The discovery of the electron by Thomson gave proof that atoms must have a structure, that is, that atoms are not simple, indivisible particles of matter. But the electron could not account for the mass of atom, since the mass of the electron is very small in comparison to that of the atom, even the lightest atom known, hydrogen. And in addition the electron is negatively charged, but atoms are ordinarily neutral. J.J. Thomson and others at first thought that the mass of the atom was due to the presence of many electrons, 1840 of them in the case of hydrogen, many more for the atoms of heavier elements, and that they were embedded in some almost massless plasma of positive charge that would make the atom as a whole electrically neutral. As early as 1898 Thomson had proposed a model of the atom in which the electrons were embedded in successive rings within a masive matrix of positive charge, which was thought to occupy a volume of about one atomic diameter, which was long known to be between 1 and 10 angstrom units (= 1 × 10-10 m.), a fact inferred from the sizes of molar volumes and Avogadro's number. This model has been called the "plum pudding" model because the electrons were imagined to be arranged like raisins in a pudding. But this concept of the structure of the atom was refuted by Ernest Rutherford's (1871-1937) with his discovery of the nucleus of the atom in 1911.
"It was quite the most incredible event that has ever happened to me in my life. It was as though you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you."Rutherford found that he could account for this result, if he assumed that all the positive charge repelling the positive alpha particles, and nearly all the mass of the atom as well, were confined to a very small volume at the center of the atom, which he called the nucleus of the atom. Rutherford showed that a tremendous amount of force necessary to produce the backward scattering would be exerted only at a distance of about 10-12 cm, and by a positive charge which for case of gold he first estimated to about 100 times the charge on the electron. The correct value was later determined to be 79 and not 100. He concluded that the atomic diameter must be in the order of 10-8 cm. This experiment clearly indicated that the negatively charged electrons must be distributed throughout the volume of this diameter and that the positive charge and most of the mass of the atom are much more concentrated. The mass of the electrons being almost negligible compared with that of even the lighest atom as a whole, most of the mass is concentrated within a radius that compared to the total atomic radius is analogous to that between the sun's radius and the radius of the total solar system at Pluto's orbit. This view of the atom is that the atom is mostly empty space. Only this account for the relative ease that they are penetrated by the alpha particles through thousands of atoms. On the basis of the atomic dimensions deduced from kinetic theory, the gold foil consisted of about 103 layers of atoms.
It occurred to Rutherford and others that each atom is like a miniature solar system, with the electrons orbiting about the nucleus like the planets orbiting about sun, and with electrical attraction between the positive nucleus and the negative electrons instead of the gravitational attraction between the sun and the planets in the solar system. But there were objections to this view. The electrons in their circular orbit about the nucleus would be accelerating, and according to the principles of electricity and magnetism any accelerated charge would emit electromagnetic radiation and thus loses energy. Calculations showed that electrons would lose their kinetic energy and would spiral into the attracting nucleus in a very short period of time, but the matter that we are experiencing have not collapsed nor radiated electromagnetic radiation as that would be given off by a spiraling electron. This raises the problem how the radiation is emitted by atoms. Any satisfactory model of the atom must account for the radiation that is actually observed.
The simplest line spectrum comes from hydrogen gas.
There is certain regularity in the line arrangement, but the difference in the
wavelengths of the various lines diminishes rapidly toward the shorter
wavelengths. This suggests that they could be represented by an algebraic
or geometric series. But there is no simple series corresponding to the
measured lines. In 1883 after many attempts, a German high school teacher,
Johann Balmer (1825-1898), worked out a purely empirical formula that seemed
to fit the measured lines very well.
λ = 3645.6[n2/(n2 -
4)], (1)
where λ is the wavelength in Angstroms = 10-8 cm, and
n is a whole number equal to 3, 4, 5, 6, ... infinity. When n
is placed equal to 3 in the formula, a wavelength is obtained very close to
that of the first line in the hydrogen series. The following table shows the
observed wavelengths and the calculated wavelengths using the Balmer formula.
| n | Observed Wavelength | Calculated Wavelength |
|---|---|---|
| 3 | 6562.84 | 6562.79 |
| 4 | 4861.46 | 4861.33 |
| 5 | 4340.58 | 4340.47 |
| 6 | 4101.76 | 4101.74 |
| etc. |
The first half-dozen or so of these wavelengths are visible to the human eye.
And when the spectrum is examined with an instrument sensitive to the
ultraviolet and infrared light from hydrogen, more lines are found, and many
of these do not fit with values given by the Balmer formula. In 1890
Johannes Rydberg showed that if the Balmer formula is modified, then the
Balmer formula can be extended to include lines observed in the spectra of
other elements. The following is the Rydberg formula:
1/λ = R(1/m2 -
1/n2), (2)
where
λ is the wavelength of a line measured in meters,
R is Rydberg's constant =
1.09678 × 107 m-1 for hydrogen,
m is an integral number,
n is an integral number =
(m + 1), (m + 2), ... infinity.
If m is set equal to 2 in the above formula, then it becomes equivalent
to the formula given by Balmer.
In 1908 Louis C. Paschen observed another series of lines at shorter
wavelengths to those in the Balmer series. He showed that they correspond
to the wavelengths calculated by the Rydberg formula when m = 3.
Later another series still further in the infrared was found by Brackett,
and yet a further one by Pfund. The lines in these series are given
precisely by the Rydberg formula, if m is given the value of 4 and 5.
Is there a series for m = 1? Yes, there is a series in the ultraviolet,
and it was found by Theodore Lyman during the period from 1906 to 1916.
| Name of Series | Date of Discovery | Region of Spectrum | Values in Rydberg Formula |
|---|---|---|---|
| Lyman | 1906-1914 | Ultraviolet | m = 1, n = 2, 3, 4, ... |
| Balmer | 1885 | Uv & visible | m = 2, n = 3, 4, 5, ... |
| Paschen | 1908 | Infrared | m = 3, n = 4, 5, 6, ... |
| Brckett | 1922 | Infrared | m = 4, n = 5, 6, 7, ... |
| Pfund | 1924 | Infrared | m = 5, n = 6, 7, 8, ... |
Among those theorist who had been working, without success, to develop an adequate theory of emission of continuous spectra was the German physicist Max Planck (1858-1947). Since Theory I and II each led to the correct curve for part of the spectrum, one for short wavelengths, the other for very long wavelengths. He thought that an equation could be set up empirically that would combine the best features of each. This Planck succeeded in doing. Finally, there was a single equation which fitted the observed data for the complete energy distribution curve at any temperature. This equation was announced by Planck on October, 1900, to the Deutsche Physikalische Gesellschaft.
Planck's method of obtaining this equation was not just a blind search for an equation that would fit the data; he was guided in part by the results of Theories I and II, which were based on two different classical models. Two month later, as Planck tells us, "after a few weeks of the most strenuous labor in my life, the darkness lifted and a new unimagined prospect began to dawn." His solution was announced by him on December 14, 1900. This was the birth of the Quantum theory and the beginning of a new physics.
In developing his solution he consider the energy distribution of what is called (somewhat misleadingly) a "blackbody radiator." By a blackbody is meant a body from which no light, or practically no light, from the outside is reflected by the body under consideration. That is, a blackbody absorbs all the radiation falling on it. According to Kirchhoff's law of thermal radiation, a good absorber is a good radiator, while a body that radiates poorly will also absorb very little radiant energy. A perfect radiator is defined as a body that absorbs all radiation falling upon it. Since it reflects nothing, such a body would appear black when illuminated with visible light. The perfect radiator is called the blackbody. A hollow box or cavity with a small hole in it approximates this condition. Very little light from the outside can be reflected through this hole. If, therefore, the whole box is heated to a very high temperature, then the resulting light coming from the interior and given off through the hole can be studied, the condition of blackbody radiation is approximated.
To explain the empirical data of blackbody radiation, Planck proposed a radical postulate. Considering Hertz's discovery of the electromagnetic waves emitted by oscillating charges, Planck became convinced that the radiation of light originates in submicroscopic electric oscillators in the cavity walls. He assumed that each oscillator has its own fixed frequency f and emits radiation at that frequency. All frequencies are represented in the enormous number of different oscillators, and hence all frequencies are present in the emitted radiation; the spectrum is continuous. While the oscillators are radiating, they must be losing energy, and thus the emitter will cool unless energy of some form, say heat, is continuously suppied to it. This is the function of the oven around the cavity. The resulting kinetic energy imparted to the particles of the emitter, including oscillators, is further distributed among them because of their incessant collisions. Now when any osciallator absorbs or radiates energy, only the amplitude of vibration changes, for its frequency is fixed. The higher its natural frequency and the greater its amplitude at the moment, the larger is the energy possessed by an oscillator. Now this is consistent with classical principles. But the classical model also implies two other postulates.
(1) the energy E of an individual oscillator can be any amount, from zero upward, and the oscillator can radiate or absorb any amount of energy, with its amplitude continuously changing while it does so.It was from these two classical postulates that Planck now broke away in order to get a theoretical basis. Planck replaced them with two new postulates.
(2) Since any electric charge emits radiation whenever it is accelerated, an electric oscillator must radiate all the time that it is vibrating.
(1) Each oscillator can have only certain definite energies. These allowed energies are integral multiples of a quantity hf, where h is a new universal constant now known as Planck's constant, and f is the frequency of the oscillator. Thus the oscillator's energy E at any moment may be 0, 1hf, 2hf, 3hf, or in general nhf, where n is any integer, but will never be a fraction of hf. That is,The quantization of energy can be graphically pictured by the concept of energy levels. If any individual oscillator has constant frequency f, with only its amplitude changing, then the allowable energy levels which it can reach at one time or another may be pictured as equally spaced rungs on a ladder. At any moment the oscillator, with regard to its energy content, is located on one another of these energy levels, if it is not changing levels. Each step or changes of levels happens by the absorption or emission of energy corresponding to a single quantum of energy of the magnitude hf.
E = hf, (4)
where hf is called the quantum of energy corresponding to the frequency f. In other words, the total energy of oscillator is "quantized" in lumps of energy of magnitude hf each. This means that the change ΔE in the energy of an oscillator must occur, not gradually and continuously, but suddenly and discontinuously.(2) An oscillator radiates only when it changes from one allowable energy value to the next smaller one, and the energy change ΔE that it loses with this sudden decrease in amplitude is emitted as a pulse of electromagnetic radiation of energy hf. Similarly, an oscillator can absorb a quantium of energy hf of incident radiation and consequently it changes immediately to its next higher allowable energy value. According to this postulate, an oscillator does not radiate as long as it remains in any quantum state nhf, even though it is all the time accelerating.
Planck's son later tells that during a walk in the Grunewald in Berlin in 1900 his father announced to him, "Today I have made a discovery as important as that of Newton." Try as he would, Planck could not find a way to explain the experimental data other than by the revolutionary concept of a basic discontinuity in nature. At first, other scientist remained skeptical, unwilling to accept such a departure from the well established principle of continuity of energy. It was not until Einstein applied Planck's hypothesis to photoelectric effects and Bohr applied it to atomic structure that the resistance to the conception disappeared. It was the younger scientists who accepted, applied, and extended Planck's hypothesis.
The photoelectric effect was discovered incidently by Heinrich Hertz in 1887 during the course of his experimental research that at the time furnished the most convincing proof of Maxwell's classical electromagnetic theory. Hertz noticed that electric sparks would jump more readily across the air gap between the metal spheres of the spark gap in the receiving circuit, if they were polished. He soon found that the sparks were influence by the light coming from the sparks in the spark gap of the transmitting circuit. Upon further investigation, he concluded that ultraviolet radiation was responsible for the phenomena and that the effect was greatest when the light fell on the negative terminal (cathode) of the spark gap. Being concerned with other problems, Hertz abandoned further study of the effect. Many others carried on a more detailed investigation of the phenomena. Wilhelm Hallwach showed that this emmission consisted of negative electricity and in 1899 J.J. Thomson published a paper in the Philosophical Magazine in which he showed that the negative electricity ejected by the ultraviolet light have the same e/m ratio as cathode rays. P. Lennard also obtained the same result. These electrons were called photoelectrons.
The photoelectric effect consists in the fact that light upon striking
certain metals cause electrons to be given off, transforming the radiant
energy absorbed by the metal into kinetic energy of emitted electrons.
According classical physics, it would be expected that as the intensity
(thus energy) of light increases, the kinetic energy of the electrons
emitted will increase. This would imply that, as the intensity of the
beam increases, the velocity of the electrons emitted would increase.
The classical view requires that the kinetic energy equal
Ek = ½mev2,
where me is the mass of the electron, and v its
velocity. But experimentally this is not what happens. Instead, the maximum
velocity given to the electron is constant, regardless of the intensity of the
light, but varies in direct proportion to the frequency.
Einstein explained this in the following way. Electrons are embedded in the
atoms of the metal and are held there by attractive forces. When light is
shined on the surface of the metal of sufficient energy, the energy is used
in two ways:
(1) to overcome the attractive forces holding the electron to the metal
atoms, and,
(2) if there is any energy left over, it imparts that energy to the
electron as kinetic energy.
Therefore, for those electrons near the surface of the metal, less energy
is required to overcome the "binding force," and there is more kinetic
energy available. The "surface" electrons therefore acquire the maximum
kinetic energy. Einstein expressed this relation by the following formula:
hf = W +
½mev2, (7)
where W is the work required to free the electrons from the metal and is
equal hf0 where f0 is the threshhold
frequency below which light will not free the electron.
Thus there is a linear relation between the frequency and the maximum kinetic
energy of the electrons emitted. Note that there is nothing in the
equation relating to the intensity of the light causing electron emission.
But although the velocity of the elctrons does not depend in any way on
the intensity of light, the number of electrons emitted does so depend.
From this, Einstein, following Planck, concluded that the light is made
up of quanta or, as Einstein called them, photons. The energy of a
single photon is given by hf. However, the more intense of the light,
the greater the number of electrons emitted. The photoelectric emission of a
single electron depends upon the absorption of a single photon. If the
number of photons is increased, the number of emitted electrons is increased,
but no change occurs in the maximum kinetic energy and hence the velocity of
electrons. The Planck's constant h is the same for all metals. The
threshold energy W required to release electrons, does vary from metal
to metal, being so large in the case of some metals, for example, platinum,
that no photoelectric effect occurs.
Now on the basis of the classical wave theory of light, all these phenomena are impossible to be explained. According that theory the energy of a wave of given frequency is dependent on its amplitude. If light consists of waves, the kinetic energy given to the electrons depends upon the amphlitude of the light waves. This explanation fails to hold true for the photoelectric effect.
Einstein introduced modification of this classical theory of electromagnetic waves to account for photoelectric effect. Planck had modified the classical theory by quantizing the atomic oscillators. But Einstein further modified the theory by quantizing the waves or radiation. Light consists of quanta of energy, called photons. This theory of Einstein that light consists of photons, seemed to be a return to Newton's corpuscular theory. But light is not material particles but is now regarded as composed of bundles of energy, called photons, each photon having a certain amount of energy depending on its frequency or color. In addition, it can be shown that each photon has certain momentum, equal to hf/c, and that as the photons strike a substance and are reflected, the change of momentum causes a small but measurable pressure. But according to the older wave theory light also had momentum and causes pressure. Is light, waves or particles? Since the wave theory cannot explain the photoelectric effect and similar quantum phenomena, and since the particle theory cannot explain many phenomena of interference and diffraction, both theories are needed to explain all these phenomena; they complement each other. Just as light waves have properties like particles, so light particles have properties like waves.
Bohr's original paper was published in the Philosophical Magazine in
July 1913 under the title "On the Constitution of Atoms and Molecules"; other
papers followed from time to time. We will not follow Bohr's own line of
thought exactly because of its advanced mathematical level, and partly because,
as Max Planck himself once exclaimed, Bohr's work is written "in a style that
is by no means simple." We will here faithfully develope his theory with
algebra only.
Bohr begins reviewing Rutherford's atomic model and the difficulties it
raised concerning electronic orbits and light radiation. Then he sets
forth the postulates of his theory that he proposed as a solution of
those difficulties. Bohr at first confined his attention to the element
hydrogen, whose atom is the simplest of all the atoms.
The following are the postulates of Bohr's theory.
Postulate 1. The orbit of electrons are circular with the nucleus at their
center. The force keeping them in orbit is the Coulomb force of attraction
between the electron and the positively charged nucleus. The charge on
the nucleus is equal to the product of Z, the atomic number, times
e1, the unit of charge. Let e2 is the
charge on the electron. Thus the force of attraction is
F = KQ1Q2 / r2 =
K(Ze1)e2 / r2,
where K is the constant of proportionality and is equal
9 × 109 newton-meter2 per coulomb2
in MKS units.
This force is equal to the force accelerating the electron in a circular
orbit, that is,
F = ma = m(v2 / r) =
mv2 / r.
Equating the right side of these equations we get
mv2 / r =
KZe2 / r2, or
mv2 = KZe2 / r, (9)
since e1 = e2 = e, the charge
on an electron.
Thus far, Bohr has introduced nothing new in his model. These equations
are based on known laws of classical mechanics.
Postulate 2. Only certain orbits are permissible and that these orbits
are determined by the law that the angular momentum times 2π must be equal
to a whole number times Planck's constant h. The angular momentum of an
object traveling in circle of radius r is the linear momentum times
r. Thus
2πmvr = nh, (10)
where n is a whole number 1, 2, 3, ... etc.
Bohr here in this postulate has introduced a radical innovation. This is a
quantum condition. Bohr gave no justification for this postulate;
he merely stated it, and showed that it led to verifiable results.
One immediate consequence of these postulates is that
equations (9)
and
(10)
contain two unknowns: v and r. The whole number n is
assigned for the different orbits being discussed. Every other constant is
known. We shall solve
equation (9)
for r,
r = KZe2/(mv2),
and
equation (10)
for v,
v = nh/(2πmr).
We will substitute for v in the first equation from the second equation,
eliminating v from the first equation. We get
r = (n2h2) /
(4π2KZe2m). (11)
For hydrogen of atomic number 1, Z = 1. The other constants are
h = 6.62 × 10-24 joules-sec.
K = 9 × 109 newton-m2/coul2.
e = 1.6 × 10-19 coul.
m = 9.1 × 10-31 kg.
Substituting these values, we get
r = (n2)(5.3 × 10-10 m).
This tells us that for n = 1, when the radius is the smallest and the
atom is in its lowest energy or ground state, the radius of the hydrogen atom
is 0.53 × 10-10 m or 10-8 cm, which agrees with the
values previously obtained for other estimates of the size. And in addition,
it means that the radius can be 1, 4, 9, 16, etc. times as great as its ground
state, when in an excited state.
Postulate 3. An electron may "jump" from one orbit to another and in doing so emits or absorbs energy in quanta. The energy emitted or absorbed must be whole quanta or hf to agree with Planck's theory. This was necessary to explain the discrete spectral emissions. The electron would have to "jump" instantaneously from one orbit to the next, never occupying any intermediate position. At the same instant the electron disappeared from one orbit it would appear in the next. Bohr in this postulate was carrying out Planck's hint of discontinuity in nature one step further.
This postulate was used by Bohr to explain the Balmer seies of spectral lines.
This explanation can be developed in following way. In each orbit the
electron has a total energy which is the sum of its kinetic energy and its
potential energy. That is,
E = Ek + Ep.
These energies are completely determined by the first and second postulates.
Solving
equation (9)
for v2,
v2 = KZe2 / mr,
hence, the kinetic energy
Ek = ½mv2 =
(m/2)[(KZe2) / mr] =
KZe2 / 2r.
Analysis shows that the proper expression of the potential energy
Ep of the electron at a distance r from the nucleus is
Ep = -KZe2 / r.
Thus the total energy E is
E = Ek + Ep =
KZe2/2r +
(-KZe2/r) =
[KZe2/2r - KZe2/r] =
-KZe2/r[(1/2)-1] =
KZe2/r.
Subsituting into this equation the value of r in
equation (11)
and simplifying, we get
En =
-(2π2K2Z2
e4m) /
(n2h2), (12)
where En are the energy levels or states that is
possible for an electron to have and they are determined by the integers
n = 1, 2, 3, ..., called the total or principle quantum number.
The values of n determines the energies of the state. Where n
is large, the energy of the state is large, that is, less negative than
those for small integer n.
Now using postulate 3, the energy difference between any two orbits should
equal to a quantum of energy hf. That is,
hf = En2 - En1 =
[(2π2K2Z2
e4m) /
(nn22h2)] -
[(2π2K2Z2
e4m) /
(nn12h2)].
Thus between an outer orbit where n = n2 = n2
and an inner orbit where n = n1 = n1.
hf =
[2π2K2Z2
e4m / h2]
[(1/n22) -
(1/n12)],
which is the energy difference between an outer orbit where
n = n2
and an inner orbit where
n = n1.
Solving for the frequency of the radiation, we get
f =
[2π2K2Z2
e4m / h3]
[(1/n22) -
(1/n12)]
and since c = fλ, or λ = c/f,
then the wave number k = 1/λ = f/c.
k = f/c =
[2π2K2Z2
e4m / ch3]
[(1/n22) -
(1/n12)] =
RZ2[(1/n22) -
(1/n12)]. (13)
Evaluating the coefficient
R = 2π2K2e4m
/ ch3
(letting Z = 1, for hydrogen),
the value is exactly the same as the Rydberg constant R, that is,
3.26 × 1015 sec-1, within a fraction of one
percent. This equation (13) is the Rydberg formula,
equation (3),
k = R/m2 - R/n2, or
k = R(1/m2 - 1/n2).
where m = n2 and n = n1.
Here is a tremendous triumph for the Bohr model. It gives a numerical
confirmation and systematic explanation of the spectral lines of hydrogen.
But more than this, it also could be used for predicting other spectral lines.
We get the Balmer series when m = 2. And if m = 1, we get a
series that was discovered a year after Bohr announced his model, and was
called the Lyman series. The other series for m = 3 (Paschen series in
1908) and m = 4 (Brackett series in 1922) were already known. Later
the series for m = 5 (Pfund series in 1924) was discovered.
Bohr's model was applied to the spectra of other elements. The spectral lines of a singly ionized helium atom were calculated by Bohr and found to agree with experimental measurement of them. The helium atom, which is normally has two electrons, is singly ionized, if one of the two electrons is stripped from it. But in the case of the heavier elements the exact calculations of frequencies become extremely difficult. There are so many electrons involved that interact with each other, causing such large perturbations in each other's orbit, that ordinary mathematical analysis breaks down. Other methods had to be devised.
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