ACKNOWLEDGEMENT
“THE BEST USE OF PEN IS FOR EXCELLENCE”. With these words I RAJINDER KAUR I would like to thank the almighty GOD for showing all his blessings and giving me strength for completing this project. He is the without whom this all would never had been possible. I would also like to my teacher Prof J.P. Aggarwal for providing me with this opportunity. The Vice Chancellor of LOVELY PROFESSIONAL UNIVERSITY Dr. Vijay Gupta for providing his encouraging words. Lastly my parents without whom my life would never had been possible and also making me part for this esteemed institution.
RAJINDER KAUR
INDEX
S.No TOPIC PGE NO.
1 Acknowledgement 2
2 Bohr s model 3
3 Shortcomings 9
4 Bibliography 11
Bohr model
The Rutherford-Bohr model of the hydrogen atom (Z = 1) or a hydrogen-like ion (Z > 1), where the negatively charged electron confined to an atomic shell encircles a small positively charged atomic nucleus, and an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy hν. The orbits that the electron may travel in are shown as grey circles; their radius increases as n2, where n is the principal quantum number. The transition depicted here produces the first line of the Balmer series, and for hydrogen (Z = 1) results in a photon of wavelength 656 nm (red).
In atomic physics, the Bohr model created by Niels Bohr 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. This was an improvement on the earlier cubic model (1902), the plum-pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911). Since the Bohr model is a quantum physics-based modification of the Rutherford model, many sources combine the two, referring to the Rutherford-Bohr model.
Introduced by Niels Bohr in 1913, the model's key success lay 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. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, but it provided a justification for its empirical results in terms of fundamental physical constants.
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, before moving on to the more accurate but more complex valence shell atom. A related model was originally proposed by Arthur Erich Haas in 1910, but was rejected. The quantum theory of the period between Planck's discovery of the quantum (1900) and the advent of a full-blown quantum mechanics (1925) is often referred to as the old quantum theory.
Origin
In the early 20th century, experiments by Ernest Rutherford established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. Given this experimental data, Rutherford naturally considered a planetary-model atom, the Rutherford model of 1911 — electrons orbiting a solar nucleus — however, said planetary-model atom has a technical difficulty. The laws of classical mechanics (i.e. the Larmor formula), predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it would gradually spiral inwards, collapsing into the nucleus. This atom model is disastrous, because it predicts that all matter is unstable.
Also, as the electron spirals inward, the emission would gradually increase in frequency as the orbit got smaller and faster. This would produce a continuous 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 only emit light (that is, electromagnetic radiation) at certain discrete frequencies.
To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. He suggested that electrons could only have certain classical motions:
The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies.
The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined by the energy difference ΔE = E2 − E1 of the levels according to Bohr's formula where h is Planck's constant.
the frequency of the radiation emitted at an orbit with period T is as it would be in classical mechanics--- it is the reciprocal of the classical orbit period:
The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the electron about the nucleus only when restricted by a quantum rule. Although rule 3 is not completely well defined for small orbits, because the emission process involves two orbits with two different periods, Bohr could determine the energy spacing between levels using rule 3 and come to an exactly correct quantum rule: the angular momentum L is restricted to be an integer multiple of a fixed unit:
where n = 1,2,3,… and is called the principal quantum number. The lowest value of n is 1. This gives a smallest possible orbital 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. Starting from the angular momentum quantum rule Bohr[1] was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogenlike atoms and ions.
Electron energy levels
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. This not only includes one-electron systems such as the hydrogen atom, singly-ionized helium, doubly ionized lithium, but it includes positronium and Rydberg states of any atom where one electron is far away from everything else. It can be used for K-line X-ray transition calculations if other assumptions are added (see Moseley's law below). In high energy physics, it can be used to calculate the masses of heavy quark mesons.
To calculate the orbits requires two assumptions:
1. Classical mechanics
The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.
where me is the mass and e is the charge of the electron. This determines the speed at any radius:
It also determines the 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. The total energy is half the potential energy, which is true for non circular orbits too by the virial theorem.
For larger nuclei, the only change is that ke2 is everywhere replaced by Zke2 where Z is the number of protons. For positronium, me is replaced by the reduced mass me / 2.
2. Quantum rule
The angular momentum is an integer multiple of :
, in which .
Substituting the expression for the velocity gives an equation for r in terms of n:
so that the allowed orbit radius at any n is:
The smallest possible value of r
is called the Bohr radius.
The energy of the n-th level is determined by the radius:
So an electron in the lowest energy level of hydrogen (n = 1) has 13.606 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level at (n = 2) is -3.4 eV. The third (n = 3) is -1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom.
The combination of natural constants in the energy formula is called the Rydberg energy RE:
This expression is clarified by interpreting it in combinations which form more natural units:
: the rest energy of the electron (= 511 keV)
: the fine structure constant
For nuclei with Z protons, the energy levels are:
(Heavy Nuclei)
When Z is approximately 137 (about 1/α), the motion becomes highly relativistic. Then the Z2 cancels the α2 in R, so the orbit energy begins to be comparable to rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei[citation needed].
For positronium, the formula uses the reduced mass. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.
(Positronium)
Rydberg formula
The Rydberg formula, which was known empirically before Bohr's formula, is now in Bohr's theory seen as describing the energies of transitions or quantum jumps between one orbital energy level, and another. Bohr's formula gives the numerical value of the already-known and measured Rydberg's constant, but now in terms of more fundamental constants of nature, including the electron's charge and Planck's constant.
When the electron moves from one energy level to another, a photon is emitted. Using the derived formula for the different 'energy' levels of hydrogen one may determine the 'wavelengths' of light that a hydrogen atom can emit.
The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen energy levels:
where nf is the final energy level, and ni is the initial energy level.
Since the energy of a photon is
This is known as the Rydberg formula, and the Rydberg constant R is RE / hc, or RE / 2π in natural units. 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.
Shell model of the atom
Bohr extended the model of Hydrogen to give an approximate model for heavier atoms. This gave a physical picture which reproduced many known atomic properties for the first time.
Heavier atoms have more protons in the nucleus, and more electrons to cancel the charge. Bohr's idea was that each discrete orbit could only hold a certain number of electrons. After that orbit is full, the next level would have to be used. This gives the atom a shell structure, in which each shell corresponds to a Bohr orbit.
This model is even more approximate than the model of hydrogen, because it treats the electrons in each shell as non-interacting. But the repulsions of electrons is taken into account somewhat by the phenomenon of screening. The electrons in outer orbits do not only orbit the nucleus, but they also orbit the inner electrons, so the effective charge Z that they feel is reduced by the number of the electrons in the inner orbit.
For example, the lithium atom has two electrons in the lowest 1S orbit, and these orbit at Z=2. Each one sees the nuclear charge of Z=3 minus the screening effect of the other, which crudely reduces the nuclear charge by 1 unit. This means that the innermost electrons orbit at approximately 1/4th the Bohr radius. The outermost electron in lithium orbits at roughly Z=1, since the two inner electrons reduce the nuclear charge by 2. This outer electron should be at nearly one Bohr radius from the nucleus. Because the electrons strongly repel each other, the effective charge description is very approximate, the effective charge Z doesn't usually come out to be an integer. But Moseley's law experimentally probes the innermost pair of electrons, and shows that they do see a nuclear charge of approximately Z-1, while the outermost electron in an atom or ion with only one electron in the outermost shell orbits a core with effective charge Z-k where k is the total number of electrons in the inner shells.
The shell model was able to qualitatively explain many of the mysterious properties of atoms which became codified in the late 19th century in the periodic table of the elements. One property was the size of atoms, which could be determined approximately by measuring the viscosity of gases and density of pure crystalline solids. Atoms tend to get smaller as you move to the right in the periodic table, becoming much bigger at the next line of the table. Atoms to the right of the table tend to gain electrons, while atoms to the left tend to lose them. Every element on the last column of the table is chemically inert (noble gas).
In the shell model, this phenomenon is explained by shell-filling. Successive atoms get smaller because they are filling orbits of the same size, until the orbit is full, at which point the next atom in the table has a loosely bound outer electron, causing it to expand. The first Bohr orbit is filled when it has two electrons, and this explains why helium is inert. The second orbit allows eight electrons, and when it is full the atom is neon, again inert. The third orbital contains eight again, except that in the more correct Sommerfeld treatment (reproduced in modern quantum mechanics) there are extra "d" electrons. The third orbit may hold an extra 10 d electrons, but these positions are not filled until a few more orbitals from the next level are filled (Filling the n=3 d orbitals produces the 10 transition elements). The irregular filling pattern is an effect of interactions between electrons, which are not taken into account in either the Bohr or Sommerfeld models, and which are difficult to calculate even in the modern treatment.
Shortcomings
The Bohr model gives an incorrect value for the ground state orbital angular momentum. The angular momentum in the true ground state is known to be zero. Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may be thought of as not to rotate "around" the nucleus at all, but merely to go tightly around it in an ellipse with zero area (this may be pictured as "back and forth", without striking or interacting with the nucleus). This is only reproduced in a more sophisticated semiclassical treatment like Sommerfeld's. Still, even the most sophisticated semiclassical model fails to explain the fact that the lowest energy state is spherically symmetric--- it doesn't point in any particular direction.
In modern quantum mechanics, the electron in hydrogen is a spherical cloud of probability which grows denser near the nucleus. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree, is considered a "coincidence." (Though many such coincidental agreements are found between the semi-classical vs. full quantum mechanical treatment of the atom; these include identical energy levels in the hydrogen atom, and the derivation of a fine structure constant, which arises from the relativistic Bohr-Sommerfield model (see below), and which happens to be equal to an entirely different concept, in full modern quantum mechanics).
The Bohr model also has difficulty with, or else fails to explain:
Much of the spectra of larger atoms. At best, it can make predictions about the K-alpha and some L-alpha 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. Also, if the empiric electron-nuclear screening factors for many atoms are known, many other spectral lines can be deduced from the information, in similar atoms of differing elements, via the Ritz-Rydberg combination principles (see Rydberg formula). All these techniques essentially make use of Bohr's Newtonian energy-potential picture of the atom.
The theory does not predict the relative intensities of spectral lines; although in some simple cases, Bohr's formula or modifications of 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, which are known to be due to a variety of relativistic and subtle effects, as well as complications from electron spin.
BIBLEOGRAPHY
• www.google.com
• www.ieee.com
• www.wikipedia.com
• www.phyforums.com
• APPLIED CHEMISTRY
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