Bohr model

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In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus — similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity.

Introduced by Niels Bohr in 1913, the model's key success was in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen; while the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced.

The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics, and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics.

1 History
2 Electron energy levels in hydrogen
- 2.1 Energy in terms of other constants
3 Rydberg formula
4 Moseley's law
5 Shortcomings
6 Refinements
7 See also
8 References
- 8.1 Historical
- 8.2 Further reading

[edit] History

In the early 20th century, experiments by Ernest Rutherford and others had established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. Given this experimental data, it was quite natural for physicists to consider a planetary model for the atom, with electrons orbiting a sun-like nucleus. However, the planetary model for the atom has several difficulties. For example, the laws of classical (Newtonian) mechanics predict that the electron will release electromagnetic radiation as it orbits a nucleus. Because the electron would be losing energy, it would be predicted to gradually spiral inwards and collapse into the nucleus. As this occurred, the emission would change in frequency and would be predicted to produce a smear, in frequency, of electromagnetic radiation. However, late 19th century experiments with electric discharges through various low-pressure gasses in evacuated glass tubes had shown that atoms will emit light (that is, electromagnetic radiation), but only at certain discrete frequencies.

To overcome this and other difficulties in explaining electron motion in an atom, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. Two key ideas were:

The electrons travel in orbits and have discrete quantized energies. That is, not every orbit is possible but only certain specific ones, at certain distances away from the nucleus.
The electrons will not lose energy as they travel in the orbit, and hence will remain in a stable, non-decaying orbit.

The great significance of the model is that it states that the laws of classical mechanics do not apply to the motion of the electron about the nucleus. Bohr proposed that a new kind of mechanics, or quantum mechanics, describes the motion of the electrons around the nucleus. This model of electrons traveling in orbits around the nucleus, however, was replaced with a more accurate model of electron motion about ten years later by the German physicists Erwin Schrodinger and Werner Heisenberg.

Other points are:

When an electron makes a jump from one orbit to another, the energy difference is carried off (or supplied) by a single quantum of light (called a photon) which has an energy equal to the energy difference between the two orbits.
The allowed orbits depend on quantized (discrete) values of orbital angular momentum, L according to the equation
<math> \mathbf{L} = n \cdot \hbar = n \cdot {h \over 2\pi} </math>
Where n = 1,2,3,… and is called the principal quantum number, and h is Planck's constant.

Point (2) states that the lowest value of n is 1. This corresponds to a smallest possible radius of 0.0529 nm. This is known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton.

[edit] Electron energy levels in hydrogen

The Bohr model is accurate only for one-electron systems such as the hydrogen atom or singly-ionized helium. It can also be used for K-line X-ray transition calculations, if other assumptions are introduced (see Moseley's law below). This section uses the Bohr model to derive the energy levels of hydrogen.

The derivation starts with three simple assumptions:

1) The energy of an electron in an orbit is the sum of its kinetic and potential energies:

<math>E \,</math>	<math>=E_{kinetic} + E_{potential} \,</math>
	<math>= \begin{matrix} \frac{1}{2} \end{matrix}m_e v^2 - \frac{k q_e^2}{r}</math>

where <math>k = 1 / ({4 \pi \epsilon _0})</math>, and <math>q_e</math> is the charge of the electron.

2) The angular momentum of the electron can only have certain discrete values:

where n = 1,2,3,… and is called the principal quantum number, h is Planck's constant, and <math>\hbar=h/(2\pi)</math>.

3) The electron is held in orbit by the coulomb force. That is, the coulomb force is equal to the centripetal force:

To begin, multiply both sides of eq (3) by r to see

The term on the left hand side is the potential energy. So the equation for the energy becomes

<math>E = \begin{matrix} \frac{1}{2} \end{matrix}m_e v^2 - \frac{k q_e^2}{r} = -\begin{matrix} \frac{1}{2} \end{matrix} m_e v^2 \,</math>

Now we just need to figure out what the velocity, v is equal to, so solve eq (2) for r,

Plug this into eq (4),

Then divide both sides by m_ev to see

Now we can put in this value for v into the equation for energy, and then also plug in the values for k and <math>\hbar</math>, and we'll obtain the energy of the different levels of hydrogen:

<math>E _n \,</math>	<math>= \frac{-1}{2} m_e \left( \frac{k q_e^2}{n \hbar} \right)^2 \,</math>
	<math>= \frac{-1}{2} m_e \left(\frac{1}{4 \pi \epsilon_0} q_e^2 \frac{2 \pi}{n h} \right)^2 \,</math>
	<math>= \frac{-m_e q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} \,</math>

Or, after substituting values for the constants,

<math>E_n = (-13.6 \ \mathrm{eV}) \frac {1}{n^2} \,</math>

Thus, the lowest energy level of hydrogen (n = 1) is about -13.6 eV. The next energy level (n = 2) is -3.4 eV. The third (n = 3) is -1.51 eV, and so on. Note that these energies are less than zero, meaning that the electron is in a bound state with the proton. Positive energy states correspond to the ionized atom where the electron is no longer bound, but is in a scattering state.

[edit] Energy in terms of other constants

Starting with what we found above,

<math>E_n = \frac{-m_e q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} \,</math>

We can multiply top and bottom by <math>c^2</math>, and we'll arrive at

<math>E_n = \frac{-m_e c^2 q_e^4}{8 h^2 c^2 \epsilon_{0}^2} \frac{1}{n^2} \,</math>

or re-grouping them to make it more clear:

<math>E_n = -\frac{1}{2} m_e c^2 \left(\frac{q_e^4}{4 h^2 c^2 \epsilon_{0}^2} \right) \frac{1}{n^2}</math>

From here we can now write the energy level equation in terms of other constants to:

<math>E_n = \frac{-E_r\alpha^2}{2n^2}</math>

where,

<math>E_n \ </math> is the energy level

<math>E_r \ </math> is the rest energy of the electron

<math>\alpha \ </math> is the fine structure constant

<math>n \ </math> is the principal quantum number.

[edit] Rydberg formula

The Rydberg formula describes the transitions or quantum jumps between one energy level and another. When the electron moves from one energy level to another, a photon is given off. Using the derived formula for the different 'energy' levels of hydrogen one may determine the 'wavelengths' of light that a hydrogen atom can give off.

The energy of photons that a hydrogen atom can give off are given by the difference of two hydrogen energy levels:

<math>E=E_i-E_f=\frac{m_e q_e^4}{8 h^2 \epsilon_{0}^2} \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right) \,</math>

where q_e is the charge of an electron (1.60 × 10⁻¹⁹ C), n_f is the final energy level, and n_i is the initial energy level. It is assumed that the final energy level is less than the initial energy level.

Since the energy of a photon is

<math>E=\frac{hc}{\lambda}, \,</math>

the wavelength of the photon given off is

<math>\frac{1}{\lambda}=\frac{m_e q_e^4}{8 c h^3 \epsilon_{0}^2} \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right). \,</math>

The above is known as the Rydberg formula. This formula (with all of the numerical constants lumped into a single empirically measured Rydberg constant number R), was known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical justification for the value of R, or for the form of the formula itself, until Bohr derived them, more or less along the lines above.

[edit] Moseley's law

Niels Bohr said in 1962, "You see actually the Rutherford work [the nuclear atom] was not taken seriously. We cannot understand today, but it was not taken seriously at all. There was no mention of it any place. The great change came from Moseley."

In 1913 Henry Moseley found an empirical relationship between the strongest X-ray line emitted by atoms under electron bombardment, and their atomic number Z. Moseley's empiric formula was found to be derivable from Bohr's formula, with the two additional assumptions that [1] this X-ray line came from a transition between energy levels with quantum numbers 1 and 2, and [2], that the atomic number Z when used in the formula for atoms heavier than hydrogen, should be diminished by 1, to :(Z-1)², apparently to compensate for the screening effect of the remaining electron remaining in the lowest atomic energy level.

In Bohr's formula for hydrogen above, the charge q⁴ is a product of the electron charge q² and the nuclear charge (Zq)² = q² Z². The nuclear charge Z² may then be factored out as a pure number.

Moseley's law is given by the following changes in Bohr's formula:

<math>E= h\nu = E_i-E_f=\frac{m_e q_e^4 (Z-1)^2}{8 h^2 \epsilon_{0}^2} \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \,</math>

<math>f = \nu = \frac{m_e q_e^4 }{8 h^3 \epsilon_{0}^2} \left( \frac{3}{4}\right) (Z-1)^2 = (2.46 * 10^{15} Hz)(Z-1)^2 \,</math>

This latter relationship had been empirically derived by Moseley, in a simple plot of the square root of X-ray frequency against atomic number. Moseley's law not only established the objective meaning of atomic number (see Henry Moseley for detail) but, as Bohr noted, it also did more than the Rydberg derivation to establish the validity of the Rutherford/Bohr nuclear model of the atom, with atomic number as nuclear charge.

[edit] Shortcomings

The Bohr model gives an incorrect value <math> \mathbf{L} = \hbar </math> for the ground state orbital angular momentum. The angular momentum in the true ground state is known to be zero.

The Bohr model also has difficulty with or fails to explain:

Much of the spectra of larger atoms. At best, it can make predictions about the K-line X-ray emission spectra for larger atoms, if two additional ad hoc assumptions are made (see Moseley's law above). Emission spectra for atoms with a single outer-shell electron (atoms in the lithium group) can also be approximately predicted.
The relative intensities of spectral lines; although in some simple cases, it was able to provide reasonable estimates (for example, calculations by Kramers for the Stark effect).
The existence of fine structure and hyperfine structure in spectral lines.
The Zeeman effect - changes in spectral lines due to external magnetic fields.

[edit] Refinements

Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model or Bohr-Sommerfeld model, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. This model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the Sommerfeld-Wilson quantization condition

where p is the generalized momentum conjugate to the angular generalized coordinate q; the integral is the action of action-angle coordinates.

The Bohr-Sommerfeld model proved to be extremely difficult and unwieldy when its mathematical treatment was further fleshed out. In particular, the application of traditional perturbation theory from classical planetary mechanics led to further confusions and difficulties. In the end, the model was abandoned in favour of the full quantum mechanical treatment of the hydrogen atom, in 1925, using Schrödinger's wave mechanics.

However, this is not to say that the Bohr model was without its successes. Calculations based on the Bohr-Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. For example, up to first-order perturbation, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect. At higher-order perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model.

The Bohr-Sommerfeld quantization condition as first formulated can be viewed as a rough early draft of the more sophisticated condition that the symplectic form of a classical phase space M be integral; that is, that it lies in the image of <math>\check{H}^2(M,\mathbb{Z})\to \check{H}^2(M,\mathbb{R})\to H^2_{DR}(M,\mathbb{R})</math>, where the first map is the homomorphism of Čech cohomology groups induced by the inclusion of the integers in the reals, and the second map is the natural isomorphism between the Čech cohomology and the de Rham cohomology groups. This condition guarantees that the symplectic form arise as the curvature form of a connection of a Hermitian line bundle. This line bundle is then called a prequantization in the theory of geometric quantization.

[edit] See also

Franck-Hertz experiment provided early support for the Bohr model.
Moseley's law provided early support for the Bohr model. See also Henry Moseley
Inert pair effect is adequately explained by means of the Bohr model.
Lyman series
Schrödinger equation
Theoretical and experimental justification for the Schrödinger equation

[edit] References

[edit] Historical

Niels Bohr (1913). "On the Constitution of Atoms and Molecules (Part 1 of 3)". Philosophical Magazine 26: 1-25.
Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus". Philosophical Magazine 26: 476-502.
Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part III". Philosophical Magazine 26: 857-875.
Niels Bohr (1914). "The spectra of helium and hydrogen". Nature 92: 231-232.
Niels Bohr (1921). "Atomic Structure". Nature.
A. Einstein (1917). "Zum Quantensatz von Sommerfeld und Epstein". Verhandlungen der Deutschen Physikalischen Gesellschaft 19: 82-92. Reprinted in The Collected Papers of Albert Einstein, A. Engel translator, (1997) Princeton University Press, Princeton. 6 p.434. (Provides an elegant reformulation of the Bohr-Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.)

[edit] Further reading

Linus Pauling (1985). General Chemistry, Chapter 3 (3rd ed). Dover Publications. A great explainer of Chemistry describes the Bohr model, appropriate for High School and College students.
George Gamow (1985). Thirty years that shook Physics, Chapter 2. Dover Publications. A popularizer of physics explains the Bohr model in the context of the development of quantum mechanics, appropriate for High School and College students
Walter J. Lehmann (1972). Atomic and Molecular Structure: the development of our concepts, chapter 18. John Wiley and Sons. Great explanations, appropriate for High School and College students
Paul Tipler and Ralph Llewellyn (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-7167-4345-0.ar:نموذج بور<span class="FA" id="ar" style="display:none;" />