Download The structure of atoms

The structure of atoms
Atomic models
The Rutherford experiment
Bohr's theory of the hidrogen atom
The Frank-Hertz experiment
Atomic models
Early ideas
One of the most intriguing questions, to which already the ancient greeks seeked the
answer, is: what is the world composed of?
The greek philosopher Democritus is considered the first who proposed the existence of
atoms. He argued that the world is made up of indivisible particles (atomos = indivisible,
greek), and the vaccum inbetween.
The real turn in this view was brought by the end of the XIX. century: it turned out that
atoms are not indivisible. Based on his experiments J. J. Thomson found that cathode
rays emerging from a metal electrode consist of negatively charged identical particles,
that hence must be a component of every atom. In 1897 the particle was named electron.
Thomson’s atomic model consisted of a positively charged material filling the volume of
the atom, and the point-sized electrons embedded in this material like plums in a pudding
(“plum-pudding model”). However, this model was unable to explain some
experimentally determined properties of atoms.
Atomic models
Atomic models are pictures created about the structure of the atoms. A model is valid, if
it theoretically explains the experimental findings, like:
1) atoms are stable;
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2) their chemical properties show periodicity (periodic table of elements, Mendeleev,
1869);
3) light emitted by excited atoms has a line spectrum, e.g. it contains only components of
certain frequencies.
Already in 1802 it was discovered that black lines appear in the spectrum of the Sun.
Fraunhofer identified nearly 576 such lines, which he named by the letters A, B, C, D, ...
This is where the sodium-D name of the yellow line of sodium comes from. Fraunhofer
also found that spectral lines of sodium placed in a flame match exactly some of the black
lines found in the spectrum of the Sun. In 1822 Herschel found that composition of salts
can be identified from their emission spectrum. This was the beginning of spectral
analysis.
When excited, atoms emit light with a line spectrum characteristic to the given element.
Moreover, they can absorb light of the very same frequencies (or wavelengths) with the
emitted one. This explains the origin of the black lines in the spectrum of the Sun.
Helium was identified in 1866 based on its lines from the Sun’s spectrum.
One of the greates challenges for atomic models was to explain why atoms have line
spectra, and what defines the wavelengths of spectral lines. In the beginning only
empirical formulas were derived which predicted some mathematical order in these
wavelengths. A famous formula given by J. J. Balmer, described perfectly the
wavelengths ( λ ) of the visible lines of hidrogen, but it had no theoretical background:
æ1 1 ö
1
÷÷ ,
= R çç −
λ
è 4 n2 ø
where n = 3, 4, 5, ..., an integer number, with each value defining one wavelength, and R
is a constant, named the Rydberg constant.
It is an interesting coincidence that in 1885, the year when the formula was created, was
born Niels Bohr, whose atomic model explained this formula.
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The Rutherford experiment
By studying the scattering of alpha-particles (2+ He nuclei) on a thin metal foil, Ernest
Rutherford proved that the positive charge is not evenly distributed in the atom, as
Thomson thought before, rather it is concentrated into a very small volume, the atomic
nucleus.
Figure 1. Rutherford’s experiment.
a. experimental setup;
b. trajectory of alpha-particles in the electric field of the nucleus.
When directed onto the gold foil, most of the alpha-particles penetrated the foil without
being deflected at all. But most surprisingly, there were particles scattered at very high
angles, even "reflected". Rutherford said: "It was quite the most incredible event that ever
happened to me in my life. It was almost as incredible as if you had fired a 15-inch shell
at a piece of tissue paper and it came back and hit you."
He could only explain these results by assuming that the positive charge is highly
concentrated in the middle of the atom, the atomic nucleus.
Rutherford’s atomic model (1911) resembles a miniature solar system. The nucleus is
situated in the center of the atom:
- its dimensions, as estimated by Rutherford, are in the order of 10-15 m (for comparison:
the radius of the atom is approximately 10.000 times greater),
- its charge is Z×e (Z is the atomic number, e is the electron charge),
- its mass is practically equal to the mass of the atom.
Electrons move on circular orbits around the nucleus, held on their orbit by the
electrostatic attraction of the nucleus.
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Although it was a huge step in deciphering the structure of the atom, Rutherford’s model
had a great deficiency: such an atom was theoretically unstable. According to the laws of
electrodynamics an electron moving on a circular orbit must radiate energy, thus slow
down and fall into the nucleus on a spiral trajectory. This contradiction was solved by
Niels Bohr, danish physicist, who was maybe the greatest genius of the first part of the
XX. century.
Bohr's theory of the hidrogen atom
Bohr's atomic model described the structure of electronic shells for the simplest atom, the
hidrogen. To resolve the contradictions emerging from the Rutherford-model, Bohr
introduced his postulates.
Bohr’s postulates
1) Electrons in an atom can only have defined orbits. The formula defining the radius of
the allowed orbits is:
L = mvr = n
h
,
2π
where L is the angular momentum of the electron (L=mass×velocity×radius), n is the
principal quantum number, and h is the Planck constant. According to the formula, the
value of L can only be the multiple of h / 2π . The formula can be understood if we
rewrite it using the de Broglie formula for the electron wave:
2π r = nλ = n
h
,
mv
where λ is the wavelength of the electron. According to the formula the wavelength of
the electron must fit on the circumference of the orbit integral times (see Figure 2), e.g.
the wave must join itself in the same phase after going once around the orbit. If this does
not occur, the wave would gradually destroy itself by interference. As Bohr postulated, an
electron on these defined orbits does not radiate.
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Figure 2. Example: the wavelength of the electron fits on the circumference 5 times.
2) When the electron jumps from one allowed orbit to another, the energy difference of
the two states is emitted as a photon with the energy of hf:
hf = E 2 − E1 ,
where E2 and E1 are the energies of the electron in the two states.
Consequences of Bohr’s theory
Based on his theory, Bohr calculated the parameters of the electronic orbits in the
hidrogen atom. His theory predicted the following:
1) In the hidrogen atom the radius of the first orbit is r1=5.3×10-11 m, called the Bohrradius. The next orbits are situated farther from the nucleus:
r2 = 4r1
r3 = 9r1
…
rn = n 2 r1 .
2) The energy of the electron on the first orbit is E1 = –13,6 eV. The energy is negative
because the electron in the atom is in a bound state, and energy must be supplied to bring
the electron to the free, unbound state (a free electron has zero energy). The energies of
the next orbits are:
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E1
4
E
E3 = 1
9
E2 =
…
En =
E1
n2
.
Figure 3.
a. Radii of electron orbits in a hidrogen atom according to Bohr’s model.
b. Graphical representation of the energies of the orbits. The free state corresponds to n = ∞ .
Arrows show possibile electron transitions.
One of the greatest achievements of the Bohr-model was that it explained the line
spectrum of hidrogen, and predicted exactly the observed wavelengths. It also gave a
theoretical explanation for the Balmer-formula: the wavelengths correspond to the
“jump” of the electron to the second state from higher energy states (see Figure 3b.).
According to Bohr’s theory:
æ 1
1 ö
÷÷ ,
hν = E 2 − E n = E1 çç
−
è 22 n2 ø
From the Balmer formula:
hν = h
structure of atoms
æ 1
c
1 ö
÷÷ .
= hcRçç
−
λ
è 22 n 2 ø
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The match was perfect. From his theory Bohr was also able to calculate the value of the
Rydberg constant, which was the same as the experimentally determined values.
Although Bohr's postulates were questionable, the predictions of his theory were in a
very good aggreement with the experimental findings.
The Franck-Hertz experiment
Franck and Hertz studied the collision of electrons with mercury (Hg) atoms.
Figure 4. The experimental setup of the Franck-Hertz experiment.
The electrons were accelerated by an electric field (Figure 4.) between the cathode (C)
and the grid (G) in a tube filled with Hg vapour. After passing through the grid, an
opposite field slowed down the electrons and prevented them from reaching the anode
(A), unless they gained enough kinetic energy in the previous acceleration. The electrons
reaching the anode formed a measurable current, I.
Observations
1) When increasing the accelerating voltage, the kinetic energy of the electrons also
increases. Then, at first the current increases (see first part of Figure 5.), because the Hg
atoms, although the electrons collide with them, do not absorb energy from the electrons.
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Figure 5. Dependence of the current from the accelerating voltage.
2) When the accelerating voltage reaches 4.9 V, the current suddenly drops. In this case
the energy of the electrons reaches 4.9 eV near the grid, and at this point they loose their
energy when colliding with Hg atoms. The electrons will have no remaining kinetic
energy, hence they cannot reach the anode, because of the decelerating field.
3) When further increasing the voltage, the "energy-loosing" collision occurs closer to the
cathode, and the remaining distance is enough for the electrons to acquire enough energy
to reach the anode: the current increases again.
4) However, when reaching 9.8 V, after the first collision when they lose their energy, in
the second part of their path the electrons acquire again the 4.9 eV energy, which they
loose again in a second collision near the grid: the current drops.
Conlclusion
The Hg atoms can not absorb any energy, just well defined values, namely 4.9 eV. This
energy excites the ground state electron to the first excited state, e.g. it equals exactly to
the energy difference between the E1 and E2 states of the Hg atom. Thus, this experiment
gives an excellent proof of the validity of Bohr’s theory.
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