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Unformatted text preview: 20th Century Atomic Theory  Hydrogen Atom Rutherford’s scattering experiments (pp. 4041) in 1910 led to a nuclear model of the atom
where all the positive charge and most of the mass were concentrated in a small nucleus. Elec—
trons were pictured as revolving around the nucleus in a volume whose radius was ~ 100,000
times that of the nucleus. Classical physics dictates that motion is in a straight line unless some
force exists to change the direction. For circular motion there must be a constantly changing
force. For an electron orbiting a nucleus the force is the coulombic force of attraction between
the nucleus of charge Ze (number of protons = Z) and the electron of charge —e. (Consider force F", velocity 13?, and acceleration [2’ to be vectors having both magnitude and direction.) The
coulombic force is radially directed inwards towards the nucleus and it changes its direction as
does the velocity F coul 2 —~ — (magnitude of I?) (1) Stability requires that all of the forces acting upon the electron balance (i. e., Newton — the resul—
tant of the forces is zero). Thus the coulombic force must just .be balanced by the centrifugal
(ﬁctitious) force of the orbiting electron’s motion which is radially directed outwards (let go a
stone tied to a string which you twirl around your head and it ﬂies off away from you): muz F cent = —— (magnitude) (2)
r There was a huge ﬂaw to this otherwise very appealing nuclear model — the model was unstable
(p. 215). Classically an accelerating charged particle radiates energy. Viewed vectorally, an electron
orbiting a nucleus has a constantly changing acceleration. It is easy to picture that this motion
gives rise to an oscillating electron. Picture yourself at the nucleus laying down in the plane of
the electron’s orbit. As the electron completes one cycle of its orbit you observe the electron to
appear to oscillate up when it is closest to your head and then down when it is closest to your
feet: 2 Such an oscillating electron induces oscillating electric and magnetic ﬁelds and generates an
eleCtromagnetic wave. By so doing it emits electromagnetic radiation whose frequency corre—
sponds to the number of revolutions the electron makes about the nucleus per second. In emit—
ting radiation the atom loses some energy and the electron must move in closer to the nucleus. .
To see that this is so let us ﬁnd the energy (classically) of a hydrogen atom and see how it
depends upon r, the distance between the nucleus and the electron. Balancing the coulombic (Eq. 1) and centrifugal (Eq. 2) forces: ' zz 2 ' _%+¥=0 (3)
zz => “$32 (4) The total energy is the sum of the kinetic and potential energies
Etot : Ekin + Epot where the potential energy is the coulomb potential due to the coulombic force of Eq. 1
Ze2 
Epot : Vcoul : _ J Fcouldr : J— dr Ze2
:> Vcoul : _——r—' and the kinetic energy is the familiar 1 Ze2
E . : — 2 = — 6
km 2 m” 2r ( )
where Eq. 3 was used to express mu2 in terms of r. So I
E _ Ze2 Ze2 _ Ze2 _ (7)
mt _ 2r r _ 2r Hence if the system loses energy, AE < 0, ‘ AE . ;Z_62_(_Ze_2]=§i(l_ij
' 2rl 2 rl rf The new radius r f must be less than the original radius ri. Note from Eq. 4 that the velocity
increases as r decreases so the electron spirals into the nucleus faster and faster, emitting radia—
tion of increasing frequency. II
D1
N,
l
31
II In 1912 Bohr Was able to reconcile the stability of the nuclear model by simply saying that
classical physics was wrong in its prediction. Certain stationary orbits were stable. These orbits
are characterized by a particular radius and energy which he found by arbitrarily postulating that the angular momentum I: of the electron was quantized. The general idea of quantization was
not new: 1) in 1900 Planck quantized the material oscillators in a solid — those that could emit
blackbody radiation in multiples of hv (p. 219) and 2) in 1905 Einstein quantized radiation in his
explanation of the photoelectric effect (pp. 219—220). However, Bohr was the rst to apply quan
tization to the structure of the atom and his theory correctly predicted the emission spectral lines of hydrogen (pp. 221—227).
To understand angular momentum cogsider the ﬁgure accompanying Eq. 1 and view it 'side on L
A
m Up 69—? ‘5'
P For counterclockwise circular motion in the x,y—plane (this piece of paper) the angular momen— tum vector f: would point in the positive z direction (coming out of the paper towards you ).
Angular momentum is deﬁned as the cross product E = ?X(ml,7) = m(?><17¢’)
= m(ru sin 0) (magnitude) 2 mar (when sin 19 = 1) where 6 is the angle between 17’ and the momentum mi (900 for circular motion). Bohr’s condi—
tion nh ‘
L=mur=.— n=1,2, (8)
. 27: v
We can use the quantization condition of Eq. 8 with Eq. 4 to ﬁnd Bohr’s stable orbits
Ze2 Ze2 mr2 Zezmr2 47rzZe2mr2
r = — : — — : ~ _—
mu2 mu2 mr2 (ml/tr)2 nzh2
11th I n2 i => rn=m=zao “1’2"”. (9) where a0 is the Bohr radius. To ﬁnd the quantized energy levels we need to only substitute this
ﬁnal value for r into the total energy of Eq. 7 Ze2 Z _ z2 _ e2
2 nzao _ n2 2ao 22 [’12 . . . I 2 .
_ __. (e11m1nat1ng e w1th Eq. 9) En: n2 Sﬂzmag z2 .
=—_2Ry 71:1,2’... (10*) n where Ry is the Rydberg constant. In explaining the emission spectrum of hydrogen, Bohr postulated that the hydrogen atom
could only make a transition to a lower energy quantized state by emitting a photon whose
energy exactly matched the difference in energy between the two states: hc
EPhoton = hVPhoton : "AEatom : _(Ef T Ei)
photon
z2 Z2 1 1
—————R———R =Z2R ————
l: y ( yj] yn} 2 1 l *
=> hv=ZRyn—2—? niznf+1,nf+29'” f i 
or, as in the week 9 lab,
1 _ ZZRy 1 1
A — hc Summary of Bohr’s postulates for hydrogenic atoms: 1. Electron travels in a circular orbit 2. A set of discrete orbits exist which are stable 3. Angular momentum is an integral multiple of h/27r 4. Transitions occur between stable orbits only upon absorption or emission of a photon
where IAEI = hvpfloton Bohr’s theory was immensely successful in explaining hydrogenic spectra (Eq. 11) and
predicting the energy levels (Eq. 10). However it is now known that the orbits are not circular
and the angular momentum is not integral (though it is still quantized). His theory could not be
applied to any atom having more than one electron. Furthermore it provided no rationale for
covalent chemical bonding or reason for quantization or why the atom should not radiate its
energy. Nevertheless Bohr opened the way for the quantization of the energy of atoms and mole
cules. Modern quantum theory was developed during the 1920’s. In 1924 de Broglie proposed
that any particle which has linear momentum mu has wave—like properties and a wavelength 2
associated with it. The de Broglie relation gives their mathematical relationship (p. 228230) mini 2 h (12*) In 1925 Heisenberg presented a consistent theory based on matrix mechanics. At the time
it appeared somewhat obscure as it involved the mathematics of matrices. Later in 1925 Dirac
introduced a theory based on Hamilton’s classical equations of motion which were developed a
century earlier. Dirac translated his theory into a series of postulates, creating a versatile system
of quantum mechanics though somewhat abstract. 1926 saw the introduction of Schrodinger’s
wave mechanics (section 7.4), still popular today. He took de Broglie’s wave idea and Bohr’s
stationary states and concluded that the equation of motion must be a wave—like equation with
boundary conditions which fix the energy levels (just like a differential equation). The mathe—
matical statement of his equation is [qt/IZEl/l where E is the energy, PI is the operator for kinetic and potential energy, and y) is the solution to 5 the equation — the wavefunction. To see a simple H, consider a one—dimensional particle of
mass m conﬁned to move in a onedimensional box on the x—axis. It’s Schrodinger equation is hz dzwx) _ .
—8ﬂ2m dxz — Eel/(x) (13) where dzl/l/dx2 is the second derivative of the wavefunction y/(x) (the Schrodinger equation for
the hydrogen atom can be found on the bottom of 'p. 232). The solutions to this equation, the l//, I
we have already encountered when we saw the standing waves for the spring in the class demon—
stration. We interpret the wavefunction as suggested by Max Born (p. 232): the square of the
wavefunction, ly/(x)2, is proportional to the probability of ﬁnding the particle at x. Notice that
we never know exactly where the particle is only its probability. This accords very nicely with
Heisenberg’s uncertainty principle of 1927 (p. 231) Flu/(x) = h .
A(mu)Ax 2 Z; (14*) which states that we can never precisely and simultaneously know (or measure) the momentum
and position of a particle. There will always be an uncertainty of at least 11/47: in the product of
the uncertainty in mu and the uncertainty in x. So Bohr’s stable orbit does not exist. Since we
know the w solutions to Eq. 13, how about the energies? Without solving the Schrodinger equa—
tion one can solve for E. Let L be the length of the box that the particle moves in. An integral
number of half wavelengths must ﬁt into the length L in order to have the constructive interfer—
ence necessary to create a standing wave (or to. produce a pleasant sounding harmonic on a
stringed instrument) 11/1
2 From the de Broglie relationship of Eq. 12 we can relate this wavelength )1 to the particle’s linear
momentum mu =L n=1,2, . (15) h "h 1 2 16)
mu :  2 ~— :  ~ 
2 2L " ’ ’ (
Using the formula for kinetic energy
1
E  = — 2
km 2 m“
and substituting into Eq. 16 for the momentum
(mu)2 1 nh 2 r1th .
Em: =—— = =1,2, 17
1‘ 2m 2m 2L 8mL2 ” _ .( ) The last expression for the energy is exactly what we would have obtained had we solved Eq. 13. Finishing the development of quantum theory: in 1926 Schrodinger shows that his
approach is identical to those of Heisenberg and Dirac. In 1928 Dirac develops relativistic quan
tum mechanics where the spin quantum number ms emerges. In the two cases that we have looked at so far, Bohr’s hydrogen atom with the electron
traveling on the circumference of a circle (Eq. 10) or the particle in a one—dimensional box (Eq.
17), a quantum number n appears upon which the energy is dependent. An electron in a real
hydrogenic atom moves in three—dimensional space so three quantum numbers (p. 234) are 6 necessary to specify the state of the electron (one quantum number for each degree of freedom or
dimension). When fully treated by considering relativistic effects, a fourth quantum number
becomes necessary to fully specify the state. n = l, 2,    is the principal quantum number which determines the energy of the elec—
tron. n indicates the effective volume in space in which the electron moves. If n = l the electron
is more likely to be closer to thenucleus than if n = 2, 3,   l = 0, l, . . . , n — l is the angular momentum (azimuthal) quantum number which deter
mines the magnitude of the electron’s angular momentum. l designates the shape of the volume
or region in space that the electron occupies. The integer specifying l is generally replaced by a
letter: ' II II
VV II
V H
#UJNl—‘O
II
V
GOVsafes“ II
V ml 2 — l, — l + l, . . . , O, ..., l — 1, l is the magnetic quantum number which determines
the orientation in space of the angular momentum and heme the magnetic moment. ml indicates
the orientation in space of the volume or region that the electron occupies. _ The wavefunction for a hydrogenic atom depends upon n, l, and ml and is referred to as an
orbital such as the three 2p orbitals of Figure 7.17 (n = 2, l = l). The three 2p orbitals corre—
spond to three different values for ml (—l,0,l). ' Dirac’s relativistic treatment showed that a fourth quantum number was needed to fully
specify the state of an electron, the spin quantum number ms with values —l/2 or +1/2 corre— 
sponding to two different orientations of spin (p. 247). ...
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