9.8.A - Bohr's Model of the Atom


With the discovery of the electron in the late 1800s, Thompson proposed the plum pudding model of the atom whereby a low density positively charged, spherical atom had units of negative charge dispersed through it. in 1911, Rutherford discovered that the atom has a pvery dense nucleus and was mostly empty space. He proposed that electrons orbited the nucleus like the planets did around the sun.

Some years later, Niels Bohr used the spectra of the hydrogen atom to propose a new model that had a nucleus at the centre but electrons orbiting a precise and discrete radii that were proportional to the energy possessed by the electron. Electrons could only have certain allowed values for energy which corresponded to the radius of the stable orbits.

Bohr's model of the atom paved the wave for a new way of think in physics and heralded the birth of quantum physics.

In this unit you will:

  • Explain the relationship between the spectra of elements and Bohr's model of the atom
  • Describe the Rutherford's model of the atom and its limitations
  • Describe the Bohr's model of the atom and its limitations
  • Use the Rydberg equation to predict the wavelength of spectral lines for hydrogen


Spectroscopy

In the 1860's, chemists were using spectroscopic techniques to analyse the light energy released when atoms of different elements were 'excited' in some way. What was particularly interesting was that the spectrum of each different element was unique and chemists started cataloguing the spectra of all of the known elements. At the time, no one knew what was responsible for producing the unique spectrum of each element but it was suspected that it had something to do with atomic structure.

There are three types of spectra all produced when white light is differentially refracted through a prism. The longer the wavelength, the more diffraction (bending of light) that occurs. Longer wavelength red light is refracted to a greater extent than blue light and this differential refraction produces the characteristic spectrum of visible light that looks like a rainbow.

The three different types of spectra: continuous, emission and absoprtion
spectra. Note that the spectral lines in the absorption and emission spectrum
of the same element appear at the same wavelengths.

There are three types of spectra that are produced in different ways:

A continuous spectrum is produced when light is split into its component wavelengths (colours) by a glass prism.

An emission spectrum is produced when the energy radiated by a hot (excited) gas shows coloured lines on a dark background.

An absorption spectrum is produced when the energy radiated by a light source passes through a cold gas showing black lines on a continuous spectrum.

Spectroscopic analysis allowed chemists to work out which wavelengths of light corresponded to the lines in the spectra. Each element had its own unique set of wavelengths where the bands appeared and these were recorded at the time for all of the known elements. You should note that for the same element, the bands in the absorption and emission spectra appeared at the same wavelength. It was only the method used to obtain the spectra that was different.

Hydrogen Spectrum

The emission and absorption spectrum for hydrogen showing the Balmer series and the
corresponding wavelength of each spectral line.

When hydrogen gas is excited by a large potential difference, it produces the characteristic emission and absorption spectra. A series of spectral lines that were found in the visible light section of the electromagnetic spectrum were called the Balmer series. Spectral lines were also found to exist in other sections of the EM spectrum and these were given other names. In cataloguing the spectra of the elements, spectral lines were given a letter, such as H designating the Balmer series, and a subscript using a Greek letter for each different line in the series.

You should remember that towards the end of the 1800s, scientists were still unsure as to what was responsible for the spectral lines.

 

Atomic Structure

Thompsons's Model of the Atom

Thompson's plum pudding model of the atom.

With the discovery if the electron in 1897, Thompson proposed a model of the atom known as the 'plum pudding model'. In this model, the atom was composed of electrons (which Thomson still called corpuscles), surrounded by a soup of positive charge to balance the electrons' negative charges, like negatively-charged "plums" surrounded by positively-charged "pudding".

The electrons (as we know them today) were thought to be positioned throughout the atom, but with many structures possible for positioning multiple electrons, particularly rotating rings of electrons. Instead of a soup, the atom was also sometimes said to have had a cloud of positive charge. Most notably, Thompson's model assumed that the atom was spherical and that it had a low density.

Rutherford's Model of the Atom

In an experiment conducted by Geiger and Marsden (students of Rutherford) in 1904, a piece of thin gold foil was bombarded by alpha radiation (helium nuclei). Because of the assumed low density of the atom, it was expected that the particles would pass straight through or be deflected slightly due to interaction between the charges in the atom. For most particles, they did pass straight through, but about one in 8000 were deflected backwards as detected by the screens at C in the diagram. There was something dense and very small that was causing the alpha particles to bounce back in this way.

In 1911 as a result of this experiment, Rutherford proposed a new model for the atom that was primarily empty space with the mass of the atom concentrated in a very small central, positively charged nucleus.

His new model for the atom proposed:

  • a heavy, dense and positively charged nucleus at the centre

  • much lighter electrons in orbit around the nucleus

  • the radius of the nucleus was in the order of 10-15 m and the radius of the whole atom was of the order of 10-12 m. If the atom was as big as a football feild, the nucleus would be the size of a pea!

 

The Rutherford model of the
atom.

The Rutherford model was a great step forward in our understanding of atomic structure but it still had its limitations. To overcome for force of attraction between the positive nucleus and the electrons, electrons must be in circular motion (much like the planets around the Sun). Since electrons were in circular motion, they would be experiencing acceleration and accelerating charges were known to emit electromagnetic radiation. This loss of energy would cause the electrons to spiral closer to the nucleus (like orbital decay for satellites) and matter would be destroyed. This was clearly not happening so there were problems with the model.

Rutherford's model could not explain spectral lines either. As electrons spiralled toward the nucleus with increasing speed, they should emit radiation of all frequencies, not just the one frequency corresponding to the wavelength of the spectral line. Thus, the observed spectrum of the element should have been continuous and not a line spectrum.

 

Bohr Model of the Atom

The Bohr model, like Rutherford, also has an atom consisting of a small, positively-charged nucleus orbited by negatively-charged electrons, however, the orbiting electrons were only allowed to orbit at discrete radii from the nucleus.

Niels Bohr proposed the Bohr Model of the Atom in 1915. Because the Bohr model is a modification of the earlier Rutherford model, some people call Bohr's model the Rutherford-Bohr model. The Bohr model contains some errors, but it is important because it describes most of the accepted features of atomic theory without having to understand and apply quantum mechanics. Unlike earlier models, the Bohr model was able to explain the existence of spectral lines in the spectrum of atomic hydrogen.

The Bohr Model is a planetary model in which the negatively-charged electrons orbit a small, positively-charged nucleus similar to the planets orbiting the Sun (except that the orbits are not planar). Bohr realised that there was a serious problem with having electrons circle the nucleus. Any charged particle accelerating in an electric or magnetic field radiates energy. This energy is known as bremsstrahlung or 'braking radiation'. When electrons orbit the nucleus in circular orbits they are being accelerated (anything travelling in a circular path is accelerating with the acceleration directed towards the centre of the orbit) and so they radiate energy in the form of bremsstrahlung radiation. This causes the kinetic energy of the electron to decrease. As the amount of kinetic energy decreases the electron slows down and this forces the radius of the orbit to decrease. This, in turn, causes the electron to spiral into the nucleus. This led Bohr to the second postulate which is outlined below.

The main postulates of the Bohr model are:

  1. An electron can only have certain allowed, discrete orbits defined by radius and energy.

  2. An electron in an allowed orbit does not emit electromagnetic radiation.

  3. Quantised radiation (a photon) is absorbed or emitted when an electron moves from one orbit to another.


Bohr Model of Hydrogen

The emission and absorption of a photon is shown above when electrons
transition between energy levels. The different combinations of
transitions are shown in the second diagram with each transition
representing a spectral line.

The simplest example of the Bohr model is for the hydrogen atom (Z = 1) in which a negatively-charged electron orbits a small positively-charged nucleus. When the hydrogen atom absorbs energy, as it does in a gas discharge tube, the electron is raised from the orbit n = 1 to a higher orbit such as n = 2 or n = 3 or even higher. Then when the electron drops back to a lower orbit, energy is emitted in the form of light (photons). Since the energy of the electron in a given orbit is fixed, a drop from one particular orbit to another, say from n = 2 to n = 1, always releases the same amount of energy and the frequency of light emitted because of this change in energy is always precisely the same.

The size of the photon being released or absorbed could be calculated from the spectral lines and their corresponding wavelengths. Using a combination of the wave equation and Planck's formula, the energy carried by a photon is given by:

Each of the lines in the spectrum of hydrogen corresponded to a transition between energy levels as shown on the right and a spectral line in the absorption or emission spectra. The Balmer series could be seen in the visible spectrum but the others were not visible because they corresponded to wavelengths of ultraviolet or infrared light.

 

 

Problems with the Bohr Model

The Bohr model of the atom was specifically developed with hydrogen in mind; it succeeded in explaining the hydrogen spectrum. However, when it was used to try to explain the spectra of·other elements several problems emerged.

The primary reason was that hydrogen has only one electron orbiting the nucleus (in its neutral form). All other elements have multiple electrons orbiting the nucleus (when they are neutral). As all electrons are negative the electrons repel each other. In larger atoms an effect known as 'screening' occurs. The inner electrons screen the outer electrons from the positively charged nucleus. As a result, the outer electrons do not feel all the positive charge. The Bohr model does not take into account this interaction and so cannot predict the spectral lines of large atoms with many electrons.

The Bohr model also cannot predict or explain the relative intensity (brightness) of spectral lines. There are several laws in quantum mechanics that forbid certain transitions between orbits. Rules known as the selection rules allow certain transitions to go ahead but forbid or suppress others. The allowed transitions showed up in spectra with brighter or more intense spectral lines and at the time, this could not be explained by the Bohr model as the selection rules were not known.

Bohr was also unable to predict the hyperfine structure seen in atomic spectra, even for hydrogen. Hyperfine spectral lines are faint, thin lines that exist as a cluster around a main spectral line. This splitting of the spectral lines comes about because of the interaction between the angular momentum of the protons and the electrons. Bohr did not know this at the time so the model could not explain these spectral observations.

As the electrons circulate the nucleus they create a magnetic field. Any moving charge creates a magnetic field. When an external magnetic field is applied to the atom this adds to the magnetic field caused by the motion of the electrons. The interaction of the electron with this altered magnetic field produces a different energy to that produced when the external magnetic field is turned off. This changes the spectrum produced by the atom by causing the spectral lines to split in the presence of an external magnetic field (known as the SZeeman effect). Bohr did not consider atoms and their magnetic fields in his model.



An oldie but a goodie! A video summarising the main features and shortcomings of the Bohr model of the atom.


Rydberg Formula

The Rydberg formula was known empirically before Bohr used it to describe the energies of transitions or quantum jumps between one orbital energy level and another.

When the electron gets moved from its original energy level to a higher one, it then jumps back each level until it comes to the original position, which results in a photon being emitted. Using the derived formula for the different energy levels of hydrogen, the 'wavelengths' of light that a hydrogen atom can emit can be determined.

The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen energy levels as shown below.

Equation - Rydberg Equation

λ

R

nf

ni

wavelength of the photon

Rydberg constant

final energy level

initial energy level

metres (m)

1.097 x 107 m-1

integer

integer

You should use this equation to calculate the wavelength of a photon emitted when an electron transitions from a higher energy level to a lower energy level.


This is known as the Rydberg formula, and the Rydberg constant R is 1.097 x 107 m-1. This formula was known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical explanation for this form or a theoretical prediction for the value of R, until Bohr. In fact, Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the Lyman (nf = 1), Balmer (nf = 2), and Paschen (nf = 3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted.