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Samenvatting Physical Chemistry, alles wat je voor het tentamen moet weten
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- Vrije Universiteit Amsterdam (VU)
Samenvatting Physical Chemistry, alles wat je voor het tentamen moet weten
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Topic 7: Quantum Theory7A: The origins of quantum mechanics
Classical mechanics developed by Newton is a successful theory for describing motion.
However, some observations were made that could not be explained by classical
mechanics. In these cases, classical mechanics was replaced by quantum mechanics.
Electromagnetic radiation: oscillating electric and
magnetic disturbances that propagate as waves. The
two components are perpendicular to each other,
and to the direction of propagation. In a vacuum
these waves travel at the speed of light, c =
2.99792458 x 108 m s-1. A wave is characterized by
wavelength . The properties of a wave can also be
expressed in terms of frequency, (Greek letter nu).
These are related as follows: c =
It is also common to describe a wave in terms of its
wave number, ṽ = 1/ = /c
All objects emit electromagnetic radiation over a range of frequencies with an intensity
dependent on the temperature. The radiation emitted by hot objects is described in
terms of a black body.
Black body: a body that emits and absorbs electromagnetic radiation without favoring
any wavelengths. At each temperature T there is a wavelength, max, at which the
radiation intensity is at a maximum. T and max are related by Wien’s law: max T =
constant = 2.99 mmK
The energy density, E(T), is the total energy inside the container divided by its volume.
∞
E ( T )=∫ ρ ( λ , T ) dλ . The units of E(T) are joules per meter cubed (Jm-3).
0
Stefan-Boltzmann law: E(T) = constant x T4 = 7.567 x 10-16 J m-3 K-4 x T4
According to classical physics, every oscillator is excited to some extent, and according to
the equipartition principle every oscillator, regardless of its frequency, has an average of
kT. On this basis, the Rayleigh-James law was deduced:
8 πkT
ρ ( λ , T )= 4 (where k is the Boltzmann’s constant, k = 1.381 x 10-23 J K-1)
λ
The Rayleigh-James law is inconsistent with Wien’s law. It implies that the concentration
of radiation at very short wavelengths is intense and becomes infinitely intense as the
wavelengths tend to zero. This concentration at short wavelengths is called the
ultraviolet catastrophe and is an unavoidable consequence of classical physics.
Planck found that the experimentally observed intensity distribution of black-body
radiation could be explained by proposing that the energy of each oscillator is limited to
discrete values. He assumed that for an electromagnetic oscillator of frequency v, the
permitted energies are integer multiples of hv. (h = Planck’s constant = 6.626 x 10-34 Js)
The limitation of energies to discrete values is called energy quantization. On this basis,
Planck was able to derive an expression for the energy spectral density which is now
called the Planck’s distribution. This expression fits the experimental data very well at all
wavelengths and looks as follows:
8 πhc
ρ ( λ , T )= hc
5
λ (e λkT −1)
, Heat capacity: when energy is supplied as heat to a substance its temperature rises. The
heat capacity is the constant of proportionality between the energy supplies and the
temperature rise. (C = dq/dT and at constant volume: CV,m =((Um/T)V))
At room temperature it was found that the molar heat capacity is 3R, where R is the gas
constant. At much lower temperatures, it was found that the heat capacity decreased,
tending to zero as the temperature approached zero. This was not explicable with
classical physics. Einstein came up with a different expression, which was able to explain
this phenomenon:
θE
( )
2
θE e2T 2
C V ,m ( T )=3 R f E (T ), with f E ( T )= ( θ )
T
E
T
e −1
Within this formula E is the Einstein temperature (E =hv/k)
The physical reason is that as the temperature is lowered, less energy is available to
excite the atomic oscillations. At high temperatures, many oscillators are excited into
high energy states leading to classical behavior. However, one discrepancy arises from
Einstein’s formula: this is because he assumes that all atoms oscillate with the same
frequency.
The most compelling evidence for the quantization of energy comes from spectroscopy,
the detection and analysis of the electromagnetic radiation absorbed, emitted, or
scattered by a substance. The record of variation in the intensity of this radiation is called
its spectrum.
If the energy of an atom decreases by E, and this energy is carried away as radiation,
the frequency of the radiation v and the change in energy are related by the Bohr
frequency condition: E = hv
Wave-particle duality: the blending together of the characteristics of waves and
particles, lies at the heart of quantum mechanics. The Planck treatment of black-body
radiation introduced the idea that an oscillator of frequency v can have only the energies
0, hv, 2hv, etc… This leads to the suggestion that the resulting electromagnetic radiation
of that frequency can be thought of as consisting of 0, 1, 2, etc… particles, each particle
having an energy hv. These particles of electromagnetic radiation are now called
photons.
Thus, if an oscillator of frequency v is excited to its first excited state, then one photon of
that frequency is present, if it is excited to its second excited state, 2 photons of that
frequency are present and so on.
The observation of discrete emission spectra from atoms and molecules can be pictured
as the atom or molecule generating a photon of energy hv when it discards an energy of
magnitude E, with E = hv.
Note that one transition generates 1 photon, not a shower of photons.
Each photon has an energy hv, so the total number N of photons needed to produce
energy E is N = E/hv. To use this equation, you need to know the frequency of the
radiation (from v = c/) and the total energy emitted. The latter is given by the product
of the power (in Watts) and the time interval, t, thus: E = Pt.
E Pt λP t
N= = =
Thus: hv c hc
h( )
λ
So far, the existence of photons is only a suggestion. Experimental evidence for their
existence comes from the measurement of the energies of electrons produced in the
, photoelectric effect, the ejection of electrons from metals when they are exposed to
ultraviolet radiation. The experimental characteristics of the photoelectric effect are as
follows:
o No electrons are ejected, regardless of the intensity of the radiation, unless its
frequency exceeds a threshold value characteristic of the metal.
o The kinetic energy of the ejected electrons increases linearly with the frequency
of the incident radiation but is independent of the intensity of the radiation.
o Even at low radiation intensities, electrons are ejected immediately if the
frequency is above its threshold value.
These observations strongly suggest that in the photoelectric effect a particle-like
projectile collides with the metal and, if the kinetic energy of a projectile is high enough,
an electron is ejected. If the projectile is a photon of energy hv (v is the frequency of
radiation), the kinetic energy of the electron is Ek, and the energy needed to remove an
electron from the metal, which is called the work function, is , then the conservation of
energy implies that:
hv = Ek + or Ek = hv -
This model explains the three experimental observations:
o Photo ejection cannot occur if hv < , because the photon brings insufficient
energy.
o The kinetic energy of an ejected electron increases linearly with the frequency of
the photon.
o When a photon collides with an electron, it gives up all its energy, so electrons
should appear as soon as the collisions begin, provided the electrons have
sufficient energy.
De Broglie relation: any particle, not only photons, travelling with a linear momentum
h
p = mv should have in some sense a wavelength given by: λ=
p
Meaning, a particle with a high linear momentum has a short wavelength. Macroscopic
bodies have such high momenta even when they are moving slowly, that their
wavelengths are undetectably small, and the wave-like properties cannot be observed.
This is why classical mechanics can be used to describe the behavior of macroscopic
bodies. It is only necessary to use quantum mechanics for microscopic bodies, such as
atoms and molecules, in which the masses are small.
To use the De Broglie relation, we need to know the linear momentum, p, of the
electrons. To calculate this, note that the energy acquired by an electron accelerated
through a potential difference is e, where e is the magnitude of its charge. At the
end of the period of acceleration, all the acquired energy is in the form of kinetic energy,
Ek = ½ mev2 = p2/2me. You can therefore calculate p by setting p2/2me equal to e.
7B: Wavefunctions
In classical mechanics an object travels along a definite path or trajectory. In quantum
mechanics a particle in a particular state is described by a wavefunction , which is
spread out in space, rather than being localized. The wavefunctions contain all the
dynamical information about the object in that state, such as its position and
momentum.
Schrödinger equation: An equation was proposed for finding wavefunctions of any
system. The time-independent Schrödinger equation for a particle of mass m moving in
, one dimension with energy E in a system that does not change with time (for instance, its
volume remains constant) is:
−ℏ2 d 2 ψ
+V ( x ) ψ=Eψ
2m d x 2
The constant ℏ=h/2 π is a convenient modification of Planck’s constant, used widely in
quantum mechanics. V(x) is the potential energy of the particle at x. Because the total
energy E is the sum of potential and kinetic energies, the first term on the left must be
related to the kinetic energy of the particle.
The plausibility of the Schrödinger equation can be demonstrated with the De Broglie
relation: the first step is to note the potential energy V(x) is zero everywhere, so E is
equal to the kinetic energy E = p2/2m, thus:
2 2 2
−ℏ d ψ p
= ψ ( x)
2m d x 2 2 m
A wave of wavelength has the form (x) = sin(2x/), so the left-hand side of the
equation is:
( )
d 2 sin
2 πx
λ
( ) ( )
2 2 2
−ℏ d ψ −ℏ −ℏ2 2 π 2 πx h2
= = sin = ψ (x)
2m d x 2 2m dx
2
2m λ λ 2mλ
2
So, p2/2m = h2/2m2, and therefore p2 = h2/2, which implies the De Broglie relation: =
h/p.
The Born interpretation: the square of the amplitude of an electromagnetic wave in a
region is interpreted as its intensity and therefore (in quantum terms) as a measure of
the probability of finding a photon present in the region. The interpretation is as follows:
If the wavefunction of a particle has the value at x, then the probability of finding a
particle between x and x + dx is proportional to ||2dx. ||2 is the complex conjugate of
, allowing for the possibility of being complex. If the wavefunction is real: ||2 = 2
Because ||2dx is a dimensionless probability, ||2 is the probability density, with the
dimensions of 1/length. The wavefunction itself is called probability amplitude. For a
particle free to move in three dimensions, the wavefunction depends on the coordinates
x, y, and z and is denoted (r). In this case the Born interpretation is as follows: If the
wavefunction of a particle has the value (r) at r, then the probability of finding the
particle in an infinitesimal volume d = dxdydz at that position is proportional to |(r)|
2
d. The Born interpretation gets rid of any worry about the significance of a negative
value of , because ||2 is always real
and nowhere negative. Only the square
modulus is directly physically
significant, and both negative and
positive regions of a wavefunction may
correspond to a high probability of
finding a particle in a region. However,
the presence of positive and negative
regions of a wavefunction is of great
indirect significance because it gives
rise to the possibility of constructive
and destructive interference between
different wavefunctions.
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