Geoscience Reference
In-Depth Information
0
OH
OD
3
2
1
0
Zero-point
energy
Nuclei separation
Energy of the oxygen
----
hydrogen bond. The potential of the O
----
H bond as a function of the
distance between the two nuclei is shown as a continuous curve. The zero represents the energy
of the system when O and H are fully dissociated. Because the heavy nucleus hardly budges when
the light electrons wiggle, this curve is independent of the mass of the nuclei (Born-Oppenheimer
approximation) and is therefore the same for all isotopes. The quantized energy levels (
n
Figure 3.2
=
0, 1,
)areshownfortheO
----
H(oxygen
----
hydrogen) and O
----
D (oxygen-deuterium) bonds. The
first energy levels are nearly equally spaced. The isotopically heavier molecules are in a lower
state of energy than the lighter molecules. Zero-point energy is the elevation of the ground state
(
n
2,
...
0) above the minimum of the potential well. Differences in zero-point energies between
molecules with different isotopes (e.g. O
----
HandO
----
D) account for all stable isotope fractionation
effects.
=
at
r
0
and show that it measures the force of the spring. Stretching the bond by applying a
force
F
increases its potential energy and this takes the form:
d
V
d
r
F
=−
(3.4)
with the minus sign ensuring that energy increases when the force stretches the bond.
Comparing the two equations leads to Hooke's law, which we already met in the previous
chapter:
F
=−
k
(
r
−
r
0
)
(3.5)
The constant
k
is therefore a measure of the “hardness” of the spring, i.e. of how
much energy is needed to achieve a particular change in the bond length. Now,
let us summon our college physics and remember one of Newton's great discover-
ies:
F
M
d
2
r
d
t
2
, where
M
is the mass, or force and acceleration are proportional.
=
/
