COBISS: 1.01 letter: Problems in using the approach of Rayleigh distillation to interpret the 13C and 18O isotope compositions in stalagmite calcite UDK 551.435.843 Wolfgang Dreybrodt1,2 Introduction When calcite is precipitated from a water film on top of a stalagmite to its surface, the carbonate in the solution and consequently also the carbonate in the calcite depos­ited becomes enriched in the heavy isotopes 18O and 13C (Hansen et al. 2016; Dreybrodt & Scholz 2011; Romanov et al. 2008). This isotope signal is added to the isotope imprint resulting from climate variability. Therefore a physical model of the evolution of the isotope composi­tion of carbonate in a water film, either flowing down the surface of the stalagmite at high drip rates or stagnant, when the drip rate is low, is necessary to discriminate the climate signal from the signal resulting from physical processes in the cave. Currently two models are proposed. In the first one (Scholz et al. 2009) one assumes that the isotope evolu­tion can be described by a Rayleigh distillation process (Mook 2000) where during the entire process of pre­cipitation the fractionation factor for carbon or oxygen, respectively remains constant. In another model one as­sumes that precipitation is a uni-directional non-equi­librium process, where the constants in the rate equa­tions are slightly different for the light and the heavy isotope (Dreybrodt 2008; Dreybrodt & Scholz 2011). Re­sults from these models have been compared for various scenarios by Scholz et al. (2009), Dreybrodt and Scholz (2011), Dreybrodt and Deininger (2014) and Dreybrodt and Romanov (2016). Differences of opinion clearly exist in the research community regarding which of these models best repre­sents reality. In this letter I discuss the meaning of the Rayleigh equation for precipitation of calcite under equi­librium conditions and for precipitation of calcite gov­erned by uni-directional rate equations. Rayleigh distillation We consider a reservoir containing N molecules with both rare and abundant isotopes. The number of the rare isotopes is NA, that of the abundant ones NB and NA << NB. Now by some physical or chemical process, such as evap­oration or precipitation to a solid, a small number dNA of rare isotopes and dNB of abundant isotopes are removed irreversibly from the reservoir step by step. We now ask for the evolution of the isotope composition in the res­ervoir. The isotope ratio of the molecules remaining in the reservoir is R = NA/NB, the isotope ratio of the re­moved isotopes dNA and dNB is Rrem= dNA/dNB. Now in isotope equilibrium, Rrem = .iso.R where .iso is the isotope fractionation factor, which is a constant during the entire process. Then Integration yields NA0 and NB0 are the initial numbers, when removal of the molecules starts. From this one finds Dividing by NB / NB0 and observing NB N and NB0 N0, because NB NA, one gets the Rayleigh equation or by using concentrations because concentrations C are proportional to the number of the molecules N in the reservoir. Note that R does not depend on time, but only on the fraction f = N/N0 of molecules remaining in the res­ervoir, independent on the pathway of removal. The rea­son for this is the condition that .iso is a constant dur­ing the entire process. Therefore the concept of Rayleigh distillation must be used with care, always regarding the processes in the background. An example is evaporation of water. If evapora­tion proceeds in isotope equilibrium with a vapor at­mosphere, from which vapor is removed slowly, then, .iso = .eq where .eq is the equilibrium fractionation co­efficient between liquid water and vapor. On the other hand if water evaporates irreversibly into a dry atmo­sphere with humidity of zero, the conditions of evapora­tion are constant, no matter, how much water has been already removed. Therefore also in this unidirectional non-equilibrium process the Rayleigh equation is valid. Here the different diffusion coefficients of the isotopes determine the value of .. For equilibrium evaporation .eq-1 = .eq = - 0.01 whereas for evaporation into dry air . = - 0.035 (Horita et al. 2008). 1 Faculty of Physics and Electrical Engineering, University of Bremen, Germany 2 Karst Research Institute ZRC SAZU, Postojna, Slovenia e-mail: dreybrodt@t-online.de Received/Prejeto: 01.09.2016 ACTA CARSOLOGICA 45/3, 285–293, POSTOJNA 2016 Wolfgang Dreybrodt Classical Rayleigh equation: Equilibrium reaction The isotope evolution of carbonate during precipitation of calcite proceeding slowly under conditions of chemi­cal and isotope equilibrium of all species involved has been described by Mook (2000) using mass conservation between the species in the solution and the environment (calcite and the cave atmosphere). Such conditions exist when calcite precipitates from stagnant water layers with depths of more than 1 cm, as they exist in cave pools. In this case out gassing of CO2 is slow with time constant .D =4d2/.2D, where d is the water depth and D is the con­stant of molecular diffusion of CO2 in water (Dreybrodt & Scholz 2011). The equilibrium concentration Ceq of HCO3- then changes slowly and precipitation proceeds close to equilibrium. Under equilibrium conditions the isotope ratio R = NA/NB = A/B for the heavy and the light isotope in the HCO3- ions is described by the Rayleigh equation. A, B stand for the concentration of the corre­sponding isotope. R0 = A0/B0 is the isotope ratio at the onset of precipi­tation, C0 is the initial hydrogen carbonate concentration and C is the actual one. .eq = .eq-1 is the equilibrium fractionation factor for carbon or oxygen, respectively, determined solely by the corresponding mass action constants (Mickler et al. 2004, 2006). Eqn. 5 is valid only, if for all times t the solution is in thermodynamic and isotope equilibrium or at least very close to it. Then .eq remains constant during the entire process. For equi­librium fractionation, .eq is derived by thermodynam­ics and does not depend on the details of the reaction mechanisms. Extended Rayleigh equation: Uni-directional reaction If out gassing, however, is fast, Ceq which depends on the pco2 in the surrounding atmosphere, within several 10 s, rapidly reaches its final value (Dreybrodt 2011; Hansen et al. 2013) and precipitation of calcite proceeds under disequilibrium until finally equilibrium with respect to calcite is reached and precipitation stops. Precipitation in this case is ruled by the rate laws (Dreybrodt 2008; Drey­brodt & Scholz 2011) and where A and B are the concentrations of the heavy (rare) and light (abundant) isotope in the HCO3- respectively, Aeq and Beq are the equilibrium concentrations when pre­cipitation stops, d is the depth of the water film on top of the stalagmite, and .pA, .pB, are rate constants, slightly different for the heavy and the light isotope. Aeq and Beq are related by Aeq/A0 = .•Beq/B0, . 1. This relation needs some explanation: Let us present the rare isotopes in the carbonate of the solution by red nuclei with concentration A, and the abundant ones by white nuclei with concentration B. First we assume iden­tical properties of both. Then the chance of some nucleus to be incorporated into the calcite deposited is equal for a red and a white nucleus. If the initial concentrations in the aqueous solution are A0 and B0, the ratio NAc /NBc of the number of rare (red) and abundant (white) iso­topes in the calcite deposited must be equal to the ra­tio A0/B0 at all times. Therefore at equilibrium we have NAc,eq/NBc,eq = A0/B0. In reality the chance of the rare isotope to be incorporated into the crystal is slightly different from that of the abundant one and may differ from the chance of the abundant one by a factor ß 1. Then one has NAc,eq/NBc,eq = ßA0/B0 at all times. On the other hand, in equilibrium with respect to calcite one has fractionation for the isotope ratios Reqc = .Req. With Reqc = NAc,eq/NBc,eq = .•Aeq/Beq = ß•A0/B0 one finally gets Aeq/A0= ß•Beq/.•B0= .•Beq/B0 with . 1. The isotope evolution can now be de­rived from equation (6). By division one gets with .A and .B are the exponential decay times of precipita­tion. Eqn. 7 can be rewritten as Here .iso clearly is a function of A and B and the condition that .iso must be a constant during the entire process is violated. Integration of equation (7) and some algebraic ma­nipulations yields with .kin = .B/.A (8) Using B(t) C(t), B0 C0, and Aeq/A0 = .•Beq/B0 = .•Ceq/C0 one finds which is further on called the extended Rayleigh equa­tion. This extended Rayleigh equation has been derived already from the time evolution of R(t) by Dreybrodt (2008). It must be stressed here that C(t) is the solution of the differential equation which is If any other function is introduced instead of C(t), eqn. 9 is no longer valid. If Ceq is zero, not possible for calcite precipitation, eqn. 9, reads This equation is valid only for reactions with pure linear kinetics with rate laws dA/dt = -A/.A and dB/dt = -B/.B where C(t) = C0.exp(-t/.A). One obtains a Rayleigh equation, formally identical with the classical Rayleigh equation, eqn. 4 for equilibrium. It describes, however, a quite different kinetic process and is valid only for a pure linear reaction with C(t) = C0exp(-t/.A), (White 2007, 2013). The difference to the equilibrium re­action is hidden in .kin =.B/.A, which is a ratio of reaction times and in no way is related to .eq. To call .kin kinetic partition coefficient for calcite precipitation is misleading for two reasons: a) The reaction of calcite precipitation is not pure linear, because the rate law is dA/dt = -(A-Aeq)/.A and dB/dt = -(B-Beq)/.B, b) .kin is a constant related to rate constants and not to mass action constants. To describe the isotope evolution for calcite precipi­tation the full kinetic equation must be used. To conclude: When using the Rayleigh equation in its time dependent version R = R0·(C/C0).eq, one has to be sure that the underlying process supports its applica­tion. Problems in using the approach of Rayleigh distillation to interpret the 13C and 18O isotope ... Wolfgang Dreybrodt Comparing the results from the classical and the extended Rayleigh equation Now we compare the results using the classical Rayleigh (eqn. 5) and its extended version (eqn. 9) by inserting into both the time evolution C(t) (eqn. 11), which has been verified experimentally by Hansen et al. (2013). Fig. 1 presents the result of the classical Rayleigh equation (5) by introducing (11) into it. Fig. 1a depicts the isotope evolution in time for three values of .. The parameters determining precipitation, , and .A are also listed in the Fig. 1. For times below 200 s, one observes an increase of R(t) linear in time. Then the curves bend to reach a constant value. Fig. 1b illustrates the same situation as a function of C(t)/C0. Again R(C) rises linearly with de­creasing C/C0 for C/C0 > 0.8 and then reaches its final value at C/C0 = 0.2 when precipitation stops. In comparison to Fig. 1 we show the results for the same case when one uses the extended Rayleigh (eqn. 9) and inserts eqn. 11. Fig. 2a shows R(t) as a function of time. The full lines depict . = -0.005 and . = 1.003 or . = 1.000 respectively. The dashed ones show the combi­nation . = -0.003 and . = 1.003 or . = 1.000 respectively and the dotted ones illustrate . = -0.001 and . = 1.003 or . = 1.000 respectively. In all curves for times t < 200s one finds a linear increase in time. The final value reached is R= .. Fig. 2b shows the results of Fig. 2a plotted as function of C/C0. For C/C0 > 0.8 one observes a linear in­crease with decreasing C/C0. The linear increase in time can be explained by expanding R(t) to first order in t. For t < 0.2 .A one finds (Dreybrodt & Scholz 2011; Dreybrodt & Romanov 2016) for the classical Rayleigh equation and for the extended one. . is the delta-notation used in small numbers in the order of 0.001 instead of ‰. Note that for . = 1, . = 0, eqns. 13 and 14 become identical. The linear increase of R with decreasing C/C0 for values 1 > C/C0 > 0.8 reads for the classical Rayleigh equation and for the extended one. Therefore fitting experimental data from the early linear increase of R(C) by the classical Rayleigh equation is always possible (Hansen et al. 2016) but determining the value of . from it leads to values, which are flawed. Figs. 3a and 3b present two examples, how data generated by the extended Rayleigh equation with . = -0.003, . = 1.003 and . = -0.005, . = 1.000, respec­tively, can be fitted satisfactorily for values 1 > C/C0 < 0.6 by the classical Rayleigh equation with . = -0.0046 for the first example and . = -0.0038 for the second one. To find out experimentally, which of the two approaches, the classical or the extended version of the Rayleigh equation is valid, one has to measure the dependence of the initial slope for various values of C0/Ceq. Problems in using the approach of Rayleigh distillation to interpret the 13C and 18O isotope ... Fig. 1: Evolution of the isotope ratio R for the classical Rayleigh equation R=R0(C/C0). for various values of .. a) as function of time with C(t)=(C0-Ceq)exp(-t/.b)+Ceq and b) as function of concentration. Wolfgang Dreybrodt Fig. 2: Evolution of the isotope ratio R for the extended Rayleigh equation (eqn. 9) for various values of . and .. a) as function of time with C(t)=(C0-Ceq)exp(-t/.b)+Ceq and b) as function of concentration C. Problems in using the approach of Rayleigh distillation to interpret the 13C and 18O isotope ... Fig. 3: Comparison of the evolution of the isotope ratio R for the extended Rayleigh equation (eqn. 9), full lines and the classical Ray­leigh equation, dotted curves as function of a) time and b) concentration. The values of . for the classical Rayleigh equation have been chosen to match the corresponding curves of the extended Rayleigh equation for times t < 500 s. Wolfgang Dreybrodt Conclusion In modeling the isotope evolution of DIC during pre­cipitation of calcite many researchers (Deininger et al. 2012; Polag et al. 2010; Mühlinghaus et al. 2007, 2009; Romanov et al. 2008; Scholz et al. 2009) have used a time dependent modification of the classical Rayleigh equa­tion R(t) = R0•(C(t)/C0). by inserting the time evolution C(t) of, HCO3-, C(t) = (C0-Ceq) exp (-t/.B) + Ceq. In view of the discussion above this is not a valid approach. Instead of using the classical Rayleigh equa­tion the extended version must be used. It is always possible to fit experimental data in their linear range for C(t)/C0 < 0.6 by using the classical Ray­leigh equation. The resulting fitting parameter .fit, how­ever, does not have any physical meaning and cannot be used for any reasonable interpretation of the data. It is just a parameter to describe the experimental data by some equation. One might as well use a linear function in t or in C to do this. Acknowledgement I highly appreciate funding by the Deutsche Forschungs­gemeinschaft (DFG), grant DR 79/14-1. References Deininger, M., Fohlmeister, J., Scholz, D. & A. 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Problems in using the approach of Rayleigh distillation to interpret the 13C and 18O isotope ...