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Chapter 8
In this case, the terms for bond length also cancel, so the expression involves only the reduced masses. So the expression for the rotational equilibrium constant becomes:
K rot =
q 16 O
2
qC 18 O
q 18 O 16 O qC 16 O
I 16 O
2
I C 18 O
2 I 18 O 16 O I C 16 O
& =^
(ignore the 1/2, it will cancel out later). If rotation were the only mode of motion, CO would be 8‰ poorer in 18 O. The vibrational equilibrium constant may be expressed as:
K vib =
q 16 O
2
qC 18 O
q 18 O 16 O qC 16 O
= e
" h ( # (^16) O 2 + # C (^18) O "# C (^16) O "# (^18) O (^16) O )
2 KT 8.
Since we expect the difference in vibrational frequencies to be quite small, we may make the ap- proximation e x^ = x + 1. Hence:
K vib " 1 +
h
2 KT
{^ # C 16 O $# C 18 O } $ #^16 O
2
[ {^ $#^18 O^16 O }] 8.
Let's make the simplification that the vibration frequencies are related to reduced mass as in a simple Hooke's Law harmonic oscillator:
k
where k is the forcing constant, and depends on the nature of the bond, and will be the same for all iso- topes of an element. In this case, we may write:
! C 18 O =! C 16 O
μ C (^16) O μ C (^18) O
=! C 16 O
= 0.976! C 16 O 8.
A similar expression may be written relating the vibrational frequencies of the oxygen molecule:
Substituting these expressions in the equilibrium constant expression, we have:
K vib = 1 +
h
2 kT
" C 16 O [1# 0.976] #" 16 O
2
( [1#^ 0.9718])
The measured vibrational frequencies of CO and O 2 are 6.50 × 10 13 sec -1^ and 4.74 × 10 13 sec -1. Substi- tuting these values and values for the Planck and Boltzmann constants, we obtain:
K vib = 1 +
T
At 300 K (room temperature), this evaluates to 1.0185. We may now write the total equilibrium constant expression as:
K = K tr K rot K vib!
M 16 O
2
M C 18 O
M 18 O 16 O M C 16 O
3/
μ 16 O
2
μC 18 O
2 μ 18 O 16 O μC 16 O
h
4 ) kT
k
μ C 16 O
k
μ C 18 O
' *^
k
μ 16 O 2
k
μ 18 O 16 O
! 16 O^18 O = 0.9718! 16 O 2
Chapter 8
Evaluating this at 300 K we have:
K = 1.0126!
Since α = 2K, the fractionation factor is 1.023 at 300 K and would decrease by about 6 per mil per 100° temperature increase (however, we must bear in mind that our approximations hold only at low temperature). This temperature depend- ence is illustrated in Figure 8.02. Thus CO would be 23 permil richer in the heavy isotope, (^18) O, than O 2.^ This illustrates an important rule of stable isotope fractionations: The heavy iso- tope goes preferentially in the chemical compound in which the element is most strongly bound. Translational and rotational energy modes are, of course, not available to solids. Thus iso- topic fractionations between solids are entirely controlled by the vibrational partition function. In principle, fractionations between coexisting solids could be calculated as we have done above. The task is considerably complicated by the variety of vibrational modes available to a lattice. The lattice may be treated as a large polyatomic molecule having 3N-6 vibrational modes, where N is the number of atoms in the unit cell. For large N, this approximates to 3N. Vibrational frequency and heat capacity are closely related because thermal energy in a crystal is stored as vibrational energy of the at- oms in the lattice. Einstein and Debye inde- pendently treated the problem by assuming the vibrations arise from independent harmonic oscillations. Their models can be used to pre- dict heat capacities in solids. The vibrational motions available to a lattice may be divided into 'internal' or 'optical' vibra- tions between individual radicals or atomic groupings within the lattice such as CO 3 , and Si–O. The vibrational frequencies of these groups can be calculated from the Einstein function and can be measured by optical spec- troscopy. In addition, there are vibrations of the lattice as a whole, called 'acoustical' vibra- tions, which can also be measured, but may be calculated from the Debye function. From ei- ther calculated or observed vibrational frequen- cies, partition function ratios may be calculated, which in turn are directly related to the frac- tionation factor. Generally, the optical modes are the primary contribution to the partition
Figure 8.02. Fractionation factor, α= (^18 O/^16 O)CO/ (^18 O/^16 O)O 2 , calculated from partition functions as a function of temperature.
Figure 8.03. Calculated temperature dependencies of the fractionation of oxygen between water and quartz. After Kawabe (1978).
Chapter 8 Spring 2011
R 13 C 18 O − r =
[
13
C
18
O ]
[
12
C
16
O ]
⎠⎟^ r
[
13
C ][
18
O ]
[
12
C ][
16
O ]
It gets a little more complex for molecules with more than 2 isotopes. In most cases, we are interested in combinations of isotopes rather than permutations, which is to say we don’t care about order. This will not be the case for highly asymmetric molecules such as nitrous oxide, N 2 O. The structure of this molecule is N-N-O and 14 N^15 N^16 O will have different properties than 15 N^14 N^16 O, so in that case, order does matter. The CO 2 molecule is, however, symmetric and we cannot distinguish 12 C^16 O^18 O from (^12) C (^18) O (^16) O. Its random abundance would be calculated as:
R 13 C 16 O 18 O − r =
[^13 C^16 O^18 O ]
[^12 C^16 O 2 ]
r
2 [^13 C ][^16 O ][^18 O ]
[^12 C ][^16 O ]^2
2 [^13 C ][^18 O ]
[^12 C ][^16 O ]
The factor of 2 is in the denominator to take account of both 12 C^16 O^18 O and 12 C^18 O^16 O.
Kinetic Fractionation
Kinetic effects are normally associated with fast, incomplete, or unidirectional processes like evapora- tion, diffusion and dissociation reactions. As an example, recall that temperature is related to the aver- age kinetic energy. In an ideal gas, the average kinetic energy of all molecules is the same. The kinetic energy is given by:
E =
mv^2 8.
Consider two molecules of carbon dioxide, 12 C^16 O^2 and 13 C^16 O 2 , in such a gas. If their energies are equal, the ratio of their velocities is (45/44)1/2, or 1.011. Thus 12 C^16 O 2 can diffuse 1.1% further in a given amount of time at a given temperature than 13 C^16 O 2. This result, however, is largely limited to ideal gases, i.e., low pressures where collisions between molecules are infrequent and intermolecular forces negligible. For the case of air, where molecular collisions are important, the ratio of the diffusion coeffi- cients of the two CO 2 species is the ratio of the square roots of the reduced masses of CO 2 and air (mean molecular weight 28.8):
D 12 CO
2
D 13 CO
2
μ (^13) CO 2 μ (^12) CO 2
Hence we would predict that gaseous diffusion will lead to only a 4.4‰ fractionation. In addition, molecules containing the heavy isotope are more stable and have higher dissociation en- ergies than those containing the light isotope. This can be readily seen in Figure 8. 01. The energy re- quired to raise the D 2 molecule to the energy where the atoms dissociate is 441.6 kJ/mole, whereas the energy required to dissociate the H 2 molecule is 431.8 kJ/mole. Therefore it is easier to break bonds such as C-H than C-D. Where reactions go to completion, this difference in bonding energy plays no role: isotopic fractionations will be governed by the considerations of equilibrium discussed in the pre- vious section. Where reactions do not achieve equilibrium the lighter isotope will be preferentially concentrated in the reaction products , because of this effect of the bonds involving light isotopes in the reactants being more easily broken. Large kinetic effects are associated with biologically mediated reactions (e.g., bac- terial reduction), because such reactions generally do not achieve equilibrium. Thus 12 C is enriched in the products of photosynthesis in plants (hydrocarbons) relative to atmospheric CO 2 , and 32 S is en- riched in H 2 S produced by bacterial reduction of sulfate. We can express this in a more quantitative sense. The rate at which reactions occur is given by:
R = Ae " Eb^ /^ kT^ 8.
Chapter 8 Spring 2011
where A is a constant called the frequency factor and Eb is the barrier energy. Referring to Figure 8.01, the barrier energy is the difference between the dissociation energy, ε, and the zero-point energy. The constant A is independent of isotopic composition, thus the ratio of reaction rates between the HD molecule and the H 2 molecule is:
RD
RH
e "(^ #"^1 2 h $^ D^ )/^ kT
e
"( #" 1 2 h $ H )/ kT^8.
or
RD
RH
= e
( " (^) H # " (^) D ) h / 2 kT
Substituting for the various constants, and using the wavenumbers given in the caption to Figure 8.0 1 (remembering that ω = cν where c is the speed of light) the ratio is calculated as 0.24; in other words we expect the H 2 molecule to react four times faster than the HD molecule, a very large difference. For heavier elements, the rate differences are smaller. For example, the same ratio calculated for 16 O 2 and (^18) O (^16) O shows that the 16 O will react about 15% faster than the 18 O (^16) O molecule. The greater translational velocities of lighter molecules also allows them to break through a liquid surface more readily and hence evaporate more quickly than a heavy molecule of the same com- position. The transition from liquid to gas in the case of water also involves breaking hydrogen bonds that form between the hydrogen of one molecule and an oxygen of another. This bond is weaker if 16 O is involved rather than 18 O, and thus is broken more easily, meaning H 216 O is more readily available to transform into the gas phase than H 218 O. Thus water vapor above the ocean typically has δ^18 O around – 13 per mil, whereas at equilibrium the vapor should only be about 9 per mil lighter than the liquid. Let's explore this example a bit further. An interesting example of a kinetic effect is the fractionation of O isotopes between water and water vapor. This is another example of Rayleigh distillation (or con- densation), as is fractional crystallization. Let A be the amount of the species containing the major iso- tope, H 216 O, and B be the amount of the species containing the minor isotope, H 218 O. The rate at which these species evaporate is proportional to the amount present:
dA=kAA 8 .62a and dB=kBB 8 .62b
Since the isotopic composition affects the reaction, or evaporation, rate, kA ≠ kB. We'll call this ratio of the rate con- stants α. Then
dB
dA
B
A
Rearranging and integrating, we have
ln
B
B ˚
= " ln
A
A ˚
or
B
B ˚
A
A ˚
(
where A° and B° are the amount of A and B originally present. Dividing both sides by A/A°
B / A
B ˚/ A ˚
A
A ˚
() 1
Since the amount of B makes up only a
Figure 8.04. Fractionation of isotope ratios during Rayleigh and equilibrium condensation. δ is the per mil difference be- tween the isotopic composition of original vapor and the iso- topic composition as a function of ƒ, the fraction of vapor remaining.
Chapter 8 Spring 2011
discussion is given here, a fuller discussion of the causes of mass independent fractionation can be found in Thiemens (2006). There is at least a partial theoretical explanation in the case of ozone (Heidenreich and Thiemens, 1986, Gao and Marcus, 2001). Their theory can be roughly explained as follows. Formation of ozone in the stratosphere typically involves the energetic collision of monatomic and molecular oxygen, i.e.:
O + O 2 → O 3 *
The ozone molecule thus formed is in a vibrationally excited state (designated by the asterisk) and, con- sequently, subject to dissociation if it cannot loose this excess energy. The excess vibrational energy can be lost either by collisions with other molecules, or by partitioning to rotational energy. In the strato- sphere, collisions are comparatively infrequent hence repartitioning of vibrational energy represents an important pathway to stability. Because there are more possible energy transitions for asymmetric spe- cies such as 16 O^16 O^18 O and 16 O^16 O^17 O than symmetric ones such as 16 O^16 O^16 O, the former can repartition its excess energy and form a stable molecule. At higher pressures, such as prevail in the troposphere, the symmetric molecule can readily lose energy through collisions, lessening the importance of the vi- brational to rotational energy conversion. Gao and Marcus (2001) were able to closely match observed experimental fractionations, but their approach was in part empirical because a fully quantum me- chanical treatment is not yet possible. Theoretical understanding of mass independent sulfur isotope fractionations is less advanced. Mass independent fractionations similar to those observed in Archean rocks (discussed in a subsequent chap- ter) have been produced in the laboratory by photo-dissociation (photolysis) of SO 2 and SO using deep ultraviolet radiation (wavelengths <220 nm). Photolysis at longer wavelengths does not produce mass independent fractionations. Current explanations therefore focus on ultraviolet photolysis. However,
Figure 8.05. Oxygen isotopic composition in the stratosphere and troposphere show the effects of mass independent fractionation. A few other atmospheric trace gases show similar effects. Essentially all other material from the Earth and Moon plot on the terrestrial fractionation line. Af- ter Johnson et al. (2001).
Chapter 8 Spring 2011
there as yet is no theoretical explanation of this effect and alternative explanations, including ones that involve the role in symmetry in a manner analogous to ozone, cannot be entirely ruled out.
HYDROGEN AND OXYGEN ISOTOPE RATIOS IN THE HYDROLOGIC SYSTEM We noted above that isotopically light water has a higher vapor pressure, and hence lower boiling point than isotopically heavy water. Let's consider this in a bit more detail. Raoult's law states that the partial pressure, p , of a species above a solution is equal to its molar concentration in the solution times the standard state partial pressure, p° , where the standard state is the pure solution. So for ex- ample:
pH
2 16 O^
= pH
2 16 O
o H
2
16 O
[ ] 8.70a^ and
pH
2 18 O^
= pH
2 18 O
o H
2
18 O
[ ] 8.70b Since the partial pressure of a species is proportional to the number of atoms of that species in a gas, we can define α, the fractionation factor between liquid water and vapor in the usual way:
" l / v =
pH
218 O^
/ pH
216 O
[ H 2
18
O ]/[ H 2
16
O ]
By solving 8.70a and 8.70b for [H 216 O] and [H 218 O] and substituting into 8.71 we arrive at the rela- tionship:
" l / v =
pH
218 O
o
pH
216 O
o 8.
Interestingly enough, the fractionation factor for oxygen between water vapor and liquid turns out to be just the ratio of the standard state partial pressures. The next question is how the partial pressures vary with temperature. According to classical thermodynamics, the temperature dependence of the partial pressure of a species may be expressed as:
d ln P
dT
" H
RT^2
where T is temperature, ∆H is the enthalpy or latent heat of evaporation, and R is the gas constant. Over a sufficiently small range of temperature, we can assume that ∆H is inde- pendent of temperature. Rear- ranging and integrating, we obtain:
ln p =!
" H
RT
+ const 8.
We can write two such equa- tions, one for [H 216 O] and one for [H 218 O]. Dividing one by the other we obtain:
ln
pH
2 (^18) O o
pH
216 O
o =^ A^ "^
B
RT
Figure 8. 06. Temperature dependence of fractionation factors be- tween vapor and water (solid lines) and vapor and ice (dashed lines) for various species of water.
