Stable Isotope
Geochemistry:
Theory
Lecture 35
Fractionations
• Isotope fractionation can
originate from both kinetic
effects and equilibrium
effects.
• Equilibrium Fractionations:
o
Quantum mechanics predicts that
the mass of an atom affects its
vibrational motion, and therefore
the strength of chemical bonds. It
also affects rotational and
translational motions. From an
understanding of these effects of
atomic mass, it is possible to predict
the small differences in the
chemical properties of isotopes
quite accurately.
• Kinetic Fractionations
o
Lighter isotopes form weaker bonds
and therefore react faster. They
also diffuse more rapidly. These also
lead to isotopic differences.
Equilibrium Fractionations
•
•
•
•
•
Equilibrium fractionations arise from
translational, rotational and vibrational
motions of molecules in gases and
liquids and atoms in crystals.
Isotopes will be distributed so as to
minimize the vibrational, rotational,
and translational energy of a system.
All these motions are quantized (but
quantum steps in translation are very
small).
Of the three types of energies,
vibrational energy makes by far the
most important contribution to isotopic
fractionation. Vibrational motion is the
only mode of motion available to
atoms in a solid.
These effects are small. For example,
the equilibrium constant for the
reaction:
½C16O2 + H218O ⇄ ½C18O2 +
H216O
is only about 1.04 at 25°C and the ∆G
of the reaction, given by -RT ln K, is only
-100 J/mol
Predicting Fractionations
• Boltzmann distribution law states that the probability of a
molecule having internal energy Ei is:
gi e- Ei /kT
Pi =
å gne- Ei /kT
n
o g is a weighting factor to account for degenerate states.
• The denominator is the partition function:
Q = å gn e- Ei /kT
n
• From partition functions we can calculate the
equilibrium constant:
K = Õ Qini
i
• We can divide the partition function into three parts:
Qtotal = QtransQrotQvib
o The separate partition functions can be calculated separately.
Partition Functions
• The following applies to diatomic molecules, for which the
sums in the partition function fortunately have simple solutions.
Principles are the same for multiatomic molecules and crystals,
but the equations are more complex because there are many
Note error in book:
possible vibrations and rotations.
remove 2 in
• Vibrational Partition Function: - hn /2kT
Qvib =
o
e
1- e- hn /kT
exponential term in
denominator
where h is planks constant (converting frequency to energy) and ν is the vibrational
frequency of the bond.
• Rotational Partition Function
8p 2 IkT
Qrot =
s h2
o
where I is the moment of inertia I = µr2; i.e., the reduced mass (µ) of atoms times
bond length.σ is a symmetry factor; σ= 1 for a non-symmetric molecule (18O16O) and
2 for a symmetric one (16O16O)
• Translational partition function for molecule of mass m is
(derived from Schrödinger’s equation for particle in a box):
Qtrans
(2p mkT )3/2
=
V
h3
Partition Function Ratios
K = Õ Qin
• Since
i
• what we really want is the ratios of partition functions for
isotope exchange reactions. Most terms cancel (including
bond length).
• This ratio for two isotopic species (isotopologues) of the same
diatomic molecules, e.g., 16O18O and 16O16O, will be:
i
- hn
Q16 O 18 O
Q16 O
2
hn 16 O 18 O e 16 O 18 O
3/2
- hn 16 /kT
-hn 16 18 /2 kT
H O
- hn 16 18 /kT µ16 O 18 O m16 O 18 O
)µ16 O 18 Os 16 O m163/2O 18 O
s 16 O 18 O kT 1- e O O
n 16 O 18 O e O O (1- e 2
2
=
=
- hn 16 /2 kT
- hn 16 /2 kT
- hn 16 18 /kT
O
O
2
2
hn 16 O e
n 16 O e
(1- e O O )µ16 O s 16 O 18 O m163/2O
3/2
2
2
2
2
µ16 m16
s 16 O kT 1- e-hn 16 O2 /kT O2 O2
2
/2 kT
• We see the partition function ratio is temperature dependent
(which arises only from the vibrational contribution:
temperature canceled in other modes).
• We also see that we can predict fractionations from
measured vibrational frequencies and atomic and molecular
masses.
Temperature Dependence
•
•
•
•
The temperature dependence is:
e- hn /2kT
Qvib =
1- e- hn /kT
At low-T (~surface T and below), the
exponential term is small and the
denominator approximates to 1.
Hence
hn
- hn /2kT
Qvib » e
= 1-
a = A+
•
O isotope fractionation
between CO2 and H2O
B
T
At higher temperature, however, this
approximation no longer holds and α
varies with the inverse square of T:
a = A+
•
2kT
and the fractionation factor can be
expressed as
B
T2
The temperature dependence leads to
important applications in
geothermometry & paleoclimatology
Kinetic Fractionations:
Reaction Rates
•
•
•
•
Looking again at the hydrogen
molecular bond, we see it takes
less energy to break if it is H-H
rather than D-H.
This effectively means the
activation energy is lower and the
rate constant, k, will be higher: DH
will react faster than H2.
We can calculate a kinetic
fractionation factor from the ratio
of rate constants:
kD e-(e -1/2hn D )/kT
a=
=
kH e -(e -1/2hn H )/kT
This will make no difference if the
reaction goes to completion, but
will make a difference for
incomplete reactions. (good
example is photosynthesis, which
does not convert all CO2 to
organic carbon).
Kinetic Fractionation:
Diffusion
• Lighter isotopic species will diffuse more rapidly.
o Energy is equally partitioned in a gas (or liquid). The translational kinetic
energy is simply
E = ½mv2.
o Consider two molecules of carbon dioxide, 12C16O2 and 13C16O2, in a gas.
If their energies are equal, the ratio of their velocities is (45/44)1/2, or 1.011.
Thus 12C16O2 can diffuse 1.1% further in a given amount of time at a given
temperature than 13C16O2. But this applies to ideal gases (i.e., low
pressures where collisions between molecules are infrequent.
o For the case of air, where molecular collisions are important, the ratio of
the diffusion coefficients of the two CO2 species is the ratio of the square
roots of the reduced masses of CO2 and air (mean molecular weight
28.8):
µ12 CO -air 4.1906
D12 CO
o
2
2
=
=
= 1.0044
D13CO
µ13CO -air 4.1721
2
2
o leading to a 4.4‰ fractionation (actually observed).
Rayleigh
Distillation/Condensation
•
•
Different isotopologues of water
evaporate at different rates and
have different condensation
temperatures.
We can imagine two ways in which
condensation occurs:
o
o
•
•
droplets remain in isotopic equilibrium with vapor
droplets do not remain in equilibrium: fractional
condensation
If the fractionation between vapor
and liquid is α, for fractional
condensation, the fractionation, ∆,
varies with fraction of vapor
remaining, ƒ, as:
∆ = 1000(ƒα-1-1)
For equilibrium condensation it is:
æ
ö
1
∆ = ç 1´1000
è (1- f ) / a + ƒ ÷ø
•
Fractional condensation can lead to
quite extreme compositions of
remaining vapor.
Isotopic composition of vapor when
the fraction of original vapor, ƒ,
remains.
Isotope Fractionations
• As a rule, heavy isotopes partition preferentially into
phases in which they are most strongly bound (because
this results in the greatest reduction in system energy).
o Covalent bonds, and bonds to heavier atoms, are generally strongest and
hence will most often incorporate the heavy isotope.
• Largest fractionations will occur where the atomic
environment or bond energy differences are greatest
o So, for example, fractionation of O between silicates are not large, because
the O is mainly bound to Si.
• Fractionations tend to be large between different
oxidation states of an element (e.g., for C, N, S).
• Lighter isotopes are likely to be enriched in the products
of incomplete reactions and also reactions where
diffusion is important.
Mass Dependent
Fractionation
•
•
•
•
•
if a 4‰ fractionation of δ18O is observed in a particular sample,
what value of δ17O do we predict? We might guess it would ½ as
much.
Mass occurs in a variety of ways in the partition function, as m3/2,
as reduced mass, and in the exponential term. Consequently, the
ratio of fractionation of 17O/16O to that of 18O/16O in most cases is
about 0.52.
Nevertheless, the fractionation between isotopes predicted by
this equation is proportional to the difference in mass – this is
referred to as mass-dependent fractionation.
There are some exceptions where the ratio of fractionation of
17O/16O to that of 18O/16O is ≈1. Since the extent of fractionation in
these cases seems independent of the mass difference, this is
called mass-independent fractionation.
Examples
o
o
o
Oxygen in meteorites
Sulfur in Archean sulfides
Oxygen in stratospheric gases
Mass Independent
Fractionation
•
•
•
•
•
•
Most examples of ‘MIF’ seem to be related
to photochemical reactions.
Formation of ozone in the stratosphere
involves the energetic collision of
monatomic and molecular oxygen:
O + O2 → O3
The ozone molecule is in a vibrationally
excited state and subject to dissociation if it
cannot lose this excess energy. The excess
vibrational energy can be lost either by
collisions with other molecules, or by
partitioning to rotational energy.
In the stratosphere, collisions are infrequent,
hence repartitioning of vibrational energy
represents an important pathway to stability.
Because there are more possible energy
transitions for asymmetric species such as
16O16O18O and 16O16O17O than symmetric
ones such as 16O16O16O, the former can more
readily repartition its excess energy and form
a stable molecule.
In the troposphere, collisions more frequent,
reducing this effect.
Isotope Geothermometry
•
One of the principal uses of
stable isotopes is
geothermometry. Stable isotope
geothermometers are based on
the temperature dependence of
the fractionation factor or
equilibrium constant, which can
generally be expressed as:
a = A+
o
•
B
T2
(at low temperatures, the form of changes to
α ∝ 1/T).
Temperature dependence can
be theoretically calculated or
experimentally measured.
Measuring the isotopic
composition of two phases allows
us to calculate the temperature
at which they equilibrated
(assuming, of course, that they
did equilibrate).
Isotope “Clumping”
• Consider the distribution of 18O between CO and O2 (Example
9.1). CO and O2 will consist actually of 12 isotopically distinct
molecules or “isotopologues”, such as 12C16O, 12C17O, 13C18O,
16O17O, etc.
• The distribution of isotopes between these species will not be
random but rather some of these isotopologues will be
thermodynamically favored.
o
Essentially, grouping the heavy isotopes in one molecule, e.g., 13C18O, reduces bond
energy by a bit more than twice the reduction of putting one heavy isotope in the
molecule. Thus “clumping” of heavy isotopes reduces system energy.
• This ‘clumping’ depends on temperature (greater at low T). By
analyzing the various isotopologues of the species, one can
calculate equilibrium temperatures.
o
o
The advantage is that we need analyze just one phase involved in the reaction, for
example, carbonate precipitated from water.
In addition to calculating temperature, one can also calculate the isotopic
composition of the water.
• This is a very new field, but holds great promise in isotope
geothermometry.