition.
Read this quick article for a journey in the lego bricks of matter. You will learn:

  • The history of atomic models;
  • The Bohr's model;
  • The equation for the Bohr model: how to calculate the energy difference between orbitals;
  • The calculations for the Bohr model in the hydrogen atom.

What is the Bohr model?

Bohr model is the first atomic model that introduced the principles of quantum mechanics in describing the fundamental components of matter.

Before introducing the Bohr model, let's quickly survey the previous atomic models: it will make understanding the Bohr model and its equation easier.

  • The first "modern" model for atoms was proposed by John Dalton, which postulated the "quantized" nature of matter after observing definite proportions of elements in many compounds. Dalton's model dates back to 1803.
  • Nearly a century later, Thomson introduced the newly discovered negatively charged electrons in his atomic model: with the "plum pudding mode,l" the atoms were finally divisible, but physicists were still a far shot from a realistic description of the atom.
  • At the beginning of the XX century, Rutherford and colleagues performed a simple experiment that proved the existence of atomic nuclei: small positively charged groups of particles (protons). In Rutherford's model, electrons orbit the atoms in orbits that resemble satellites revolving around their planets.

The Rutherford model has a significant set of problems that kept it from describing atoms' observed behavior.
The first issue is the lack of emitted radiation: when subjected to an acceleration, a charged particle emits electromagnetic radiation at specific frequencies. No such thing was observed, which somehow implied orbiting without energy loss.

The other fundamental issue was the unexplained nature of the emission spectra of atoms.

Niels Bohr, one of the giants of physics, addressed these issues in his Bohr model. Using the newly discovered law of quantum mechanics, the scientist postulated that:

  • Electrons orbit in specific circular orbits where they don't lose energy;
  • The energies of electrons in orbit are quantized and easily computable using a set of quantum numbers;
  • An electron can move from one orbit to the other emitting or absorbing a photon of specific energy.

In this model, Bohr joined two of the most important discoveries of the newly created quantum mechanics: the quantization of light (we talked about it in the photon energy calculator) and the existence of a set of quantum numbers (you can discover them in our quantum number calculator).

The straightforward nature of the previous conditions allows us to calculate the Bohr model's equation for electronic transitions quickly. Let's find out how to calculate the energy difference between orbits.

The fundamental equation of the Bohr model: how to calculate the energy difference between orbitals

We can calculate an electronic transition energy — the difference in energy between two allowed orbits with a straightforward formula:

ΔE=E2E1=hf\Delta E = E_2 - E_1 = h \cdot f

Where:

  • E1E_1 — The energy of the first orbit;
  • E2E_2 — The energy of the final orbit;
  • ΔE\Delta E — The energy difference calculated for the electronic transition;
  • hh — The Planck's constant; and
  • ff — The frequency of the photon emitted/absorbed to perform the transition.

The Bohr model is good but not perfect. While describing the basic spectrum of the hydrogen atom, it fails as soon as we try to apply it to bigger atoms. It also fails to explain many spectral lines associated with marginal atomic phenomena. The orbital model for atoms fixed many of these shortcomings.

🙋 Bohr model also fitted the Einstein model for the photoelectric model: it's no surprise that the equation you can meet on our photoelectric effect calculator is similar to the one above!

Calcualting the Bohr model in the hydorgen atom: simple electronic transition calculator

Bohr model fits best the hydrogen atom, thanks to its simplicity: a proton, an electron, and that's it. Let's try to calculate the energy emitted by a hydrogen atom during an electronic transition: we calculate the transition from the first two lines of the Balmer series. The lines have energy:

E1=1.89 eVE2=2.55 eV\begin{align*} E_1&=1.89\ \text{eV}\\ E_2&=2.55\ \text{eV} \end{align*}

Insert these values in our Bohr model calculator:

ΔE=E2E1=hf=(2.551.89) eV=0.66 eV\begin{align*} \Delta E& = E_2 - E_1 = h \cdot f\\ &=(2.55-1.89)\ \text{eV} = 0.66\ \text{eV} \end{align*}

The frequency of the photon associated with this transition is:

f=Eh=159.6 THzf = \frac{E}{h} = 159.6\ \text{THz}

This transition corresponds to light emitted in the far infrared region of the spectrum.

🙋 The formula you met in the Bohr model calculator is strikingly similar to the equation that first explained the spectral lines in hydrogen and hydrogen-like atoms: learn more about the first experimental observations of quantum phenomena with our Rydberg equation calculator.

Davide Borchia
Initial energy
eV
Final energy
eV
Energy difference
eV
Frequency
THz
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