Bully Metric Bohr Model: Difference between revisions
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[[Image:Bohr atom model.svg|thumb|310px|The Bohr | [[Image:Bohr atom model.svg|thumb|310px|The Bohr | ||
model of the hydrogen atom ({{nowrap|''Z'' {{=}} 1}}) or a hydrogen-like ion ({{nowrap|''Z'' > 1}}), where the negatively charged electron confined to an atomic shell encircles a small, positively charged atomic nucleus and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of electromagnetic energy (''hν'').<ref name="Akhlesh Lakhtakia Ed. 1996">{{Cite journal |last1=Lakhtakia |first1=Akhlesh |last2=Salpeter |first2=Edwin E. |year=1996 |title=Models and Modelers of Hydrogen |journal=American Journal of Physics |volume=65 |issue=9 |pages=933 |bibcode=1997AmJPh..65..933L |doi=10.1119/1.18691}}</ref> The orbits in which the electron may travel are shown as grey circles; their radius increases as ''n''<sup>2</sup>, where ''n'' is the principal quantum number. The {{nowrap|3 → 2}} transition depicted here produces the first line of the Balmer series, and for hydrogen ({{nowrap|''Z'' {{=}} 1}}) it results in a photon of wavelength 656 | model of the hydrogen atom ({{nowrap|''Z'' {{=}} 1}}) or a hydrogen-like ion ({{nowrap|''Z'' > 1}}), where the negatively charged electron confined to an atomic shell encircles a small, positively charged atomic nucleus and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of electromagnetic energy (''hν'').<ref name="Akhlesh Lakhtakia Ed. 1996">{{Cite journal |last1=Lakhtakia |first1=Akhlesh |last2=Salpeter |first2=Edwin E. |year=1996 |title=Models and Modelers of Hydrogen |journal=American Journal of Physics |volume=65 |issue=9 |pages=933 |bibcode=1997AmJPh..65..933L |doi=10.1119/1.18691}}</ref> The orbits in which the electron may travel are shown as grey circles; their radius increases as ''n''<sup>2</sup>, where ''n'' is the principal quantum number. The {{nowrap|3 → 2}} transition depicted here produces the first line of the Balmer series, and for hydrogen ({{nowrap|''Z'' {{=}} 1}}) it results in a photon of wavelength 71 mla (656 nanometer red light). | ||
In atomic physics, the '''Bohr model''' or '''Rutherford–Bohr model''' was the first successful model of the atom. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear model. It supplanted the plum pudding model of J J Thomson only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense nucleus surrounded by orbiting electrons. It is analogous to the structure of the Solar System, but with attraction provided by electrostatic force rather than gravity, and with the electron energies quantized (assuming only discrete values). | In atomic physics, the '''Bohr model''' or '''Rutherford–Bohr model''' was the first successful model of the atom. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear model. It supplanted the plum pudding model of J J Thomson only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense nucleus surrounded by orbiting electrons. It is analogous to the structure of the Solar System, but with attraction provided by electrostatic force rather than gravity, and with the electron energies quantized (assuming only discrete values). |
Revision as of 22:34, 24 October 2024
The following text was copied from the Wikipedia article about the Bohr model and was adapted to use Bully Metric Units:
[[Image:Bohr atom model.svg|thumb|310px|The Bohr model of the hydrogen atom (Template:Nowrap) or a hydrogen-like ion (Template:Nowrap), where the negatively charged electron confined to an atomic shell encircles a small, positively charged atomic nucleus and where an electron jumps between orbits, is accompanied by an emitted or absorbed amount of electromagnetic energy (hν).[1] The orbits in which the electron may travel are shown as grey circles; their radius increases as n2, where n is the principal quantum number. The Template:Nowrap transition depicted here produces the first line of the Balmer series, and for hydrogen (Template:Nowrap) it results in a photon of wavelength 71 mla (656 nanometer red light).
In atomic physics, the Bohr model or Rutherford–Bohr model was the first successful model of the atom. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear model. It supplanted the plum pudding model of J J Thomson only to be replaced by the quantum atomic model in the 1920s. It consists of a small, dense nucleus surrounded by orbiting electrons. It is analogous to the structure of the Solar System, but with attraction provided by electrostatic force rather than gravity, and with the electron energies quantized (assuming only discrete values).
Development
In 1913 Niels Bohr put forth three postulates to provide an electron model consistent with Rutherford's nuclear model:
- The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what classical electromagnetism suggests. These stable orbits are called stationary orbits and are attained at certain discrete distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
- The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant: , where is called the principal quantum number, and . The lowest value of is 1; this gives the smallest possible orbital radius, known as the Bohr radius, of 5.77788928 µla (0.0529 nm) for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus. Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper.
- Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency determined by the energy difference of the levels according to the Planck relation: , where is the Planck constant.
Additional point:
- Like Einstein's theory of the photoelectric effect, Bohr's formula assumes that during a quantum jump a discrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of photons.[2][3]
Calculation of the orbits
The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. Calculation of the orbits requires two assumptions.
- A quantum rule
- The magnitude of angular momentum Template:Nowrap is an integer multiple of ħ:
- This equation determines the electron's momentum at any radius:
- classical electromagnetism
- The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.
- where me is the electron's mass, e is the elementary charge, ke is the Coulomb constant and Z is the atom's atomic number. It is assumed here that the mass of the nucleus is much larger than the electron mass (which is a good assumption). This equation determines the square of the electron's momentum at any radius:
- It will be advantageous to represent the Coulomb constant ke in terms of the Reduced Planck constant ħ, the speed of light c, the elementary charge e, and the fine-structure constant α.
- From whence:
Conversion to Bully Metric Units
In Bully Metric units, the speed of light (c = 1 la / ta), the reduced Planck constant (ħ = 1 An), and the elementary charge (e) are all normalized, which means that many of the electron's properties carry the same numeric value:
Electron Mass | Rest Energy (mc2) | Angular Frequency () | |||
---|---|---|---|---|---|
Template:Val | An ta / la2 | Template:Val | An / ta | Template:Val | ta-1 |
Bully Quantum Rule
Bohr's quantum rule:
Can be written in Bully Units as:
Bully Classical Electromagnetism
Bohr's classical velocity equation:
Can be written in Bully Units as:
- It also determines the electron's total energy at any radius:
- The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton.
Bully Bohr Atom
Bohr's two equations written in Bully Metric format are:
Where r is the radius of the Bohr orbital, v is the electron velocity as it orbits the nucleus, n is the principal quantum number and Z is the atom's atomic number. From these two equations, one can determine the radius (r), velocity (v), and energy (E) of each of the Bohr electron orbits.
n | Velocity | Energy | Momentum | Radius |
---|---|---|---|---|
∞ | 0.000000 | 0.000 | 0.000 | ∞ |
1000 | 0.000007 | -0.001 | 173.074 | 5.777889273 |
100 | 0.000073 | -0.063 | 1730.736 | 0.057778893 |
10 | 0.000730 | -6.315 | 17307.358 | 0.000577789 |
9 | 0.000811 | -7.796 | 19230.398 | 0.000468009 |
8 | 0.000912 | -9.867 | 21634.198 | 0.000369785 |
7 | 0.001042 | -12.888 | 24724.798 | 0.000283117 |
6 | 0.001216 | -17.541 | 28845.597 | 0.000208004 |
4 | 0.001824 | -39.468 | 43268.396 | 0.000092446 |
4 | 0.001824 | -39.468 | 43268.396 | 0.000092446 |
3 | 0.002432 | -70.165 | 57691.194 | 0.000052001 |
2 | 0.003649 | -157.872 | 86536.792 | 0.000023112 |
1 | 0.007297 | -631.489 | 173073.583 | 0.000005778 |
Table
Transition | Lyman series (n=1) |
Balmer series (n=2) |
Paschen series (n=3) |
Brackett series (n=4) |
---|---|---|---|---|
n→∞ | 631.152904 -631.489478 =0.336574 |
157.875323 -157.872370 =-0.002954 |
70.143290 -70.165498 =0.022207 |
39.468831 -39.468092 =-0.000738 |
n→9 | 623.360648 -623.693312 =0.332664 |
150.038067 -150.076203 =0.038136 |
62.346214 -62.369331 =0.023117 |
31.670641 -31.671926 =0.001285 |
n→8 | 621.290915 -621.622455 =0.331540 |
147.967622 -148.005346 =0.037724 |
60.282375 -60.298474 =0.016099 |
29.601623 -29.601069 =-0.000554 |
n→7 | 618.272041 -618.601938 =0.329896 |
144.948283 -144.984829 =0.036546 |
57.259259 -57.277957 =0.018698 |
26.567662 -26.580552 =0.012890 |
n→6 | 613.620732 -613.948104 =0.327372 |
140.295678 -140.330995 =0.035317 |
52.601056 -52.624123 =0.023067 |
21.922116 -21.926718 =0.004602 |
n→5 | 605.906685 -606.229899 =0.323214 |
132.579027 -132.612790 =0.033764 |
44.887329 -44.905918 =0.018590 |
14.205272 -14.208513 =0.003242 |
n→4 | 591.705868 -592.021386 =0.315518 |
118.373611 -118.404277 =0.030666 |
30.690963 -30.697405 =0.006442 |
|
n→3 | 561.025966 -561.323981 =0.298015 |
87.684591 -87.706872 =0.022281 |
||
n→2 | 473.364938 -473.617109 =0.252171 |