1. 第二節 同位素的分餾原理 同位素效應與分餾 同位素分餾過程 平衡分餾與分餾系數 同位素比值的表示方法 動力分餾 影響分餾的原素
第二節 同位素的分餾原理
同位素效應與分餾
同位素分餾過程
平衡分餾與分餾系數
同位素比值的表示方法
動力分餾
影響分餾的原素
同位素地質溫度計

2. 同位素效應與分餾
Differences in chemical and physical properties arising from variations in atomic mass of an element are called “Isotope effects”.
Isotope Fractionation refers to the change in an isotope ratio that arises as a result of some chemical or physical process.
The electronic structure of an atom determines its chemical behavior, whereas the nucleus is responsible for its physical properties. Certain differences exist in physicochemical properties due to mass differences. The replacement of any atom in a molecule by one of its isotopes produces a very small change in chemical behavior.

3. Characteristic physical properties of H216O, D216O, and H216O8

4. There are three modes of motion available to gaseous molecules: vibrational, rotational, and translational.
It is easy to understand how some isotope fractionations can arise from vibrational motion.
ν is the ground state vibrational frequency:
Fundamental vibration frequencies are 4405 cm-1 for H2, 3817 cm-1 for HD, and 3119 cm-1 for D2.
μ is reduced mass:

5. 能量越高的核素越活躍，處於激發態，越容易躍遷到低能態而趨於穩定。化學鍵結合的越不牢固。
輕質量的核素，振動頻率高，導致振動能量高，化學鍵較重質量核素的化學鍵易斷開，同位素比值發生變化。
Vibration frequencies are 4405 cm-1 for H2, 3817 cm-1 for HD, and 3119 cm-1 for D2. Which one has the highest vibration energy?

6. 同位素分餾過程
The vibrational energy level for a given quantum number will be lower for a bond involving a heavier isotope of an element, thus bonds involving heavier isotopes will be stronger.
The energy of the system is minimized when the heavy isotope occupies the site with the stronger bond.

7. Isotope fractionation can originate from both kinetic effects and equilibrium effects.
Temperature-dependent equilibrium isotope fractionations arise from quantum mechanical
effects in vibrational motions.
Zero Point Energy (ZPE): At absolute 0oK the atoms will vibrate at a ground frequency n0. The system will have an energy of 1/2hn0, where h is Planck's constant. It depends on mass, and thus will be different for isotopes.
The zero-point energy of H2 is greater than that of HD which is greater than that of D2.

8. 平衡分餾與分餾系數 Boltzmann distribution function︰
At equilibrium, the ratio of molecules having internal energy Ei to those having the zero point energy E0 is:
where n0 is the number of molecules with zero point energy, ni is the number of molecules with energy Ei and k is Boltzmann‘s constant, T is the absolute temperature, and g is a statistical weight factor。
Boltzmann distribution function︰
The average energy (per molecule) in a system is given by the sum of the energy of all possible states times the number of particles in that state divided by the number of particles in those states
The denominator of this equation is called The partition function, Q.

9. The relationship between partition function and equilibrium constant is through the Gibbs Free Energy change:
where x is the stoichiometric coefficient(化學計量系數 ) : 在化學反應中各元素或化合物之間的比例關係,以及已配平的化學方程中反應物及反應產物的系數,反映了參與化學反應物質(包括反應物與反應產物)間數量的比例關係 .
For example:

10. The partition function can be written as the product of the translational,
rotational, vibrational, and electronic partition functions:
The electronic configurations and energies of atoms are unaffected by the isotopic differences, so the last term can be ignored in the equation.
where σ is a symmetry factor whose value is 1 for heteronuclear molecules such as 16O-18O and 2 for homonuclear molecules such as 16O-16O. m is mass, V is the volume of the box, k is the forcing constant, μ is reduced mass, I is the moment of inertia, I = μr2, where r is the inter-atomic distance. h is Plank’s constant. T is the absolute temperature. ν is the ground state vibrational frequency.

11. Calculation of the fractionation factor α from the equilibrium constant
Equilibrium fractionations arise from translational, rotational and vibrational motions of molecules in gases and liquids and atoms in crystals because the energies associated with these motions are mass dependent.
At equilibrium, α may be related to the equilibrium constant of thermodynamics by:
n is the number of atoms exchanged; K is the equilibrium constant; K∞ is the equilibrium constant at infinite temperature.

12. For a reaction
where A1 and A2 are two molecules of the same substance differing only in their isotopic composition, and a and b are the stoichiometric coefficients, the equilibrium constant is:
K∞ is simply the ratio of the symmetry factors.

13. Krot = 0.9917/2 μ is reduced mass: νC-16O = 6.505 × 1013 sec-1,
ν16O-16O = × 1013 sec-1
How will the fractionation factor, α= (18O/16O)CO/(18O/16O)O2, vary as a function of temperature?
The rotational and translational contributions are independent of temperature.
the symmetry factor, σ, is 2 for 16O2 and 1 for all other molecules.
μ is reduced mass:
Substituting values μC16O = 6.857, μC18O = 7.20, μ18O16O = 8.471, μ16O2 = 8,
Krot = /2

14. For translational motion, we noted the ratio of partition functions reduces to the ratio of molecular masses raised to the 3/2 power.
Substituting MC16O = 28, MC18O = 30, M18O16O = 34, M16O2 = 32,

15. The vibrational equilibrium constant may be expressed as:
νC-16O = × 1013 sec-1,
ν16O-16O = × 1013 sec-1
νC-18O = × 1013 sec-1,
ν18O-16O = × 1013 sec-1
k is the forcing constant, and depends on bond strength, and will be identical for all isotopes of an element。

16. ( )
For a given temperature, we can calculate K.
= 2( ) [ ]

17. 分餾系數與溫度成反比 溫度越高，同位素分餾越小
Fractionation factor, α= (18O/16O)CO/(18O/16O)O2, calculated from partition functions as a function of temperature

18. Calculated value of α18O for CO2 H2O, shown vs. 1/T and 1/T2
The α ∝ 1/T relationship breaks down around 200°C; above that temperature the relationship follows α ∝ 1/T2.

19. 同位素比值的表示方法 e = a -1 分餾系數 aA-B = RA/RB R 為重同位素與輕同位素之比值，
H218O + 1/3CaC16O3 = H216O + 1/3CaC18O3
aCaCO3-H2O = (18O/16O)CaCO3/(18O/16O) H2O
注意A和B的先後次序。
分離系數
e = a -1

20. The δ Notation
δA = (RA/RStd -1)
δB = (RA/RStd -1)
 

21. The relationship between Δ and α
The fractionation of isotopes between two phases is also often reported as ΔA-B = δA - δB.
And dB << 1000
Because a is close to 1, so ln(a – 1) = lna

22. 動力分餾
Kinetic isotope fractionations are normally associated with fast, incomplete, or unidirectional processes like evaporation, diffusion, dissociation reactions, and biologically mediated reactions.
Where reactions do not achieve equilibrium the lighter isotope will usually 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., photosynthesis, bacterial reduction), because such reactions generally do not achieve equilibrium and do not go to completion (e.g., plants don’t convert all CO2 to organic carbon). Thus 12C is enriched in the products of photosynthesis in plants (hydrocarbons) relative to atmospheric CO2, and 32S is enriched in H2S produced by bacterial reduction of sulfate.

23. Predicting Isotopic Fractionations
Considering 12C16O2 and 13C16O2, in an ideal gas, if their energies are equal, the ratio of their velocities is (45/44)1/2, or Thus 12C16O2 can diffuse 1.1% further in a given amount of time than 13C16O2.
For the case where molecular collisions are important, the ratio of their diffusion coefficients is
Gaseous diffusion will lead to a 4.4‰ rather than 11‰ fractionation.
Molecules containing the heavy isotope are more stable and have higher dissociation energies than those containing the light isotope.

24. Rayleigh distillation (or condensation)
The fractionation of O isotopes between water and water vapor
Let A be the amount of the species containing the major isotope, H216O, and B be the amount of the species containing the minor isotope, H218O. The rate at which these species evaporate is proportional to the amount present:
Whereas k is reaction rate or here evaporation rate
dA=kAA and dB=kBB
For kinetic fractionations, the fractionation factor is simply the ratio of the rate constants. Rearranging and integrating, we have:
where A° and B° are the amount of A and B originally present. Dividing both sides by A/A°

25. Since the amount of B makes up only a trace of the total amount of H2O present, A is essentially equal to the total water present, and A/A° is essentially identical to ƒ
Subtracting 1 from both sides, we have:
The left side of the equation is the relative deviation from the initial ratio. The permil relative deviation is simply:
a = 1.01
Even if the vapor and liquid remain in equilibrium through the condensation process, the isotopic composition of the remaining vapor will change continuously.

26. 影響分餾的原素
A general rule : The heavy isotope goes into the phase in which it is most strongly bound.
化學鍵︰Bonds to ions with a high ionic potential and low atomic mass are associated with high vibrational frequencies and have a tendency to incorporate the heavy isotope preferentially.
1。共價鍵強於離子鍵。E.g., the O is bound more strongly in quartz than in magnetite, so the former is typically enriched in 18O.
2。與原子核小的結合分餾大於與原子核大的。Δ18Ocal-water, between calcite and water is about 30 per mil at 25° C. The Δ18Ocarb-H2O decreases to about 25 when Ba replaces Ca (Ba has about 3 times the mass of Ca).

27. 3。晶體架構的影響。The Δ18O between aragonite and calcite is of the order of 0
3。晶體架構的影響。The Δ18O between aragonite and calcite is of the order of 0.5 permil. However, there is apparently a large fractionation (10 permil) of C between graphite and diamond.
4。壓力效應。壓力越大，分餾越小。Pressure effects on fractionation factors turn out to be small, no more than 0.1 permil over 20 kbars.
5。溫度效應。溫度越高，分餾越小。At infinite temperature, the fractionation is unity, since lnα = 0.
6。相態效應。重同位素傾向於架構緊密的相態，固相>液相>氣相。

28. 同位素地質溫度計
Stable isotope geothermometers are based on the temperature dependence of the equilibrium constant.
At low temperatures, the equation changes to K ∝ 1/T.
The constants A and B are slowly varying functions of temperature, such that K tends to zero at absolute 0, corresponding to complete separation, and to 1 at infinite temperature, corresponding to no isotope separation. At infinite temperature, there is complete disorder, hence isotopes would be mixed randomly between phases. At absolute 0, there is perfect order, hence no mixing of isotopes between phases.

29. Coefficients for Oxygen Isotope Fractionation at Low Temperatures

30. Example of Oxygen Isotope Geothermometry
A granite-gneiss contains coexisting quartz, muscovite and magnetite with the following δ18O: quartz: ; magnetite: Find the temperature of equilibration.
Substituting ΔQz-Mag for 1000 ln αQz-Mag and solving for T:
We calculate ΔQz-Mag as Substituting this into equation on the left, we find T = 505 K = 232 ° C.

31. Homework
1. What would you predict would be the ratio of the diffusion coefficients of H2O and D2O in air?
2. A granite contains coexisting feldspar (3% An content) and biotite with δ18O (SMOW) of 9.2‰ and 6.5‰ respectively. Using the information in the Table, find the temperature of equilibration.
3. Consider the condensation of water vapor in the atmosphere. Assume that the fraction of vapor remaining can be expressed as a function of temperature (in kelvins):
Also assume that the fractionation factor can be written as:
Assume that the water vapor has an initial δ18OSMOW of -9‰. Make a plot showing how the δ18O of the remaining vapor would change as a function of ƒ

32. 4. Calculate the δ18O of raindrops forming in a cloud after 80% of the original vapor has already condensed assuming (1.) the water initial evaporated from the ocean with δ18O = 0, (2.) the liquid vapor fractionation factor, α =
5. 同位素的定義是什麼？
6. 穩定同位素分為哪2類？它們之間的區別是什麼？
7. 影響同位素分餾的原素有哪些？
8. 一個樣品的d值是如何表示的？
9. 兩種物質A和B的d值之差與分餾系數a有何關係？請匯出這種關係？
10. 生物在海水中生長，試問它的有機碳d13C值比海水溶解CO2的d13C值重還是輕？為什麼？

33. Boltzmann constant
It is named after the Austrian physicist Ludwig Boltzmann, who made important contributions to the theory of statistical mechanics
(24)×10−23 joule/kelvin
= (15)×10 −5 electron-volt/kelvin

34. The Plank Hypothesis proportional to the frequency.
h=Planks constant, x 10 J Hz.
or C (speed of light) = λV to get E= h (c/λ)