Bully Metric Bohr Model
The following text was copied from the Wikipedia article about the Bohr model and was adapted to use Bully Metric Units:
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
editIn 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 5.777 889 micropan (52.917 721 picometers) for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus.
- 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.
Calculation of the orbits
editThe 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 and classical electromagnetism.
- A quantum rule
- The magnitude of angular momentum L = mevr is an integer multiple of ħ:
- This quantum rule determines the electron's momentum (p) at any radius (r), for each integer n:
- classical electromagnetism
- The electron is held in a circular orbit by electrostatic attraction. The centripetal force is therefore 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 classical equation determines the square of the electron's momentum (p) at any radius (r), for each integer n:
Calculation of energy levels
editClassical energy is the sum of kinetic and potential energy. Classical kinetic energy is equal to one half of the mass multiplied by the velocity squared. And from the previous section, the momentum squared turns out to be equal to the Coulomb potential multiplied by the electron mass.
- The total energy here is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the atom. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. 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 Bohr's three equations become:
Conversion to Bully Metric Units
editThe Quantization Rule
editBohr's quantization rule:
Can be written in Bully units as:
This rule is not a special property of the Bohr atom, but rather, is a universal property of quantum mechanics called quantization of angular momentum. This rule has an extremely simple form when momentum and radius are plotted on a log-log graph using Bully units. The quantization of angular momentum appears as a series of parallel straight lines with a slope of negative one, each line representing an integer value of the principle quantum number n. The lowest energy level (n = 1) has the property that the momentum is always equal to the numerical inverse of the radius. For example, if an electron were to orbit a nucleus at 1 micropan (0.000001 la), then the quantization of angular momentum would require the electron's perpendicular momentum to be 1 actionat per micropan, or in other words a million actionats per length apan (1000000 An / la). The slope of negative one indicates that momentum is proportional to the inverse of the radius.
Bully Classical Electromagnetism
editIn Bully Metric units, the speed of light (c = 1.0 la / ta), the reduced Planck constant (ħ = 1.0 An), and the elementary charge (1.0 e) are all normalized, which means that many of the electron's properties carry the same numeric value but with differing units as shown in Table 1.
Electron Mass (m) | Rest Energy (mc2) | (mcħ) | |||
---|---|---|---|---|---|
23717311.411 | An ta la-2 | 23717311.411 | An ta-1 | 23717311.411 | An^2 la-1 |
Bohr's classical electromagnetism equations:
Can be written in Bully units as shown below (note that 137.035999177 is the inverse fine-structure constant):
For a hydrogen atom with one proton (Z = 1), this becomes:
When momentum and radius are plotted on a log-log graph using Bully units, Bohr's classical electromagnetism momentum equation appears as a straight line with a slope of negative two (negative two indicating that momentum squared is proportional to the inverse of the radius).
Bully Metric Solutions for Bohr's Hydrogen Atom
edit- The solution
- The above two equations are sufficient to find exact r and p values (two equations in two unknowns) for each given integer n. Dividing momentum squared by momentum, the radius dependence drops out:
See Table 2 for the list of Bohr hydrogen atom energy level solutions in Bully Metric units. Table 3 provides a list of photons that are emitted or absorbed when an electron transitions to a different energy level within the Bohr hydrogen atom.
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 |
5 | 0.001459 | -25.260 | 34614.717 | 0.000144447 |
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
editTransition | 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 | 561.024872 561.323981 0.299109 |
87.684591 87.706872 0.022281 |
||
n→2 | 473.364899 473.617109 0.252210 |
- ↑ Lakhtakia, Akhlesh; Salpeter, Edwin E. (1996). "Models and Modelers of Hydrogen". American Journal of Physics 65 (9): 933. doi:10.1119/1.18691.