Chemistry Reference
In-Depth Information
So the frequency of oscillation is proportional to the square root of the spring constant
(the stiffer the spring is, the more rapid the oscillation is) and inversely proportional to the
square root of the reduced mass (the heavier the atoms are, the slower the oscillation is).
The amplitude of the oscillation can now be related to the total energy by substituting
our result for
ω
back into Equation (A6.9):
2
E
k
A
=
(A6.12)
So the amplitude is inversely proportional to the square root of the spring constant and
proportional to the total energy available. This means that (for the same total energy) the
stiffer the spring is, the lower the amplitude of vibration is; this is because the potential
energy increases more sharply with the bond extension.
We have shown that, in the classical picture, distortions due to atom motion that extends
the bond will increase the potential energy and cause a restoring force that tends to
shorten it. Also, if the bond is compressed, then the potential energy again increases,
but the restoring force now favours extension of the bond. The result is oscillation with
energy constantly switching between kinetic and potential. At the equilibrium point, the
potential energy is zero and the kinetic energy is equal to the total. At the maximum exten-
sion/compression (
x
=
A
) the kinetic energy is zero but the potential energy is equal to
the total (to show this, try substituting Equation (A6.7) with
A
from Equation (A6.12) into
Equation (A6.1)).
The total energy itself can take on any value; under thermal equilibrium, the range of
total energies available to each molecule would be set by the Boltzmann distribution and
depends on the temperature of the system.
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A6.2 Model Based on Quantum Mechanics
The classical model works well when describing macroscopic systems of masses and
springs. However, we run into difficulties when trying to apply the same mechanics to
problems at the scale of atoms and electrons. For example, in the vibrating H F molecule
in Figure A6.1 the dipole moment of the molecule would be changing over the cycle of the
oscillation. Such an oscillating dipole should radiate electromagnetic waves. In fact, this is
how radio and television signals are broadcast; the transmitting aerials set up macroscopic
oscillating fields from which radio waves emanate. If the same rules applied to the H F
molecule then it should spontaneously emit radiation at the frequency of the bond vibra-
tion, losing energy until the atoms come to rest. From experimental observation we know
that this is not the case: molecules can vibrate without radiating energy and their atoms are
never at rest.
Quantum mechanics gives us a way to deal with the behaviour of matter at the molecular
level. In this approach, molecular vibrations are stable only in specific stationary states
which define energy levels for the system. While in a stationary state the vibration still
takes place, but no radiation is emitted. Radiation is only emitted or absorbed on transition
between energy levels (see Figure 6.2). The state of a system, such as a vibrating molecule,
is described in quantum mechanics by a wavefunction, and we will see below how the use
of a wave-like description naturally gives discrete energy levels.
