Isotopic fractionations associated with phosphoric acid digestion
of carbonate minerals: Insights from first-principles theoretical
modeling and clumped isotope measurements
Weifu Guoa,*, Jed L. Mosenfeldera, William A. Goddard IIIb, John M. Eilera
aDivision of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
bMaterials and Process Simulation Center, California Institute of Technology, Pasadena, CA 91125, USA
Received 28 July 2008; accepted in revised form 27 May 2009; available online 23 June 2009
Phosphoric acid digestion has been used for oxygen- and carbon-isotope analysis of carbonate minerals since 1950, and was
recently established as a method for carbonate ‘clumped isotope’ analysis. The CO2recovered from this reaction has an oxygen
isotope composition substantially different from reactant carbonate, by an amount that varies with temperature of reaction and
carbonate chemistry. Here, we present a theoretical model of the kinetic isotope effects associated with phosphoric acid diges-
tion of carbonates, based on structural arguments that the key step in the reaction is disproportionation of H2CO3reaction
intermediary. We test that model against previous experimental constraints on the magnitudes and temperature dependences
oftheseoxygenisotopefractionations, andagainstnewexperimentaldeterminations ofthefractionationof13C–18O-containing
isotopologues (‘clumped’ isotopic species). Our model predicts that the isotope fractionations associated with phosphoric acid
digestion of carbonates at 25 ?C are 10.72&, 0.220&, 0.137&, 0.593& for, respectively,18O/16O ratios (1000lna*) and three
indices that measure proportions of multiply-substituted isotopologues ðD?
fractionations follow the mass dependence exponent, k of 0.5281 (where a17O¼ ak
to independent experimental constraints for phosphoric acid digestion of calcite, including our new data for fractionations
of13C–18O bonds (the measured change in D47= 0.23&) during phosphoric acid digestion of calcite at 25 ?C.
We have also attempted to evaluate the effect of carbonate cation compositions on phosphoric acid digestion fractionations
using cluster models in which disproportionating H2CO3interacts with adjacent cations. These models underestimate the
magnitude of isotope fractionations and so must be regarded as unsucsessful, but do reproduce the general trend of variations
and temperature dependences of oxygen isotope acid digestion fractionations among different carbonate minerals. We suggest
these results present a useful starting point for future, more sophisticated models of the reacting carbonate/acid interface.
Examinations of these theoretical predictions and available experimental data suggest cation radius is the most important fac-
tor governing the variations of isotope fractionation among different carbonate minerals. We predict a negative correlation
between acid digestion fractionation of oxygen isotopes and of
relationship to estimate the acid digestion fractionation of D?
theoretical evaluations of13C–18O clumping effects in carbonate minerals, this enables us to predict the temperature calibra-
tion relationship for different carbonate clumped isotope thermometers (witherite, calcite, aragonite, dolomite and magnesite),
and to compare these predictions with available experimental determinations. The success of our models in capturing several
of the features of isotope fractionation during acid digestion supports our hypothesis that phosphoric acid digestion of
carbonate minerals involves disproportionation of transition state structures containing H2CO3.
? 2009 Elsevier Ltd. All rights reserved.
49Þ. We also predict that oxygen isotope
18O). These predictions compare favorably
13C–18O doubly-substituted isotopologues, and use this
47for different carbonate minerals. Combined with previous
0016-7037/$ - see front matter ? 2009 Elsevier Ltd. All rights reserved.
*Corresponding author. Present address: Geophysical Laboratory, Carnegie Institution of Washington, Washington DC 20015, USA.
Tel.: +1 202 478 8993; fax: +1 202 478 8901.
E-mail address: email@example.com (W. Guo).
Available online at www.sciencedirect.com
Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
Analysis of the stable isotope compositions of carbonate
minerals is among the most common and useful measure-
ments in isotope geochemistry. For example, much of paleo-
climatology is based on the carbonate–water oxygen isotope
thermometry (Urey, 1947) and most records of the global
carbon cycle through time depend on measuring the carbon
isotope compositions of sedimentary carbonates (Des Mar-
ais, 1997). More recently, Ghosh et al. (2006) and Schauble
et al. (2006) developed a carbonate ‘clumped isotope’ ther-
mometer based on the ordering of13C and18O into bonds
with each other within the carbonate mineral lattice.
These stable isotope measurements of carbonates are
generally performed indirectly by reacting the sample car-
bonate with anhydrous phosphoric acid and then analyzing
the product CO2on a gas source isotope ratio mass spec-
trometer (McCrea, 1950). This method is relatively straight-
forward to perform, is applicable to a wide range of sample
sizes, has been automated in several different ways, and is
exceptionally precise. These features make this approach
preferable to alternative methods for most applications
(e.g., fluorination, Sharma and Clayton, 1965; secondary
ion mass spectrometry, Rollion-Bard et al., 2007; laser
ablation, Sharp and Cerling, 1996). However because only
two out of three oxygen atoms in carbonate are released
as CO2 during phosphoric acid digestion, this method
involves a kinetic oxygen isotope fractionation; i.e., product
CO2is ?10& higher in d18O than reactant carbonate (Gilg
et al., 2003 and reference therein). The exact magnitude of
temperature and appears to differ among carbonate miner-
als (Sharma and Clayton, 1965; Kim and O’Neil, 1997; Gilg
et al., 2003). Similarly, preliminary evidence suggests
that values of D47(a measure of the abundance anomaly
of13C–18O bonds in CO2; defined as D47¼
1000; Eiler and Schauble, 2004) are enriched in CO2
produced by acid digestion of calcite and aragonite relative
to values one expects in the absence of any associated
fractionation (Ghosh et al., 2006). These analytical fractio-
nations must be corrected for in any study of the oxygen
isotope or ‘clumped isotope’ compositions of carbonate
The oxygen isotope fractionations associated with phos-
phoric acid digestion of different carbonate minerals have
been experimentally studied over a range of temperatures
1969a,b; Rosenbaum and Sheppard, 1986; Swart et al.,
1991; Bottcher, 1996; Kim and O’Neil, 1997; Gilg et al.,
2003; Kim et al., 2007). However, there are significant
discrepancies among acid digestion fractionations of oxygen
isotopes determined in different studies (Kim et al., 2007 and
reference therein). For example, reported acid digestion
fractionations at 25 ?C range from 10.10& (Das Sharma
et al., 2002) to 10.52& (Land, 1980) for calcite, and from
10.29& (Sharma and Clayton, 1965) to 11.01& (Kim and
O’Neil, 1997) for aragonite. Even within a single study,
the measured acid digestion fractionation factors for the
same type of carbonate minerals can vary among different
samples of the same phase (as much as ?2& for octavite,
with acid digestion
?0.6& for witherite and ?0.5& for calcite at 25 ?C; Kim
and O’Neil, 1997). More generally, our understanding of
this phenomenon is entirely empirical, and thus provides
little basis for extrapolation to new materials or conditions
of acid digestion.
Our understanding of acid digestion fractionations is
particularly poor in relation to the carbonate ‘clumped iso-
tope’ thermometer (Ghosh et al., 2006). It has been shown
that the abundance anomaly of13C–18O bonds in CO2pro-
duced by acid digestion of carbonate differs from that in
reactant carbonate (Ghosh et al., 2006), but the exact mag-
nitude of this fractionation was poorly constrained and its
variation among different carbonate minerals is unexplored.
These gaps in our understanding limit the use of the car-
bonate clumped isotope thermometer for minerals other
than calcite and aragonite, e.g., dolomite or magnesite. Be-
cause these carbonates are difficult to synthesize in isotopic
equilibrium (especially at ambient temperatures), theoreti-
cal understandings of both13C–18O clumping within miner-
fractionations during acid digestion are important guides
to interpreting observations on natural samples.
To the best of our knowledge, no detailed theoretical
model has been proposed to explain the isotope fractiona-
tions that accompany phosphoric acid digestion of carbon-
ate minerals. Sharma and Sharma (1969b) determined the
oxygen isotope acid digestion fractionation factors for sev-
eral different carbonate minerals, and explained them as a
result of two factors: a temperature-dependent factor and
a temperature-independent factor. Sharma and Sharma
(1969b) further suggested the temperature-independent fac-
tor varies as a function of the atomic mass of the cations in
carbonate minerals, and can be explained by their proposed
structure for the transition state through which the acid
digestion reaction proceeds (Fig. 1a). However, Sharma
and Sharma’s model does not quantitatively describe the
temperature-dependent factor, and subsequent experimen-
tal results (Bottcher, 1996; Gilg et al., 2003) are inconsistent
with their model.
In this study, we present a quantitative model of the
phosphoric acid digestion reaction based on transition sate
theory and statistical thermodynamics. We use this model
to predict isotopic fractionations among all isotopologues
(including multiply substituted isotopologues) of reactant
carbonate ions, and thus the isotopic composition (includ-
ing abundances of multiply-substituted isotopologues) of
CO2produced by phosphoric acid digestion of carbonate
minerals. Finally, we test the accuracy of our model by
comparison with previous data documenting the oxygen
isotope fractionation associated with this reaction, and with
new data we have generated documenting the fractionation
(which controls the D47value of product CO2). We observe
quantitative agreements between our model predictions
and available experimental data on the magnitude and
associated with phosphoric acid digestion for calcite. We
then modify this model to explore the effect of cation
content on the acid digestion fractionation; this effort fails
to yield an accurate match to experimental data—
and clumped isotope
13C–18O bearing isotopologues during acid digestion
7204W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
presumably because our guess regarding the structure of the
carbonate-acid interface is imperfect. Nevertheless, it yields
a trend of variations of oxygen isotope fractionations with
cation identity that resembles experimental data, and thus
the approach may be a fruitful starting point for future,
more sophisticated models.
This study provides a framework for understanding iso-
tope fractionations accompanying phosphoric acid diges-
tion of carbonate minerals, and applying
conditions or materials that are not yet understood through
experimental work. Furthermore, this study demonstrates a
technique of first-principles modeling of kinetic isotope ef-
fects associated with irreversible reactions and illustrates
the utility of this technique by application to one of the
more extensively studied inorganic reactions in stable iso-
tope geochemistry, taking advantage of new constraints
provided by clumped isotope measurements. This approach
therefore serves as a model for future work of similar but
less well known kinetically-controlled fractionations.
2. THEORETICAL AND COMPUTATIONAL
2.1. Transition state theory of reaction rates
Transition state theory is long established as a tool for
studying chemical kinetics (Eyring, 1935a,b) and has been
used previously to understand irreversible reactions in
geoscience problems (Lasaga, 1998; Felipe et al., 2001).
Classical transition state theory is based on two key
assumptions (Felipe et al., 2001): (1) instead of transform-
ing directly into products, reactants in a chemical reaction
first proceed through an unstable chemical state called the
‘‘transition state”, which has a higher chemical potential en-
ergy than reactants or products; (2) the transition state may
only form from the reactants (i.e., the conversion of the
transition state to products is irreversible), and any transi-
tion state that proceeds in the reaction coordinate past its
potential energy maximum must eventually form products.
This second assumption is also called the ‘‘non-recrossing
rule”. For example, transition state theory would describe
A þ B ! C þ D as proceeding through two steps: revers-
ible transformation of reactants A and B into a transition
state, M?, after which M?transforms irreversibly into prod-
ucts C and D (i.e. A þ B ¢
rate of the overall reaction (i.e. the production rate of C
and D), R, equals the decomposition rate of the transition
state, M?and can be described through the relation:
2C þ D). Therefore, the
where [M?] is the concentration of transition state M?, s is
the average lifetime of transition state M?and my
‘decomposition frequency’ (defined as the reciprocal of
the average life time) of M?(Melander and Saunders,
1987; Felipe et al., 2001). The concentration of the transi-
tion state, [M?], can be estimated by assuming that the
reversible reaction,A þ B ¢
(Melander and Saunders, 1987):
My, is atequilibrium
½My? ¼ K½A?½B?
where [A] and [B] are the concentrations of reactant A and
B respectively, and K is the equilibrium constant for reac-
tion 1 and is evaluated using statistical thermodynamics
where Q?, QA, QBare the partition functions of transition
state M?and reactants, A and B, respectively; -y
are the vibration frequencies, in wave numbers, for the
transition state M?and reactants, A and B, respectively
(one such term is required for each mode of vibration of
each species); s?, sA, sBare the symmetry numbers for tran-
sition state M?and reactants A and B, respectively; N?, NA,
NBare the numbers of atoms within transition state M?and
reactants, A and B, respectively; h is Plank’s constant; c is
the velocity of light; k is the Boltzmann constant; and T
is the reaction temperature in Kelvin.
When one is interested in kinetic isotope effects, as in
this study, relative reaction rates (i.e., the ratios of reaction
rates of different isotopologues) are of greatest importance:
Fig. 1. Reaction mechanisms during phosphoric acid digestion of
carbonate minerals, proposed by (a) Sharma and Sharma (1969b),
and (b) this study. See Section 2.2.1 for details.
Phosphoric acid digestion of carbonate minerals7205
where subscript (1) and (2) denote different isotopologues
of the transition state.
2.2. Application of transition state theory to phosphoric acid
digestion of carbonate minerals
2.2.1. Proposed reaction mechanism and transition state
structure during phosphoric acid digestion of carbonate
In order to calculate the partition function of the transi-
tion state associated with phosphoric acid digestion of car-
bonates, we must first determine or assume the transition
state structure. This is commonly done by initially guessing
the structure of transition state and then refining on that
guess using ab inito calculations (Felipe et al., 2001).
Little is known about the exact reaction mechanism and
transition state structure during the phosphoric acid diges-
tion of carbonate minerals. Although the transition state
structure suggested by Sharma and Sharma (1969b) is intu-
itively appealing (Fig. 1a), there is no evidence to date that
supports it. Instead, recent spectroscopic studies of calcium
carbonate undergoing reactions with anhydrous acidic
gases (e.g., HNO3, SO2, HCOOH and CH3COOH) identi-
fied absorbed H2CO3on the carbonate surface and sug-
gested carbonic acid as the important intermediate species
during these reactions (Al-Hosney and Grassian, 2004;
Al-Hosney and Grassian, 2005). Though carbonic acid
decomposes rapidly in aqueous solution, it is kinetically sta-
ble in the absence of water (Hage et al., 1998; Loerting
et al., 2000 and reference therein). We suggest that these
experiments are analogous to the local environment at the
surface of a carbonate mineral during reaction with
anhydrous phosphoric acid (i.e., the 105% concentrated
phosphoric acid used in stable isotope analyses of carbon-
ates; Coplen et al., 1983). We therefore propose that
H2CO3 is also an intermediate during phosphoric acid
through two steps (Fig. 1b):
proceedsXHPO4þ CO2þ H2O,
where X is a cation, such as Ca, contained in the carbonate
We infer that the first of these two reaction steps should
be associated with little or no net isotopic fraction-
ation (including fractionations of multiply-substituted
isotopologues), for two reasons: (1) in practice, phosphoric
acid digestion is always driven to completion before collect-
ing and analyzing product CO2. Because CO2?
in the reactant carbonate are quantitatively converted into
H2CO3during step 1, it is not possible to express a net iso-
topic fractionation of C or O isotopes during that step, even
if that reaction has some intrinsic kinetic isotope effect.
And, (2) any kinetic isotope effect that might accompany
step 1 could only be expressed if the site at which the
reaction occurs (i.e., a mineral surface) can undergo isoto-
pic exchange with the unreacted mineral interior, which
we consider unlikely at the low temperatures and anhy-
drous conditions of phosphoric acid digestion. That is, we
infer that step 1 is analogous to sublimation of ice, which
generally fails to express a vapor pressure isotope effect be-
cause the reaction effectively ‘peels’ away layers of the solid
without leaving an isotopically modified residue. Similarly,
we assume no oxygen isotope exchange between the car-
bonic acid and the surrounding anhydrous phosphoric
acids and among different carbonic acid isotopologues dur-
ing the phosphoric acid digestion. In order for such ex-
change to occur, C–O bonds in the carbonic acid would
have to break and reform. We regard this as unlikely given
the kinetic stability of carbonic acid under anhydrous
environments (Loerting et al., 2000; Hage et al., 1998 and
reference therein). For these reasons, we focus on step
2—the dissociation of carbonic acid.
No prior studies exist on the exact bonding environment
of intermediate H2CO3during phosphoric acid digestion of
carbonate minerals. In the following model, we describe the
intermediate H2CO3as free isolated molecules, and take a
previously determined transition state structure for gas
phase carbonic acid decomposition (Loerting et al., 2000)
as our ‘initial guess’ (Fig. 2b). We then optimize that
structure through further ab initio calculations. It is
reasonable to suspect that the presence of unreacted car-
bonate surface and surrounding H3PO4might also affect
the decomposition of the intermediate H2CO3 during
phosphoric acid digestion of carbonate minerals. For
example, intermediate H2CO3might exist as absorbed mol-
ecules and dissociate on the unreacted carbonate surface.
These interactions would then introduce second order
effects on the kinetic isotope effects associated with
phosphoric acid digestion. We attempt to evaluate the
interactions between H2CO3
surface through a ‘cluster model’ in Section 2.2.3. We did
not consider potential interactions between H2CO3and sur-
rounding H3PO4, which requires more sophisticated model
and computational capacity beyond the scope of this study.
and un-reacted mineral
2.2.2. Fractionation of CO3
dissociation of carbonic acid
Carbonic acid, H2CO3, has 20 naturally occurring
isotopologues, not counting those containing D or
which can dissociate to produce 12 different isotopologues
of product CO2 (Table 1). Furthermore, many of the
isotopologues of H2CO3have more than one isotopomer
because the various O sites are not structurally equivalent
to one another (see Fig. 2). To distinguish the isotopologues
and isotopomers in our following discussion, formula
expressions for isotopomers are underlined, with their oxy-
gen atoms expressed in the order of position 2, 3 and 4 in
Fig. 2 (note, oxygen atoms at position 4 are the ones to
be abstracted during phosphoric acid digestion). For exam-
ple, H2C18O16O16O, H2C16O18O16O and H2C16O16O18O
isotopologue, with18O occupying the oxygen position 2, 3
and 4, respectively, (Fig. 2d–f). Similarly, the formula
expressions for transition state structures in the following
7206 W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
discussions are superscripted with ‘‘?” signs, in order to dis-
tinguish them from those for stable molecules.
Isotopic fractionations during
carbonic acid can arise for two reasons: (1) the various
isotopologues of carbonic acid differ from one another in
their rates of dissociation (e.g., the rate of dissociation of
H212C16O3is faster than the weighted average dissociation
rate of the three isotopomers of H212C18O16O2; Fig. 2c–f);
and (2) when an isotopologue has more than one isotopo-
mer (e.g., Fig. 2), the relative rates of dissociation of those
isotopomers differ from one another. However, since
phosphoric acid digestion is always driven to completion
before collecting and analyzing product CO2 (i.e., all
isotopologues of carbonic acid eventually decompose) and
we assume there is no isotope exchange between different
H2CO3isotopologues, the first type of isotope effects will
not be expressed in the final isotopic composition of prod-
uct CO2(though they might influence the temporal evolu-
tion of the isotopic composition of product CO2). Only
the second type of isotope effects contributes to the fact that
final product CO2is expected to be higher in18O/16O ratio
than the reactant H2CO3. For example, differences between
the rates of dissociation of the three isotopomers of
H212C18O16O2promote production of
12C–16O bond in order to dissociate [Fig. 2d and e] do so
more quickly than those isotopomers that must break a
12C–18O bond [Fig. 2f ]).
We calculate the proportions of CO2isotopologues pro-
duced by dissociation of each H2CO3isotopologue based
on a formulation that is exemplified as follows for the case
the dissociation of
12C16O2 (i.e., those isotopomers that must break a
12C18O16O2. This isotopologue of carbonic acid con-
sists of three isotopomers, which dissociate to form two iso-
topologues of CO2,12C16O2and
relative abundance of each of these products is calculated
through the equations:
where n12C16O2–H212C18O16O2and n12C18O16O–H212C18O16O2denote the
total numbers of molecules of12C16O2and12C18O16O pro-
duced from dissociation of H212C18O16O2; nH212C18O16O2is
the number of molecules of reactant H212C18O16O2, which
we take to equal the abundance of X12C18O16O2in the
carbonate undergoing phosphoric acid digestion; i.e., we
ignore any isotopic discrimination associated with step 1
ofthe overall reactionand
between different H2CO3 isotopologues, as defined in
Section 2.2.1. R12C18O16O–H212C18O16O16O;R12C18O16O–H212C16O18O16Oand
R12C16O2–H212C16O16O18Odenote the dissociation rates of the three
isotopomers of H212C18O16O2(Fig. 2). These R values are calcu-
lated based on statistical thermodynamic principles, as described
in Section 2.1 and exemplified below:
12C18O16O (Fig. 2). The
Fig. 2. Optimized transition state structures during phosphoric acid digestion of carbonate minerals (H2CO3dissociation model): (a) an
optimized stable structure of carbonic acid, shown for comparison; (b) the optimized transition state structure employed in the ab initio
calculations of this study. Numbers refer to atomic positions within the structure. Oxygen atom 4 is the one that is abstracted from the
reactant carbonate ion during acid digestion (i.e., oxygen atoms 2 and 3 remain bound to carbon atom 1); (c) the only carbonic acid
isotopomer (H212C16O16O16O) during the phosphoric acid digestion of12C16O3
(d–f) three possible carbonic acid isotopomers (H212C18O16O16O, H212C16O18O16O and H212C16O16O18O, respectively) during the phosphoric
acid digestion of12C18O16O2?
2?. Numbers refer to the isotopic mass, in AMU, of the atom;
2. Numbers again refer to isotopic mass of the atom.
Phosphoric acid digestion of carbonate minerals7207
Evolution of different CO3
2?isotopologues during phosphoric acid digestion of carbonate minerals.
13C18O16O 47 0.3382969
aNominal cardinal mass in amu.
bStochastic abundances, i.e., the abundances when all the isotopes are stochastically distributed within the reactant carbonate, with a bulk
isotopic composition of d13CVPDB= 0& and d18OVSMOW= 0&.
cOxygen atoms in H2CO3intermediate are expressed in the order of atom 2, 3, 4 in Fig. 2, and the last oxygen atoms (atom 4) are the ones
to be abstracted during phosphoric acid digestion.
dPredicted respective fractions of different product CO2isotopologues from the reactant CO3
2?isotopologues at 25 ?C, based on our
7208 W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
where [H212C18O16O16O], [H212C16O18O16O], [H212C16O
16O18O] are the abundances of the different isotopomers
of H212C18O16O2(Fig. 2d–f, respectively). Because the life-
time of carbonic acid is relatively long under anhydrous
conditions (Loerting et al., 2000 and reference therein),
we assume the concentrations of these various carbonic acid
isotopomers are in equilibrium (presumably through the ex-
changes of protons):
: R12C16O2?H212C16O16O18O¼ ðjmy
The relations given above for the case of H212C18O16O2
are applied to all the isotopologues of H2CO3that are capa-
ble of producing more than one CO2 isotopologue; for
H2CO3isotopologues capable of producing only one CO2
produced from that H2CO3isotopologue equal njof that
H2CO3 isotopologue, e.g., for isotopologue H212C16O3;
n12C16O2?H212C16O3¼ nH212C16O3. The summation of all nivalues
for the various isotopologues of CO2produced from all iso-
topologues of H2CO3 allow us to calculate the isotopic
fractionations (including the fractionations of multiply
substituted isotopologues) associated with acid digestion
Lj ? QyÞH212C18O16O16Oy
Lj ? QyÞH212C16O16O18Oy
Lj ? QyÞH212C18O16O16Oy : ðjmy
2.2.3. Exploration of cation effects during phosphoric acid
digestion through cluster models
The cation compositions of carbonate minerals might
exert a second-order, but measurable influence on the oxy-
gen isotope fractionation associated with phosphoric acid
digestion (e.g., Gilg et al., 2003). The data in support of
such effects are open to question, as the fluorination mea-
surements required to independently establish the d18O val-
ues of reactant carbonates generally have poorer than
expected reproducibility (and other peculiarities). Neverthe-
less, mineral-specific acid digestion fractionations seem
possible and are widely assumed. The H2CO3-dissociation
model we described above cannot account for such
effects because it describes carbonic acid, the reaction
intermediate, as free isolated molecules without any interac-
tions with the cations present in the carbonate mineral
(Section 2.2.1). Thus, while our approach has the advantage
of allowing for a relatively rigorous treatment of part of the
acid digestion process, it is an over-simplification that will
not permit full understanding of differences in fractiona-
tions between various types of carbonate minerals. We have
tried to develop an understanding of these second-order ef-
fects by constructing a ‘cluster model’ that describes the dis-
sociation of H2CO3 attached to XCO3 clusters, which
simulates the situation where H2CO3is influenced by bonds
on the surface of adjacent, un-reacted carbonate. Similar ab
initio cluster models have been used previously to investi-
gate local properties and reactions of carbonate surfaces,
such as hydration (Mao and Siders, 1997) and adsorption
(Ruuska et al., 1999). In this study, we limit our model to
small clusters comprising only two XCO3 units, i.e.
(XCO3)2? H2CO3, where X = Mg2+, Ca2+, Mn2+, Fe2+,
Zn2+, Sr2+, Pb2+, Ba2+.
Following the same method outlined in previous sec-
tions, we obtain the structures of the transition states for
these clusters (Fig. 3) and derive the isotope fractionations
during phosphoric acid digestion of different carbonate
minerals. Note, we will confine our discussion on this clus-
ter model to Section 4.4. In other parts of the text, ‘‘model”
refers to the H2CO3dissociation model described in Section
2.2.1, unless stated otherwise.
2.3. Computational methods
Molecular geometries were optimized and bond frequen-
cies were calculated for the transition state isotopologues
using the Jaguar program (Ringnalda et al., 2005), on a
workstation cluster with 79 Dell PowerEdge-2650 server
nodes (Xeno, 2.2–2.4 GHz, 512 K) in the Materials and
Process Simulation Center at Caltech. The singlet state elec-
tron wave functions of the molecular configurations were
built using a density functional theory with hybrid func-
tionals, B3LYP, and extended basis sets 6–31G? (for the
H2CO3model) and LACV3P (for the cluster model). These
were selected based on their previous success in similar ab
initio models (Foresman and Frisch, 1993; Scott and
Radom, 1996; Ringnalda et al., 2005).
Ab initio harmonic oscillator calculations typically over-
estimate vibration frequencies, mostly because they neglect
the effects of anharmonicity (Scott and Radom, 1996).
Therefore, a scaling factor based on the comparison be-
tween the calculated bond vibration frequencies and exper-
imentally measured frequencies usually needs to be applied
to harmonic frequencies derived from ab initio models. No
carbonic acid transition state are available for comparison
with our ab initio model. Therefore, we have used a univer-
sal frequency scaling factor of 0.9614, previously shown to
be appropriate for B3LYP/6-31G? calculations (Scott and
Radom, 1996). To test the effectiveness of this assumed fre-
quency scaling factor, we computed the vibration frequen-
cies for the carbonic acid molecule using the B3LYP/6-
31G? method, applied the 0.9614 scaling factor, and
compared the scaled bond-vibration frequencies to the pre-
viously published results (Tossell, 2006) from more sophis-
ticated, higher level calculations (CCSD/6-311+G(2d,p)
level) and anharmonicity corrections (B3LYP/CBSB7
level). Results from these two independent models are
generally consistent with each other (Fig. 4), suggesting that
of the assumed
Phosphoric acid digestion of carbonate minerals7209
our scaling of the B3LYP/6-31G? model adequately ac-
counts for systematic errors due to anharmonicity and re-
lated effects. No frequency scaling was employed for the
cluster model calculations, due to the absence of a universal
scaling factor for the LACV3P basis set. However, we show
below that any influence of the scaling factor is likely to be
negligible for the general conclusions we reach based on our
3. EXPERIMENTAL METHODS
Starting materials for the carbonate recrystallization
experiments conducted as part of this study consisted of
three different calcite materials: NBS19 carbonate standard
distributed from IAEA (1 aliquot); MZ carbonate pre-
heated at 1100 ?C from Ghosh et al., 2006 (1 aliquot);
and Sigma-carb purchased from Sigma–Aldrich Co. (2
aliquots). The materials were loaded into Pt capsules, which
were then sealed by welding and inserted into CaF2cell
assemblies. The experiments were conducted in a piston-
cylinder apparatus at 1550 or 1650 ?C and either 2 or
3 GPa for 24 h (Table 2). Temperature was monitored using
type C thermocouples, uncorrected for the effect of pressure
on the electromotive force of the thermocouples. The
temperature and pressure conditions were chosen to be
close to or above the melting point of CaCO3 (Suito
et al., 2001) to ensure complete stochastic distributions of
isotopes (which might not occur over laboratory timescales
due to solid-state recrystallization alone; Ghosh et al.,
2006). Experimental charges were quenched rapidly by
turning off the power to the furnace, resulting in cooling
to below 200 ?C in less than 20 s and to room temperature
within 1 minute. Carbonate crystals were recovered by care-
fully stripping off the Pt capsule. The phase of each sample
was verified using X-ray diffraction, and the aliquots were
then reacted with anhydrous phosphoric acid (q = 1.91 g/
cm3) at 25 ?C for 18–24 h. The D47values of released CO2
were analyzed on a gas source mass spectrometer config-
ured to simultaneously measure masses 44 to 49. A detailed
description of the mass spectrometer configuration and
analysis procedures was given by Ghosh et al. (2006).
4. RESULTS AND DISCUSSION
4.1. Experimentally determined acid-digestion fractionation
We assume that heating CaCO3to temperatures and
pressures above its melting point should drive its13C and
18O toward a stochastic distribution. Thus, we anticipate
that the D47value of the CO2extracted from CaCO3that
has been quenched from melt should equal 0 (the stochastic
value) plus any fractionation associated with phosphoric
acid digestion. There is no simple way for us prove that
the stochastic distribution in carbonate is preserved during
rapid quenching from a melt, but this seems like a reason-
able inference given previous evidence that isotopic redistri-
bution in crystalline calcite is inefficient at laboratory
timescales, even at high temperature (Ghosh et al., 2006).
X-ray diffraction analyses confirmed that CaCO3samples
quenched from heating experiments are all of the calcite
structure. The D47values of CO2gases derived from these
CaCO3samples average 0.232 ± 0.015& (1r), and show
no systematic difference between the experiments at
1550 ?C/2 Gpa and at 1650 ?C/3 Gpa, nor any correlation
with the D47values of the CO2extracted from these samples
before recrystalization (Table 2). Note, such a correlation
Fig. 3. Representative transition state structures during phosphoric acid digestion of carbonate minerals, as in our ‘cluster models’ that
attempt to describe interactions between disassociating H2CO3and adjacent mineral surfaces. Individual structures are: (a) (CaCO3)2?
H2CO3; (b) (MgCO3)2? H2CO3; and (c) (BaCO3)2? H2CO3. Letters denote the chemical identity of each atom.
Fig. 4. Comparison of vibration frequencies of gas-phase carbonic
acid (H2CO3) obtained in this study using B3LYP/6-31G? ab initio
models with a scaling factor of 0.9614 vs. those obtained through
more sophisticated higher level calculations and anharmonicity
corrections (CCSD/6-311+G(2d,p) with B3LYP/CBSB7 based
anharmonic calculations and corrections; Tossell, 2006). The solid
line indicates a 1:1 correlation.
7210 W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
was observed by Ghosh et al. (2006), in the products of
solid-state recrystallization experiments, leading them to
conclude that such treatment led to only partial approach
to the stochastic distribution. These results support our
inference that our heating experiments succeeded at driv-
ing these samples to a stochastic distribution, and thus im-
ply that the ‘clumped isotope’ fractionation of
bonds during phosphoric acid digestion of calcite at
25 ?C corresponds to an increase in D47of 0.23& in prod-
4.2. Model results for the oxygen-isotope and clumped-
isotope fractionations associated with phosphoric acid
Table 3 summarizes the vibration frequencies we calcu-
late for the various isotopologues and isotopomers of tran-
sition states in our H2CO3dissociation model. The negative
frequencies, -1, correspond to the decomposition frequen-
Following procedures outlined in Section 3, these frequen-
cies are used in our transition-state-based predictions of the
proportions of different CO2isotopologues that are pro-
duced by dissociation of the H2CO3intermediate, and the
temperature dependence of those proportions. Unless sta-
ted otherwise, all of our calculations assume that reactant
carbonate has a d13CVPDBvalue of 0&, a d18OVSMOWvalue
of 0& and a stochastic distribution of multiply-substituted
isotopologues (Table 1), and that the H2CO3intermediate is
identical in isotopic composition to reactant carbonate
(1000lna*) and multiply substituted species ðD?
during phosphoric acid digestion as the differences between
d18O, D47, D48, D49 in the product CO2 and d18O,
D63,D64,D65in the reactant carbonates, respectively:
Lin Section 3.1 (Melander and Saunders, 1987).
1000lna?¼ 1000lnd18OCO2=1000 þ 1
d18OXCO3=1000 þ 1;
48¼ D48? D64;
47¼ D47? D63;
49¼ D49? D65
where D48, D49, D63, D64and D65are defined, similar to D47,
following the same principle as in Eiler and Schauble
where ‘s’ in the subscript denotes the expected abundance
of an isotopologue when all the isotopes are stochastically
distributed; and X denotes the cation contained in the car-
Fig. 5 and Table 4 present the oxygen isotope fractiona-
tions that accompany phosphoric acid digestion over a
range of relevant temperatures, as predicted by our transi-
tion-state theory model. The predicted oxygen isotope frac-
tionation and its temperature dependence are broadly
similar to those determined for different carbonate minerals
in previous laboratory studies. At 25 ?C, our model pre-
dicted oxygen isotope fractionation of 10.72&, is in the
middle of the range of observed fractionations among dif-
ferent carbonate minerals (from 10.06& for MnCO3 to
11.92& for MgCO3), and is close to the experimental
Fractionation of multiply-substituted isotopologues, D?
determined through phosphoric acid digestion of high temperature and pressure equilibrated CaCO3(calcite) samples.
47(see text for the definition), during phosphoric acid digestion of CaCO3at 25 ?C
SampleBefore re-crystallizationRe-crystallization experiments After re-crystallization
a1r denotes the external standard deviation. Numbers in the bracket indicate the numbers of independent replicate extraction and isotopic
analyses of the carbonate samples after re-crystallization.
Phosphoric acid digestion of carbonate minerals7211
Scaled vibration frequencies (unit: cm?1) for different transition state structure isotopomers during phosphoric acid digestion of carbonate
minerals (H2CO3model, DFT-B3LYP/6-31G? with a frequency scaling factor of 0.9614). Oxygen atoms in the transition state structure
isotopomers are expressed in the order of atom 2, 3 and 4 in Fig. 2.
7212W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
determined fractionation for calcite (10.20&). The temper-
ature sensitivity of our predicted oxygen isotope fraction-
ation during phosphoric acid digestion (?0.055&/?C at
25 ?C) is also only slightly above the range of temperature
sensitivities experimentally determined for different carbon-
ate minerals (MnCO3appears to possess the highest tem-
perature sensitivity of oxygen isotope aid digestion
fractionation, ?0.052&/?C at 25 ?C).
Our transition-state-theory model of phosphoric acid
digestion predicts that CO2 produced by dissociation of
H2CO3intermediates has abundances of
as reflected by the D?
D63value of reactant carbonate at 25 ?C, with a tempera-
ture sensitivity of ?0.0010&/?C over the temperature range
of 25 ?C to 80 ?C. The predicted fractionation at 25 ?C is
indistinguishable from this study’s experimentally deter-
mined value of 0.232 ± 0.015& (1r) for calcite (Table 2),
and the predicted temperature dependence is close to the
experimentally measured value of ca. ?0.0016&/?C (Ghosh
et al., 2006). There are no experimental data documenting
fractionations of12C18O2 and13C18O2 isotopologues dur-
ing acid digestion of carbonates, but for future reference
we note here that our transition state theory model predicts
of ?0.0011&/?C, and D?
temperature dependence of ?0.0033&/?C (Fig. 6).
The most obvious weakness of our transition state the-
ory model is the need to choose a frequency scaling factor
(which mainly reflects the effects of anharmonicity; see
Section 2.3, above). We tested the potential effects of this
frequency scaling by repeating our calculations with no
47value, +0.220& higher than the
48to be 0.137& at 25 ?C with a temperature dependence
49to be 0.593& at 25 ?C with a
scaling factors. In this case, the predicted acid digestion
fractionations at 25 ?C are 11.32& for 1000lna*, 0.235&
are sufficiently similar to the results of our preferred model
that we do not regard the frequency scaling factors as plau-
sible sources of large systematic error.
Our transition state theory model also predicts the mass
dependency of the oxygen isotope fractionation that
accompanies phosphoric acid digestion. This is relevant
for analyses of the isotopic compositions of reactant car-
bonates because one must assume the mass dependence of
the acid digestion fractionation in order to ion-correct the
measured mass spectrum of product CO2. Generally speak-
ing, measurements of the carbon and oxygen isotope com-
positions of CO2 on a gas source isotope ratio mass
spectrometer examine CO2isotopologue ions having nomi-
nal molecular masses of 44, 45 and 46 amu. Because the
instruments commonly used for this purpose cannot mass
assumptions to correct for the contribution of12C17O16O
to the mass 45 amu ion beam. This is generally accom-
plished by assuming a relationship between
abundance of the form:
inkmeijer, 2003; Miller et al., 2007), where the value k must
be assumed or determined by independent experiments
(such as fluorination of reactant carbonate and product
CO2 followed by isotopic analyses of the resulting O2
gases). To the best of our knowledge, there are no experi-
mental determinations of k associated with phosphoric acid
digestion of carbonate minerals; a value 0.528 has been sug-
gested (Assonov and Brenninkmeijer, 2003; Miller et al.,
2007). This value of k characterizes the isotopic variations
of natural waters (Li and Meijer, 1998; Barkan and Luz,
2005), and presumably is inherited by carbonate minerals
that form in isotopic equilibrium with natural waters,
although there is no reason to suppose it is also character-
istic of the acid digestion reaction process by which carbon-
ates are analyzed. Our transition state theory of phosphoric
acid digestion predicts that the value of k associated with its
isotopic fractionations is 0.5281. Thus, our model agrees
with and provides an independent theoretical justification
for the suggested value of 0.528 for CO2extracted from
carbonate samples (Miller et al., 2007) and standards (e.g.
PDB and NBS-19; Assonov and Brenninkmeijer, 2003).
This result may be useful for the interpretation of high-
carbonates by fluorinating CO2that is generated by phos-
phoric acid digestion.
47, 0.156& for D?
48, and 0.642& for D?
49. These results
12C17O16O, one must make some
(Assonov and Brenn-
4.3. Dependence of acid digestion fractionations on the
isotopic compositions of reactant carbonate minerals?
In a recent experimental determinations of oxygen iso-
tope fractionations associated with phosphoric acid diges-
tion, Kim and O’Neil’s, 1997 observed apparent variations
of the fractionation factors within a single type of carbonate
minerals at a single temperature (25 ?C): up to 0.5& for
calcite (CaCO3), 0.6& for witherite (BaCO3), and 2.5& for
octavite (CdCO3). While Kim and O’Neil (1997) suggested
Fig. 5. Oxygen isotope fractionations (1000lna*, where a*is the
18O/16O ratio of product CO2 divided by that of reactant
carbonate) plotted vs. 105T?2in K (the upper horizontal edge
indicates T in ?C, for reference). The dashed line is the predicted
temperature-dependent fractionation based on our model of
H2CO3dissociation. Labeled solid lines are measured experimental
values for various metal carbonates (Table 4). The structurally
simple transition-state structure model we propose captures the
first-order magnitude and temperature dependence of observed
fractionations, and mostly closely approaches the best-determined
value for calcite.
Phosphoric acid digestion of carbonate minerals 7213
conditions of those carbonates, we are not aware of any de-
We performed a statistical analysis of the experimental
data presented in Kim and O’Neil (1997), and observed
statistically significant correlations between the oxygen
isotope acid digestion fractionations and the oxygen
isotope compositions of the reactant carbonates, with
proportionalities of: 0.03& change in fractionation per
permil in reactant d18O for calcite; 0.036& per permil for
octavite; and 0.06 & per permil for witherite (Fig. 7). Inter-
estingly, we find these observed correlations might help
explain some of the discrepancies between independent
determinations of acid digestion fractionation factors,
e.g., for octavite and calcite in Kim and O’Neil, 1997 and
Sharma and Clayton (1965) (Fig. 7). However, the discrep-
ancy between these two studies in the acid digestion frac-
tionation for witherite cannot be explained in this way
The possibility that acid digestion fractionations depend
on d18O of the reactant carbonate is generally neglected in
studies of carbonate stable isotope composition, though the
existence of such an effect could lead to significant system-
atic errors for some materials. We examined this issue by
recalculating our transition state theory model for a range
substituted isotopologue proportions (i.e., values of D63,
D64, etc.) in the reactant carbonate (assumed identical to
H2CO3intermediate) at a constant assumed acid digestion
temperature of 25 ?C (Table 5). The proportions of multi-
ply substituted isotopologues inside reactant carbonate
are calculated based on the equilibrium constants of isotope
exchange reactions between different carbonate ion isotopo-
logues, following the similar algorithm as presented in
Wang et al. (2004) (see Appendix A-1 for details).
We observe no dependence of the oxygen isotope acid
digestion fractionation (1000lna*) on the d18O value of the
reactant carbonate, and thus our model does not provide
an explanation of such trends in the experimental data of
Kim and O’Neil (1997). Given the general success of our
model in describing the magnitude and temperature
dependence of 1000lna*(Fig. 5), this discrepancy likely
indicates that the trends observed by Kim and O’Neil
(Fig. 7) are not an intrinsic feature of the kinetic isotope
effect that accompanies phosphoric acid digestion of
carbonate. Such trends might instead reflect systematic er-
rors in the fluorination measurements that were used to
determine the bulk d18O of reactant carbonates (e.g., as
might result from an unrecognized analytical blank or
contaminant), or, as suggested by Kim and O’Neil
(1997), some cryptic artifact particular to the synthesis
of the carbonate materials they studied.
However, the transition-state-theory models summa-
rized in Table 5 show an unexpected dependence of the
fractionations of multiply-substituted isotopologues (i.e.,
values of D?
substituted isotopologues of reactant carbonates (i.e., val-
ues of D63, D64and D65). Most importantly, D?
tionation that directly influences the results of carbonate
clumped isotope thermometry, is predicted to increase by
?0.035& for every 1& increase in the D63value of reactant
carbonate. This non-ideality in the clumped isotope fracti-
onations arises from a peculiarity in the way the Divalues
are defined (see Appendix A-2 for details).
The available experimental data do not directly test our
predicted dependence of D?
However, the predicted effect does offer a partial explana-
49) on proportions of multiply
47, the frac-
47on D63of reactant carbonate.
Comparison of model predicted and experimentally observed phosphoric acid digestion fractionations.
range ( ?C)
(&, 25. ?Cb)
(&, 25 ?Cd)
Calcite group Dolomite(CaMg(CO3)2)
4.23 + 6.84 ? 105/T2
3.96 + 6.69 ? 105/T2
3.85 + 6.84 ? 105/T2
2.29 + 6.91 ? 105/T2
3.89 + 5.61 ? 105/T2
4.24 + 5.44 ? 105/T2
5.30 + 4.59 ? 105/T2
5.13 + 4.79 ? 105/T2
5.76 + 4.58 ? 105/T2
2.58 + 7.25 ? 105/T2
0.0186+0.179 ? 105/T2
?0.0787 + 0.192 ? 105/T2
?0.0386 + 0.561 ? 105/T2
Sharma and Clayton, 1965
Das Sharma et al., 2002
Gilg et al., 2003
Rosenbaum and Sheppard, 1986
Das Sharma et al., 2002
Recalculated from Kim et al. 2007c
Sharma and Sharma, 1969a
Gilg et al., 2003
aOxygen isotope compositions of the reactant carbonate minerals employed in different experimental studies.
bEquations for experimentally determined oxygen isotope phosphoric acid digestion fractionation, as summarized in Gilg et al., 2003, where T is in the unit of Kelvin.
Isotope fractionations at 25 ?C are estimated from these equations.
cThe oxygen isotope composition of the reactant aragonite was recalculated at 100% total oxygen yield, to account for the inverse correlation between the total oxygen
isotopic composition of reactant aragonite and the total oxygen yield from decarbonation and fluorination steps (d18Oaragonite= ?0.1362 ? Yield% + 38.642, Fig. 2 and Table
4 in Kim et al., 2007). The aragonite phosphoric acid digestion fractionations were adjusted accordingly.
eCalculations in our H2CO3dissociation model assumes d13CVPDB= 0&, d18OVSMOW= 0& and stochastic distribution of multiply-substituted isotopologues for the
47for different carbonate minerals at 25 ?C, based on the inverse correlation between 1000lna*and D?
47predicted by our cluster model. See Section 4.4.3 for
7214W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
tion why the experimentally determined relationship be-
tween the D47of CO2produced by acid digestion of calcite
and calcite growth temperature (Ghosh et al., 2006) is more
sensitive to temperature than the theoretically predicted
temperature dependence for D63in carbonates (Schauble
et al., 2006). For example, over the temperature range of
0–50 ?C, Schauble et al. (2006) predicts D63 (including
contributions from both D13C18O16O2and D12C18O17O16O; see
Appendix A-2) in thermodynamically equilibrated calcite
solids decreases by 0.00279& for every degree increase of
its growth temperature; Assuming calcite has the same
dependence of D?
H2CO3model, we predict the D47of CO2produced by acid
digestion of calcite will have a temperature sensitivity of
?0.00289&/?C, which is closer to the ?0.00453&/?C deter-
mined experimentally by Ghosh et al. (2006) than the theo-
retically predicted temperature sensitive of D63(Schauble
et al., 2006).
47on D63 as predicted by our above
4.4. Cation effects on acid digestion fractionations
The transition-state model we present in preceding
sections simultaneously explains a variety of features of
the kinetic isotope effects associated with phosphoric acid
digestion of carbonates, including the magnitude and tem-
perature dependence of 1000lna*and D?
Given that all of these predictions are strictly independent
of the experimental data to which they are compared, we
contend that our model closely captures the most important
mechanistic details of this reaction. However, phosphoric
acid digestion of carbonates is also believed to exhibit a
dependence of 1000lna*on the cation chemistry (and pos-
sibly crystal structure) of reactant carbonate (Table 4;
Fig. 5). At 25 ?C, the observed oxygen isotope fractiona-
tions among different carbonate minerals vary from
10.06& (MnCO3) to 11.92& (MgCO3), and the tempera-
ture sensitivity of oxygen isotope fractionations during
phosphoric acid digestion vary from ?0.027&/?C (BaCO3)
to ?0.041&/?C (MnCO3) over the temperature range of
25 ?C to 80 ?C (Table 4; Fig. 5). Nothing in our above mod-
el of H2CO3dissociation can explain such observations. In
this section, we use a cluster model of the reacting carbon-
ate surface (Section 2.2.3 and Fig. 3) to explore the possible
causes of these effects.
4.4.1. Cluster model results on the oxygen isotope
fractionation among different carbonate minerals
Table 6 and Fig. 8 present the predictions of our cluster
model on the variations of oxygen isotope fractionation
among different carbonate minerals, and the comparisons
between these cluster model predictions and the results
determined from previous experimental studies. Our cluster
model of the carbonate surface, in which the H2CO3inter-
mediate interacts with adjacent metal-carbonate groups,
succeeds in capturing the experimentally-observed depen-
dence of 1000lna*on cation composition, but fails to de-
scribe the absolute values and absolute temperature
dependences of 1000lna*(and D?
pler H2CO3dissociation model (above).
In particular, our cluster model predicts values of
1000lna*at 25 ?C from 1.771& (PbCO3) to 3.652&
(FeCO3) for the eight different carbonate minerals studied
(Table 6). Except for PbCO3, these predicted oxygen iso-
tope fractionations are approximately one-third the experi-
mentally observed values. Nevertheless, the predicted
differences in oxygen isotope fractionation between differ-
ent minerals generally reproduce those observed in previous
experimental studies (Fig. 8a). The oxygen isotope fraction-
ation predicted by our cluster model for PbCO3is an excep-
tion, falling far below the trend defined by other carbonate
minerals. This might be related to the spin-orbit effects and
the basis set superposition error in ab initio calculations of
Pb-containing complexes with effective core potential basis
sets (Ramirez et al., 2006). We evaluate the temperature
sensitivity of oxygen isotope acid digestion fractionation
predicted by our cluster model as the ratio of predicted oxy-
gen isotope fractionation between 80 ?C and 25 ?C, and
compare themwith the
(Fig. 8b). The model prediction and experimental data fall
close to 1:1 correlation, indicating our cluster model, de-
spite its obvious failure at matching the absolute magni-
tudes of acid digestion fractionations,
variations of the temperature sensitivity of oxygen isotope
acid digestion fractionation among different carbonate min-
erals (including PbCO3; Fig. 8b).
We conclude that our cluster model captures some
essential features of the cation effects during phosphoric
acid digestion, but is quantitatively inaccurate because it
fails to describe the structural relationship between
H2CO3 and the crystal surface. This deficiency is likely
due to the small size and simple geometry of the clusters
we have modeled. It should be possible to refine this
model so that it simultaneously describes the absolute
47) characteristic of our sim-
Fig. 6. Fractionations of multiply-substituted species ðD?
dissociation model, plotted as a function of 105T?2, in K. The solid
circle is the average value of D?
phosphoric acid digestion of calcites at 25 ?C (Table 2; this study).
The bar is 1 standard deviation (1r) of multiple replicate
extractions of the calcites (the standard error of the average is
approximately the size of the symbol).
49Þ during phosphoric acid digestion predicted by our H2CO3
47experimentally determined during
Phosphoric acid digestion of carbonate minerals 7215
values and temperature dependencies of the acid digestion
fractionations and their dependence on reactant composi-
tion and structure, although we suspect this would require
a sophisticated treatment of the extended structures of
surfaces that is beyond the scope of this study (e.g., Keri-
sit et al., 2003). Nevertheless, the fact our cluster model
reproduces the observed variations in size and temperature
sensitivity of oxygen isotope fractionations among several
carbonate minerals, combined with the substantial success
of our simpler H2CO3-dissociation model in describing the
systematics of phosphoric acid fractionations generally,
suggests that the conceptual framework will provide a use-
ful foundation for a more structurally complex model of
4.4.2. Controls on the variations of acid digestion isotope
fractionations among different carbonate minerals
Previous attempts (e.g., Bottcher, 1996; Gilg et al., 2003)
to understand the variations of acid digestion fractionation
among different carbonate minerals examined empirical cor-
relations between the intercept in a plot of 1000lna*vs. 1/T2
and the radius or mass of the cation in the reactant carbon-
ate (following the suggestion of Sharma and Sharma,
1969b). These efforts failed to reveal any simple correlation
shared by all minerals. (Note that, O’Neil, 1986 made an
alternate suggestion that 1000lna*values for phosphoric
acid digestion reactions might also be controlled by reaction
rate; Kim and O’Neil, 1997) However, it is not clear what
physiochemical meaning should be attached to the intercept
in a plot of acid digestion fractionation vs. 1/T2. In this
study, we focus on the differences of 1000lna*among differ-
ent carbonate minerals at a given temperature (e.g. 25 ?C)
and its temperature dependencies separately.
We compare values of 1000lna*at 25 ?C and the temper-
ature dependencies of 1000lna*(from both model prediction
and experimental determination) for different carbonate
minerals against their respective cation radius (Fig. 9). The
theoretically predicted values of 1000lna*at 25 ?C positively
Fig. 7. Empirically observed correlations between the oxygen isotope fractionations associated with phosphoric acid digestion and the oxygen
isotopic composition of reactant carbonates. Experimental data are from Kim and O’Neil (1997) and the solid lines are the least square
regressions to the experimental data. Neither our H2CO3dissociation model nor our more complex ‘cluster’ models predict a correlation
between these two variables. Also shown for comparison (star symbols) are the data from Sharma and Clayton (1965).
7216 W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
correlate with the reciprocal of cation radius (Fig. 9a). The
theoretically predicted, as opposed to experimentally deter-
mined, 1000lna*at 25 ?C show better correlations in this
comparison, because significant uncertainties are associated
with experimental determinations of 1000lna*(more specif-
ically, associated with the determination of total oxygen iso-
topic compositions of reactant carbonates through the
fluorination methods; Kim et al., 2007). In contrast, both
theoretically predicted and the experimentally determined
temperature dependencies of 1000lna*exhibit positive
correlations with the reciprocal of cation radius (Fig. 9c
and d). The experimental determinations of the temperature
dependencies of the oxygen isotope acid fractionation
have much smaller uncertainties (since the uncertainties
associated with fluorination methods cancel out in the
determination of temperature dependence), and therefore
show better correlation in the above comparison. The exis-
tence of these correlations suggests that cation radius is
the most important mineral-specific factor controlling acid
To examine the possible effects of cation mass on the
acid digestion fractionation, we adopt a similar strategy
to that employed by Schauble et al. (2006) in their discus-
sion of equilibrium carbon and oxygen isotope fractiona-
tions among different carbonate minerals, and create two
hypothetical isotopic carbonates,40MgCO3and40BaCO3.
In these cluster models, all the optimization and calcula-
tion were performed in the same manner as outlined
Variations of acid digestion isotope fractionations and their temperature dependencies among different carbonate minerals, predicted from
our cluster model. The cation radius and mass are shown for comparison.
Cation radiusar, (A˚)
0.650.700.79 0.801.02 (cal)
Cation mass, m24 6556 55 88207137 40 40
25 ?C 1000lna*
?0.0311 ?0.0581 ?0.0538
50 ?C 1000lna*
?0.0233 ?0.0450 ?0.0414
80 ?C 1000lna*
?0.0166 ?0.0337 ?0.0308
0.0971 0.03190.0691 0.0744
aCation radii from Golyshev et al. (1981). For CaCO3, the Ca cation radii are different in its two polymorphs, calcite and aragonite, and are
1.02 A˚and 1.26A˚respectively.
Dependences of phosphoric acid digestion fractionations on the isotopic compositions and the distributions of multiply-substituted
isotopologues in reactant carbonates at 25 ?C.
Equil. Ta(K)Random Random
2?in reactant carbonate
Model prediction on product CO2
Acid digestion fractionation 1000lna*(&)
aHypothetical equilibration temperature for the distributions of multiply-substituted isotopologues inside reactant carbonates. D63, D64,
and D65(see text for definitions) are estimated based on calculations for an isolated CO3
2?group in the gas phase (see Appendix A-1 for
Phosphoric acid digestion of carbonate minerals7217
above for their isotopically normal equivalents, except the
atomic masses of Mg and Ba were both assigned as
40 amu instead of their normal masses (24.3 amu and
137.3 amu, respectively). If cation mass controls the acid
digestion fractionation, we would expect the model pre-
dicted fractionation for these two hypothetical carbonates
to closely resemble the predicted fractionation for CaCO3,
which has a cation mass of 40.1 amu. Instead, we observe
negligible differences in predicted acid digestion fractiona-
tions between these hypothetical carbonates and their iso-
topically normal equivalents (within 0.03& for MgCO3
and 0.01& for BaCO3; Table 6). Thus, the cation effect
on acid digestion fractionation most likely reflects cation
size, not mass.
Besides cation radius and mass, crystal structure has
been invoked by previous studies as another potential
factor that might place important controls on the acid
digestion fractionation (Gilg et al., 2003). The fact that
our cluster model, which did not consider the effects of
crystal structure, managed to semi-quantitatively repro-
duce the general trend of variations of acid digestion
(Fig. 8), suggests that the crystal structure, like cation
mass, exerts only very weak control on the acid digestion
4.4.3. Cluster model results on the fractionation of13C–18O
doubly substituted isotopologues during phosphoric acid
Our cluster model predicts that the D?
with phosphoric acid digestion at 25 ?C range from
0.0375& (MnCO3) to 0.0761& (BaCO3) among the eight
carbonates mineral studied (Table 6). As for oxygen isotope
acid digestion fractionations, our cluster model appears to
underestimate the absolute values of D?
associated with phosphoric acid digestion. Most of the pre-
dicted values of D?
imental observed D?
47are approximately 1/3 to 1/4 of exper-
47for calcite (0.23&, Section 4.1).
We observe that values of D?
model correlate negatively with their respective predicted
1000lna*fractionations during phosphoric acid digestion
(Fig. 10). If this correlation is general to all carbonate min-
erals, one could predict the D?
carbonate mineral, XCO3(where X is a cation), at temper-
47predicted by our cluster
47during acid digestion of any
where 1000lna*(XCO3, T) is the experimentally determined
oxygen isotope acid fractionation at temperature T, and
1000lna*(H2CO3, T) and D?
oxygen isotope and D?
perature T fromour transition-state-theory H2CO3dissocia-
tion model. Note, this relationship examines an empirical
correlation between two properties that are grossly under-
predicted by our cluster model. Nevertheless, the statistical
relationship is strong and thus might provide some basis
for predicting the currently unknown relationships between
cation content and acid digestion fractionations of13C–18O
bonds. At 25 ?C, this gives D?
This relationship predicts that the acid digestion fraction-
(MgCO3) to 0.234& (MnCO3) at 25 ?C (Table 4). More spe-
cifically, it predicts that the acid digestion fractionation, D?
for calcite is 0.231& at 25 ?C—in agreement with our
experimentally determined value of 0.232 ± 0.015&.
We combine previous theoretical estimations on the
temperature dependence of13C–18O and18O–17O clumping
in various carbonate minerals (Schauble et al., 2006) with
our predicted acid digestion fractionations, D?
the temperature calibration lines for the clumped isotope
thermometer in various carbonates (i.e., D47of CO2derived
from the phosphoric acid digestion v.s. temperature;
Fig. 11). Note that,18O–17O clumping was reported only
47(H2CO3, T) are the predicted
47acid digestion fractionation at tem-
47, for different carbonates ranges from 0.198&
47, to predict
Fig. 8. Comparison of the isotope fractionations associated with phosphoric acid digestion predicted by our ‘cluster models’ with
experimentally observed fractionations for various metal carbonates. Panel (a) depicts the oxygen isotope fractionation at 25 ?C; (b) depicts
the temperature sensitivity of the oxygen isotope fractionation, as measured by the ratio of oxygen isotope fractionations at 80 ?C and 25 ?C.
Experimental data were based on equations in Table 4. The dashed line in panel (b) indicates a 1:1 correlation.
7218W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
for calcite in Schauble et al. (2006). We therefore assume in
our estimation the same18O–17O clumping as in calcite for
all carbonate minerals. We think this is an appropriate
assumption and will not introduce any significant system
errors in our estimation, because18O–17O clumping effects
contribute only ?0.7% of D63 signal in the carbonate
minerals and furthermore are suspected to vary negligibly
among different carbonate minerals based on the reported
variations among different carbonate minerals at 25 ?C;
Schauble et al., 2006). The combination of these two theo-
retical approaches suggests that the D47value of CO2ex-
tracted at a given temperature from different carbonates
minerals grown at the same temperatures can vary by as
much as 0.05&, decreasing in the order: CaCO3(arago-
nite) > CaCO3(calcite) > BaCO3(witherite) > CaMgCO3(dolo-
mite) ? MgCO3 (magnesite). The temperature calibration
line predicted in this way for calcite and aragonite closely
13C–18O clumping effects (less than 7%
Fig. 9. Correlations between fractionations of oxygen isotopes associated with acid digestion and the ionic radius of the cation in the reactant
carbonate. Panel (a) depicts the predicted fractionations of our ‘cluster model’ for phosphoric acid digestion at 25 ?C; (b) experimentally
determined acid digestion fractionation at 25 ?C; (c) shows ‘cluster model’ predicted temperature dependence factor B for oxygen isotope acid
digestion fractionation (as expressed in the form of 1000lna*= A+B ? 105/T2; Table 4); (d) experimentally determined temperature
dependence factor B for oxygen isotope acid digestion fractionation, against cation radius. These observed correlations indicate cation radius
is the predominant factor controlling the variations of acid digestion fractionations among different carbonate minerals. Data on the cation
radii are from Golyshev et al. (1981).
Fig. 10. Inverse correlation between the oxygen isotope fraction-
ation and D?
different carbonate minerals, predicted from our cluster model.
47fractionation during phosphoric acid digestion of
Phosphoric acid digestion of carbonate minerals7219
approach the available experimental calibrations (Ghosh
et al., 2006, 2007; Came et al., 2007; Fig 12). However,
the slope of our theoretical calibration line is significantly
shallower than that of the experimental calibration line at
low temperatures (?300 K), with a temperature sensitivity
of ?0.00289&/?C over the temperature range 0–50 ?C, as
compared to the ?0.00453&/?C sensitivity determined
experimentally. Finally, for future reference, our predicted
calibration line for calcite, dolomite, and magnesite
can be represented over the temperature range 260–
1500 K by
Calcite D47¼ ?3:33040 ? 109
?2:91282 ? 103
þ2:32415 ? 107
Aragonite D47¼ ?3:43068 ? 109
þ2:35766 ? 107
?8:06003 ? 102
Dolomite D47¼ ?3:31647 ? 109
þ2:29414 ? 107
?2:38375 ? 103
Magnesite D47¼ ?3:31658 ? 109
þ2:19871 ? 107
?2:83346 ? 103
where T is the temperature in Kelvin and phosphoric
acid digestions of carbonate minerals are assumed to be at
We present the first quantitative theoretical models of
the isotope fractionations during phosphoric acid digestion
of carbonate minerals, by using classical transition state
theory and ab initio calculations to predict the relative rates
of reaction of all the isotopologues of reactant carbonate
species. These models assume that the critical reaction
intermediate during phosphoric acid digestion is H2CO3,
and thus that isotope fractionations associated with acid
digestion are controlled by kinetic isotope effects during dis-
sociation of H2CO3.The simplest form of this model (which
considers only H2CO3dissociation as isolated free mole-
cules) predicts the isotope fractionations between product
CO2and reactant carbonate to be:
1000lna?¼ 2:58 þ 7:25 ? 105=T2;
47¼ 0:0242 þ 0:189 ? 105=T2;
48¼ ?0:0825 þ 0:213 ? 105=T2;
49¼ ?0:0308 þ 0:602 ? 105=T2;
where T is the temperature of acid digestion and in the unit
of Kelvin. Both the magnitudes (10.72&, 0.220&, 0.137&,
0.592& for 1000lna?;D?
and the temperature-dependences of the these predicted iso-
tope fractionations agree well with available experimental
data on oxygen isotope fractionations and our newly deter-
mined D47fractionation of 0.232 ± 0.015& for calcite dur-
ing phosphoric acid digestion.
A subset of our models attempt to take into account
also the influence of cation composition by permitting
49respectively, at 25 ?C)
Fig. 11. Predicted temperature calibration lines for different
carbonate clumped isotope thermometers, by combining predicted
carbonate minerals (Schauble et al., 2006) and predicted D?
fractionations during phosphoric acid digestion of carbonate
minerals (this study; see text for details). Phosphoric acid digestions
of carbonate minerals are assumed to be at 25 ?C.
18O–17O clumping effects inside the
Fig. 12. Comparison between predicted temperature calibration
(this study) and the experimental temperature calibration data
acid digestions of carbonate minerals are performed at 25 ?C.
7220 W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
the H2CO3 reaction intermediate to interact with an
adjacent metal carbonate group. These ‘cluster models’
underestimate the magnitude of isotope fractionations
associated with phosphoric acid digestion by a factor of
?3, presumably because we have incorrectly described
the structure of nearest-neighbor interactions between
H2CO3 reaction intermediate and the reacting mineral
surface. Nevertheless, our cluster models reproduce the
general trend of variations (in both size and temperature
sensitivity) of oxygen isotope acid digestion fractionation
among different carbonate minerals, which therefore pro-
vides a general concept framework for future, more
sophisticated first-principles models of the carbonate-acid
We thank Edwin Schauble for helpful discussions on the
equilibrium clumped isotope effects in carbonate minerals. We
thank Juske Horita for his editorial handling of this manuscript,
and two anonymous reviewers for thoughtful and constructive
A.1. Estimation on the equilibrium distributions of multiply-
substituted isotopologues inside reactant carbonate mineral
for assumed temperatures and bulk isotopic compositions
We calculate the distributions of all singly and multiply-
substituted isotopologues inside reactant carbonate mineral
at assumed equilibration temperatures, following the method-
ology and algorithms presented by Wang et al. (2004) in their
theoretical estimations of abundances of multiply-substituted
ate mineral has a total of 20 isotopologues. To determine the
abundances of all isotopologues, we select the abundances of
non-substituted (12C16O16O16O2?) and singly-substituted
12C17O16O16O2?) as the fundamental unknowns, and express
the abundances of the other 16 multiply-substituted isotopo-
logues as functions of these fundamental unknowns and the
½13C18O16O16O2?? ¼½13C16O16O16O2?? ? ½12C18O16O16O2??
½13C17O16O16O2?? ¼½13C16O16O16O2?? ? ½12C17O16O16O2??
½12C17O18O16O2?? ¼½12C17O16O16O2?? ? ½12C18O16O16O2??
½13C17O17O16O2?? ¼½13C16O16O16O2?? ? ½12C17O16O16O2??2
½12C17O17O18O2?? ¼½12C17O16O16O2??2? ½12C18O16O16O2??
½13C17O17O17O2?? ¼½13C16O16O16O2?? ? ½12C17O16O16O2??3
½13C18O18O16O2?? ¼½13C16O16O16O2?? ? ½12C18O16O16O2??2
½12C17O18O18O2?? ¼½12C17O16O16O2?? ? ½12C18O16O16O2??2
¼½13C16O16O16O2?? ? ½12C17O16O16O2?? ? ½12C18O16O16O2??2
¼½13C16O16O16O2?? ? ½12C17O16O16O2?? ? ½12C18O16O16O2??2
½13C18O18O18O2?? ¼½13C16O16O16O2?? ? ½12C18O16O16O2??3
where K denote the equilibrium constants for the related
isotope exchange reactions at the specified equilibration
Phosphoric acid digestion of carbonate minerals7221
K3866¼Q13C18O16O16O2? ? Q12C16O16O16O2?
Q13C16O16O16O2? ? Q12C18O16O16O2?
!13C17O18O16O2?þ 2 ?12C16O16O16O2?
Q13C16O16O16O2? ? Q12C17O16O16O2? ? Q12C18O16O16O2?
Q here refer to the partition functions of different CO3
isotopologues, and can be evaluated with their respective
scaled vibration frequencies through principles of statistical
thermodynamics (Urey 1947).
Rather than performing complete calculations of these
16 equilibrium constants for multiply-substituted isotopo-
logues in carbonate mineral lattices, as Schauble et al.
(2006) did for K3866and K2876, we reduced the necessary
computation power by calculating the corresponding equi-
librium constants for isolated CO3
and use them to approximate the equilibrium constants in
carbonate lattice. The distribution of multiply-substituted
isotopologues inside the two are remarkably similar (e.g.,
at equilibration temperature of 300 K, K3866= 1.0004034
in isolated CO3
cite lattice, Schauble et al., 2006), and therefore this
approximation appears not to introduce any significant sys-
tematic errors on our model results. The geometry optimi-
zation and frequency calculation for the isolated CO3
are performed at DFT-B3LYP/6-31G? level, and the fre-
quencies are scaled with the universal scaling factor of
0.9614 as discussed in Section 2.3 (Table A1).
We combine the above equations with the constraints
from the bulk isotopic composition of the CO3
½13C16O16O16O2?? þ ½13C17O16O16O2??
þ ½13C18O16O16O2?? þ ½13C17O17O16O2??
þ ½13C17O18O16O2?? þ ½13C18O18O16O2??
þ ½13C17O17O17O2?? þ ½13C17O17O18O2??
þ ½13C17O18O18O2?? þ ½13C18O18O18O2?? ¼ ½13C?
Q13C17O18O16O2? ? ðQ12C16O16O16O2?Þ2
2?ions in the gas phase,
2?, this study; v.s. K3866= 1.0004066 in cal-
3 ? ½12C16O16O16O2?? þ 2 ? ½12C17O16O16O2??
þ 2½12C18O16O16O2?þ ½12C17O17O16O2??
þ ½12C17O18O16O2?? þ ½12C18O18O16O2?? þ 3
? ½13C16O16O16O2?? þ 2 ? ½13C17O16O16O2?? þ 2
? ½13C18O16O16O2?? þ ½13C17O17O16O2??
þ ½13C17O18O16O2?? þ ½13C18O18O16O2?? ¼ 3 ? ½16O?
½12C16O16O16O2?? þ ½12C17O16O16O2??
þ ½12C18O16O16O2?? þ ½12C17O17O16O2??
þ ½12C17O18O16O2?? þ ½12C18O18O16O2??
þ ½12C17O17O17O2?? þ ½12C17O17O18O2??
þ ½12C17O18O18O2?? þ ½12C18O18O18O2??
þ ½13C16O16O16O2?? þ ½13C17O16O16O2??
þ ½13C18O16O16O2?? þ ½13C17O17O16O2??
þ ½13C17O18O16O2?? þ ½13C18O18O16O2??
þ ½13C17O17O17O2?? þ ½13C17O17O18O2??
þ ½13C17O18O18O2?? þ ½13C18O18O18O2?? ¼ 1
where [13C], [18O], [16O] refer to the bulk
16O abundances in the CO2?
culated from its given bulk isotopic composition. By
simultaneously solving these 20 equations (using fsolve
function in MATLAB program, version 7.04), we obtain
the abundances of all 20 isotopologues at specified bulk
respectively an can be cal-
A.2. Predicted dependence of the fractionations of multiply-
substituted isotopologues on proportions of multiply
substituted isotopologues of reactant carbonates
Our H2CO3dissociation model predicted an unexpected
dependence of the fractionations of multiply-substituted
isotopologues (i.e., values of D?
tions of multiply substituted isotopologues of reactant car-
bonates (i.e., values of D63, D64, and D65) (Table 5; Section
4.3). We show below this dependence arise from a peculiar-
ity in the way the Divalues are defined.
pendent of reactant composition; that is, D13C18O16Oof prod-
uct CO2differs from D13C18O16O2of reactant carbonate by an
amount that varies with temperature but is independent of
the D13C18O16O16Oof reactant carbonate. However, for the
fractionation of total mass 47 isotopologues,
49) on propor-
ivalues for any particular isotopologue is inde-
47¼ D47? D63
? f13C18O16O? D13C18O16Oþ f12C18O17O? D12C18O17O
þ f13C17O2? D13C17O2
þ f12C18O17O16O? D12C18O17O16Oþ f13C16O17O2? D13C16O17O2
þ f12C17O3? D12C17O3
?? f13C18O16O2? D13C18O16O2
where f13C18O16O;f12C18O17O;f13C17O2are the relative abundance
fractions of isotopologues
13C17O17O in all the mass 47 isotopologues of product
CO2, f13C18O16O2, f12C18O17O16O, f13C16O17O2, f12C17O3are the rela-
tive abundance fractions of isotopologues
63 isotopologues of reactant carbonate CO3
in all the mass
7222W. Guo et al./Geochimica et Cosmochimica Acta 73 (2009) 7203–7225
13C18O16O¼ D13C18O16O? D13C18O16O2;
12C18O17O¼ D12C18O17O? D12C18O17O16O;
13C17O2¼ D13C17O2? D13C16O17O2
as the fractionations of specific isotopologue during phos-
phoric acid digestion. Substituting these definitions in the
equation above, we obtain
Scaled vibration frequencies (unit: cm?1) for different CO3
frequency scaling factor of 0.9614).
2?isotopologues (isolated CO3
2?in the gas phase, DFT-B3LYP/6-31G? with a
Estimated abundances of all CO3
2?isotopologues at different equilibration temperatures and with different bulk isotopic compositions.
Equil. T (K)
1.257521E?05 1.269955E?05 1.263916E?05 1.257710E?05 1.270146E?05 1.264106E?05
6.637827E?05 6.703460E?05 6.703764E?05 6.639792E?05 6.705445E?05 6.705748E?05
4.487818E?06 4.487320E?06 4.555747E?06 4.488118E?06 4.487619E?06 4.556051E?06
4.251179E?07 4.250706E?07 4.294809E?07 4.251298E?07 4.250826E?07 4.294930E?07
1.184398E?05 1.184267E?05 1.208125E?05 1.184530E?05 1.184399E?05 1.208260E?05
5.043864E?08 5.093736E?08 5.120209E?08 5.046478E?08 5.096376E?08 5.122862E?08
4.777666E?09 4.824907E?09 4.826700E?09 4.779286E?09 4.826542E?09 4.828336E?09
1.331213E?07 1.344375E?07 1.357881E?07 1.332149E?07 1.345321E?07 1.358836E?07
5.383609E?11 5.383010E?11 5.466880E?11 5.384187E?11 5.383588E?11 5.467467E?11
8.525078E?10 8.524131E?10 8.698698E?10 8.526678E?10 8.525730E?10 8.700330E?10
4.499842E?09 4.499342E?09 4.613632E?09 4.501037E?09 4.500536E?09 4.614857E?09
7.917198E?09 7.916318E?09 8.156559E?09 7.919962E?09 7.919082E?09 8.159407E?09
6.050678E?13 6.110505E?13 6.144267E?13 6.054144E?13 6.114005E?13 6.147787E?13
9.581874E?12 9.676617E?12 9.777016E?12 9.589474E?12 9.684291E?12 9.784770E?12
5.057908E?11 5.107919E?11 5.185810E?11 5.063019E?11 5.113081E?11 5.191051E?11
8.899522E?11 8.987519E?11 9.168583E?11 8.910502E?11 8.998607E?11 9.179895E?11
aNominal cardinal mass in amu.
Phosphoric acid digestion of carbonate minerals7223
?? ? f12C18O17O? f12C18O17O16O
(e.g. for calcite D13C18O16O2¼ 0:406& and
D12C18O17O16O¼ 0:071& at 300 K; Schauble et al., 2006),
the above equation can therefore be further approximated
As mentioned above, the D?
the reactant carbonate and are a function of only acid
digestion temperature, and thus can be regarded as con-
stant at given temperature (at 25 ?C, D?
? constant þ f13C18O16O? f13C18O16O2
For carbonates with d13C=0& and d18O=0&, as as-
sumedin our model,
f13C18O16O¼ 0:9366. Therefore, we expect the dependence
?0.0305, based on the above analyses. This agrees well with
our quantitative H2CO3 model prediction, ?0.035& in-
crease in D?
crepancy is believed to arise from the approximations
adopted in the derivation of the above equations. Note that
this slope for the dependence of D?
carbonates of different bulk isotopic compositions, since
both f13C18O16Oand f13C18O16O2are insensitive to the changes
in the bulk isotopic compositions. The slope increases by
only ?0.002 for 50& increase in d13C, and decrease by
?0.0005 for 50& increase in d18O.
Þ ? ?
47? f13C18O16O? D?
13C18O16Oþ f12C18O17O? D?
þ f13C18O16O? f13C18O16O2
ivalues are independent of Diof
47? constant þ f13C18O16O? f13C18O16O2
47on D63of the reactant carbonate to have a slope of
47for every 1& increase in D63. The small dis-
47on D63varies little for
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