contents 1..1

PROGRESS IN REACTION KINETICS AND MECHANISM An International Review Journal Editor T. J. KEMP Department of Chemistry University of Warwick Coventry CV4 7AL UK

Editorial Advisory Board H. D. Burrows, Univerdisade de Coimbra, Portugal H. B. Dunford, University of Alberta, Canada A Harriman, University of Newcastle upon Tyne, UK A Mamantov, United States Environmental Protection Agency, USA A. E. Merbach, Swiss Federal Institute of Technology, Lausanne, Switzerland D. Phillips, Imperial College London, UK P. Unwin, University of Warwick, UK D. C. Walker, University of British Columbia, Canada

Associate Editors Klaus Suhling, Department of Physics, King’s College London, Strand, WC2R 2LS, UK Mark Green, Department of Physics, King’s College London, Strand, WC2R 2LS, UK Christina Goodrick Meech, Science Reviews 2000 Ltd., PO Box 314, St. Albans, Herts AL1 4ZG, UK

Production Editor Sara Nash, 33 St John’s Court, St Albans, Herts AL1 4TS, UK Tel/Fax: +44(0) 1727 764601 E-mail:[email protected]

Contents KINETIC MODELLING OF GLOBAL EVOLUTION OF TITAN’S ATMOSPHERE

227

VASILI DIMITROV

KINETICS AND MECHANISMS OF MAMMALIAN HEME PEROXIDASE REACTIONS

245

H. BRIAN DUNFORD

KINETICS AND MECHANISM OF THE OXIDATION OF ORGANIC SULFIDES

267

JAYSHREE BANERJI, PRADEEP K. SHARMA AND KALYAN K. BANERJI

KINETICS AND MECHANISM OF 1,10-PHENANTHROLINE CATALYSED CHROMIUM(VI) OXIDATION OF D-GLUCOSE IN AQUEOUS MICELLAR MEDIA BIDYUT SAHA AND KALYAN K. PAL

SCIENCE REVIEWS

283

Progress in Reaction Kinetics and Mechanism. Vol. 30, pp. 227–243. 2005 1468-6783 # 2005 Science Reviews

REVIEW KINETIC MODELLING OF GLOBAL EVOLUTION OF TITAN’S ATMOSPHERE Vasili Dimitrov Department of Geophysics and Planetary Sciences, Tel-Aviv University, Tel-Aviv, 69978, Israel E-mail: [email protected]

Contents ABSTRACT 1.

INTRODUCTION

228

2.

STATEMENT OF THE PROBLEM

230

DISCUSSION

232

3.1 Recycling of primordial CH4 3.2 Recycling of reconverted CH4

232 236

4.

SUMMARY

238

5.

CONCLUSIONS

240

6.

ACKNOWLEDGMENTS

241

7.

REFERENCES

241

3.

ABSTRACT Methane CH4 is the only highly reactive and short-lived background component in Titan’s atmosphere, so its overall reserve predetermines both features and duration of atmospheric chemical activity. Current methane atmospheric abundance is provided by its global circulation. There are two sources of methane replenishment, i.e. recycling of the primordial reserve trapped in Titan’s interior and reconversion of non-saturated final products of the atmospheric photochemical process, reconversion being the minor constituent in the global methane balance. The total bulk of primordial methane gas hydrate depends on the packing index (cage-filling efficiency) a, the latter being limited to 227

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7.2610 45a55610 2 {kg CH4=kg clathrate}. The specification of a seems to be one of the most relevant problems of the experimental modelling of Titan’s chemistry. The total number of methane renewal cycles so far equals Np * 200. Prog React Kinet Mech 30:227–243 (c) 2005 Science Reviews

KEYWORDS: Titan, atmosphere, methane cycle, clathrate, cage-filling efficiency 1. INTRODUCTION Titan, the sixth outer moon of Saturn, is a unique astrophysical object in many respects (Table 1), including the presence of a substantial, reducing and lowpotential atmosphere. Table 1 Some astrophysical indices of Titan Astrophysical indices 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Symbol

Radius (equatorial), m Oblateness (Requ-Rpol)=Requ Mass, kg Density (averaged), kg=m3 Ice=Rock ratio (averaged by mass) Core radius, m Crust shear stress, MPa Pressure at the center, GPa Pressure at the core=mantle boundary, GPa Surface pressure, Pa Surface temperature, K Mass of atmosphere, kg Background energy of atmosphere, J=molec Efficiency of chemical activity Primary photochemical energy, W=m2 Secondary photochemical energy, W=m2 Mass of airborne aerosols (averaged), kg Aerosol density (averaged), kg=m3 Aerosol elemental ratio (averaged) Titan’s life-time (since formation until now), s Sun life-time (Titan existence period), s

Requ q MT r Y ¼ Mice=Mrock Rcor s Pc Pcor PS TS Mat Eback j Lch(1) Lch(2) Maer raer C=H=N Tlife TSun

Value

Refs 6

2.575610 3.85610 5 1.34561023 1.8816103 48=52 1.66106 2.15 6.8 1.14 1.4966105 94.1 1019 2.24610 21 0.82 12.11610 3 0.69 1.361016 1.8610 4 1=1.12=0.08 1.4261017 5.061017

[31] [31] [31] [31] [26] [9] [9] [35] [33] [35] [33] [33] [36] [36] [33]

Global governing and dependent similarity criteria 1 2 3 4 5

Rotational Mach Number Non-homogeneity Number Power Number Polytropic exponent [Retention Number]

K1 K2 K3 K4 [K5]

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0.015 2.76104 5.11 1.35 5.22

[33] [33] [33] [33] [33]

Kinetic modelling of global evolution of Titan’s atmosphere

229

Diverse external energy fluxes Lch (the UV spectral domain of the solar flux, cosmic rays, solar wind and magnetosphere high-energy particles) stimulate a specific chemical activity in its atmosphere, resulting in the formation of gaseous reaction products and an extensive haze layer (aerosols) that completely screens the satellite surface and thus precludes any direct observations. The main compositional feature of Titan’s atmosphere lies in the fact that CH4 is the only highly reactive background component; hence it is the methane resource that is the only limitation of chemical activity in Titan’s atmosphere. Another background component – molecular nitrogen N2 – is very stable and is hard to involve into chemical activity. As regards the oxy-species (CO, CO2, H2O, etc.), these are alien entities in Titan’s reducing atmosphere. Their abundances are so minute (H2O flux barely exceeds (0.8 – 2.8)61012 molec m 2s 1 [1], CO abundance is essentially less than 50 ppm [2]) that they should be disregarded within the framework of general analysis. Methane reactivity can be formalised through the model of the methane cycle. Considering Titan’s atmosphere as a closed system, one cycle is defined chemically as the period T0 for complete destruction of the observable content CH40. The Cassini – Huygens Mission produced a precise atmospheric abundance of CH40 [3 – 5], which can be recalculated into an absolute content of CH40 * 2.3361017 kg. In a temporal sense, one cycle represents the lifetime T0 of this bulk, while kinetically this is the rate of its total renewal d(CH40=dT). On this basis both the total time (duration) and reaction rate in the course of one methane cycle are completely governed by the kinetics of the chemical process, so they can be calculated directly on the basis of any kinetic models. It turns out that the duration of one cycle T0 is much less than Titan’s history

Table 2 CH4 life-time T0 and number of past cycles NP T0 s 1 2 3 4 5 5 6 7 8 9

14

7.0610 6.261014 5.761014 5.261014 1.361014 6.861014 1.661014 (3.0 – 6.0)61014 (3.0 – 30)61014 (0.3 – 3)61014

NP 200 230 250 270 110 210 80 – 90 230 – 470 50 – 470 470 – 4700

Comment

Refs

specified 237-stages kinetic model via rate of CH4 destruction via rate of aerosol formation thermodynamic equilibrium via model of CH4=C2H4 ocean via rate of CH4 destruction via CH4 dissociation rate estimation via CH4 photochemical depletion via estimation of sediments

This work [33] [36] [35] [6] [37] [4] [7] [3], [38] [39]

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Tlife * 1.4261017 s. Values of CH40 and lifetime T0, complemented by the total number of past cycles in Titan’s history, NP, are summerised in Table 2. The relevant indices of aerosol formation can be found in Table 3. Another important point, as confirmed by the Cassini – Huygens isotopic analysis [3], was that the carbon in CH4 does not show the same kind of isotopic fractionation as do nitrogen and oxygen. The short lifetime and isotopic difference prove that the methane in Titan’s atmosphere must be replenished in one way or another. This raises two relevant problems: 1.

What is the source of this replenishment?

2.

How long it will operate?

2. STATEMENT OF THE PROBLEM Evidently, it is Titan’s body that feeds the atmosphere with methane, hence a mass-exchange must occur between Titan’s surface and its lower atmosphere. The discovery of the predicted mass-exchange [6] is undoubtedly one of the most important achievements of the Cassini – Huygens Mission [7,8]. This breakthrough forces the consideration of Titan’s atmosphere as being an essentially open system and implies a close analysis of ingoing (to Titan) and outgoing (from Titan) mass fluxes. Materially, these fluxes have similar compositions, with the one essential exception being that the outgoing flux is impoverished with respect to non-evaporated condensed species, while it is enriched essentially with gaseous methane. Table 3 Some averaged aerosols indices of global chemical activity in Titan’s atmosphere after [31, 33, 35, 36, 40] Index 1 2 3 4 5 6 7 8 9 10 11 12

dimension

Unit mass-rate deposition Covering deposition density Overall reaction mass-rate Overall mass deposition Total height of deposition Aerosol yield (1 cycle) Final gas yield (1 cycle) Ratio of gas=solid yield CH4 life-time (averaged) T0 Total number of past cycles Np Estimated number of following cycles Nf Total number of cycles during Titan’s life-time Nmax

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2

kg=m s kg=m2 kg=s kg m kg kg % s

value (3.0 – 3.4)610 13 (4.0 – 4.8)6104 25 – 30 (3.5 – 4.2)61018 40 – 50 1.361016 2.261017 (93 2) : (7 2) 7.061014 200 20 500 50 700 70


231

A mass-exchange analysis and mean forecast on the resulting atmospheric chemistry can be developed on the basis of the following assumptions: 1.

There is an equivalence between CH4 destruction due to chemical activity, and CH4 replenishment caused by mass-transfer from Titan’s body through (regular=spontaneous) cryovolcanic outgassing [9]. For the mean forecast, the type of dynamics of this balance (periodic, sporadic, continuous) is of no concern, and the only requirement is that CH40 replenishment must take place within the limits of one cycle T 0. This is a natural demand, formulated early as ‘‘. . . it is only necessary . . . CH4 reservoir was erupted within the last 107 yr’’ [6]. If this demand is not fulfilled, i.e. if the off-the-shelf CH40 atmospheric bulk is consumed before any new release from Titan’s body [8], then the ‘‘continuous’’ kinetics should be broken because the new feed will occur into a clean, transparent, and chemically inert atmosphere. Consequentially, a new injection of methane will renew chemical activity, but it is unlikely that the recommenced chemical process will be similar to that from the preceding cycle. This will certainly represent kinetics of another photochemistry under other thermal conditions (new albedo, new re-emission equilibrium temperature, etc.). Features of these new kinetics are hard to predict. For such a dramatic scenario, Titan’s continuous history should have to be replaced with a ‘‘mosaic’’ history. Hereinafter only the continuous history model will be considered.

2.

The astrophysical conditions remain intact over the range of lifetime of the Sun t 4 TSun 561017 s. This assumption is natural within the framework of Titan’s continuous history and needs no further comment.

3.

The intensity of the atmospheric chemical process remains fixed within the Sun’s lifetime, TSun. If such is not the case, i.e. if the overall rate of CH4 photochemical depletion decreases for one reason or another, e.g. due to continuous escape of some atmospheric bulk (leakage), then the quantities reported below should be taken as an upper theoretical limit. Any higher approximation requires a supplementary reliable hypothesis on the quenching rate of the chemical process.

Starting from these assumptions, the global circulation of methane can be presented in the form www.scilet.com

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* ingoing flux ?CH4 ðgas : Þ ?ðnon

saturated products : Þ ?condensation ?

?ðnon saturated products ; Þ ?sedimentation þ chemical interaction ? ?CH4 ðcondensed ; Þ ?fCH4 gas ðreconverted : Þ þ CH4 ðtrapped : Þg ? ?combined recycle into atmosphere CH4 ðgas : Þ ?outgoing flux * Methane enrichment takes place at the locking stage of this circulation (combined recycle into atmosphere), the outgoing flux consisting of methane of a different origin. 3. DISCUSSION 3.1 Recycling of primordial CH4 The methane reserve in Titan’s body has formed during Titan’s accretion, primordial methane being trapped in Titan’s interior structure. The ensuing gravitational evolution causes cryovolcanism. The discovery of cryoactivity on Titan is another remarkable achievement of the Cassini – Huygens Mission [9,10]. Independent of the real scenario of Titan’s history, the trapped methane evidently can exist only in an associated form (the so-called ‘‘inclusion complexes’’, ‘‘clathrates’’ or ‘‘gas hydrates’’). Clathrates occur when the ‘‘host’’ (water molecules) forms a cage-like structure around smaller ‘‘guest’’ (methane molecules), with water crystallising in the cubic system (so-called Structure I) rather than in the conventional hexagonal structure of normal ice [11]. In this structure, the cages are arranged in body-centered cubic packing; the unit cell contains 8 CH4 per 46 H2O molecules, yet not all cages are occupied [12]. The upper theoretical limit of maximal molecular packing (cage-filling efficiency) a is of the form [(CH4)65.75(H2O)], which is equivalent to amax * 10 1 {kg CH4=kg clathrate}. However, this limit can be reached if and only if the formation conditions are optimal [13-16], namely: a slow formation rate (51 cm=yr), reduced temperature (T5270 K), increased pressure (P4100 MPa), moderate acidity (6 4 pH 4 8), no admixtures in the base two-component mixture, etc. Unfortunately, the real conditions of Titan’s accretion and evolution are poorly established. This is why the exact evaluation of the packing index a is the main problem. Nonetheless, the admissible limits of the magnitude of a can be reasonably restricted. On the one hand, the formation of methane clathrate in the three-component system H2O – NH3 – CH4 which is impure because of admixtures (like saturated hydrocarbons-see [17]) for the real conditions of www.scilet.com


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Titan’s upper interior (P * 25 – 50 MPa, T * 300 – 350 K, NH3abundance * 5 – 10 wt%) indeed proceeds with difficulty [18 – 20] and results in the formation of the so-called icy structure Ih, the clathrate density approaching that of pure ice I r * 0.92 [21]. This is why the magnitude amax * 5.0610

2

after [22] should be taken cautiously as a poorly-determined

upper limit. The specification of the lower limit requires a detailed scenario of Titan’s formation and a reliable model of Titan’s interior. Such a model will not only predetermine the total bulk and location features of the trapped methane but will also specify the cage occupancy a. As to the interior structure, following substantially refs [23 – 26], we accept slow accretion ( * 1.661013 s) [27], low temperature [28], incompressibility, thermal and mechanical equilibrium [29], and a differentiated model [30], which has been somewhat corrected in an early study [31]. Omitting details, five main zones can be listed schematically as follows (Figure 1): Zone 1 – Rock core Dh1 ¼ R1 ¼ 1.66106 m consists of heavy iron silicates (like olivine and fayalite) of averaged density r1 ¼ 4.06103 kg m 3 and mass M1 6.8661022 kg Zone 2 – High-pressured icy substrate Dh2 ¼ R2 – R1 ¼ (2.2 – 1.6)6106 ¼ 6.06105 m of averaged density r2 ¼ 1.36103 kg m 3 and mass M2 3.5961022 kg Zone 3 – Liquid=semiliquid ammonia-methane-water medium Dh3 ¼ R3 – R2 ¼ (2.5 – 2.2)6106 ¼ 3.06105 m of averaged density r3 ¼ 0.956103 kg m 3 and mass M3 1.9861022 kg Zone 4 – {46% ice=54% rock}, {10 wt% ammonia=(8 – 16) wt% methane} permafrost-like crust Dh4 ¼ R4 – R3 ¼ (2.575 – 2.5)6106 ¼ 7.56104 m consists of light-weight, fine-grained silicate (like feldspar) of averaged density r4 ¼ 1.766103 kg m 22

3

and mass M4 (ice 0.28 þ rock

22

0.79)610 1.07610 kg Zone 5 – Thin regolith layer including deposition of detached aerosols [31]. NOTE:

Actually, zones 3 and 4 have a common complex design (see enlarged area in Figure 1). It includes a united hard frame extending through both zones and a set of interior separate domains filled with a liquid-semiliquid medium. Compositionally, the frame presents a www.scilet.com

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variable mixture of a light-weight, fine-grained rock (like feldspar) and a frozen H2O=NH3=CH4 fraction, the percentage of the latter being enriched at the upper boundary of zone 4 while being reduced at the lower boundary of zone 3. Structurally, the near-surface region of the frame is a permafrost crust. The liquid–semiliquid medium in cavities is conceptually of the same H2O=NH3=CH4 composition but the abundances of species are spatially variable and differ from its frozen analogue. This complex structure is irrelevant for present purposes, and so in Figure 1 it is schematised by two separate zones 3 and 4 of the indices given above. Zones 1 and 5 are irrelevant for evident reasons (no methane present). Both the size and state of zones 3 and 4 are also unimportant within the framework of target setting. Thus, if zone 4 is not a twocomponent ice=rock crust, but rather consists of a pure icy lid of another thickness Dh4 ¼ R4 – R3 ¼ (2.574 – 2.460)6106 * 1.156105 m, and if the NH3abundance in zone 3 Dh3 ¼ R3 – R2 ¼ (2.460 – 2.200)6106 * 2.46105 m varies

In the scaleless Fig. 1 these zones shown schematically as a common continuous domains

Figure 1 Schematic representation of Titan’s interior structure Dh1 ¼ R1, Dh2 ¼ R2 – R1, DH3 ¼ R3 – R2, Dh4 ¼ R4 – R3, Dh5 ¼ R5 – R4.

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within limits (5 – 10) wt%, nevertheless, these variations still keep the integral change of bulk methane {[CH4]3 þ [CH4]4} within + 10% error bars, the cage occupancy being fixed, a ¼ Const. As to zone 2, the real problem of CH4 trapped here is not the content, but rather the accessibility of CH4 for atmospheric chemistry. The probability of cryoactivity at such depths seems to be low. At the same time, the transfer rate between zones 2 and 3 is completely unknown. It seems that the conventional mass exchange should be strongly depressed because ice in zone 2 is very uniform in all senses, being weakly-conductive, highly-dense r2 ¼ 1.36103 kg m

3

[23] and weakly-porous 53% [32]. This means that neither the concentra-

tion, nor temperature or baric diffusion gradients can provide any appreciable mass exchange. Hence, reasonable and reliable mass transfer mechanisms are difficult to access. Bearing this uncertainty in mind, the two relevant possibilities are of the form: 3.1.1 Methane in zone 2 is lost for atmospheric activity for reasons of transport In order for chemical activity during Titan’s past life T life *1:42:1014 s to be ensured, the lower estimate of the packing index should be higher than alow 3;4 ðTlife yT 0 Þ6fðCH4 Þ0 y½ðCH4 Þzone3 þ ðCH4 Þzone4 g ¼ ðTlife yT 0 Þ6½ðCH4 Þ0 yðCH4 ÞS1 ¼ 1:55610 3 ;

3.1.2 Methane in zone 2 is available for atmospheric activity (no transport obstacles) In this case the lower limit of the cage-filling efficiency should be higher than alow2

4

ðTlife yT 0 Þ6ðCH4 Þ0 y½ðCH4 Þzone2 þ ðCH4 Þzone3 þ ðCH4 Þzone4 ¼ ðTlife yT 0 Þ6½ðCH4 Þ0 yðCH4 ÞS2 ¼ 0:72610 3 ;

Two features of this estimate need to be noted. On the one hand, the variation in the range Dalow ¼ ð0:72 1:55Þ610 3 is surprisingly small even in view of the noted uncertainties. On the other hand, the reliability of the estimate is extremely high since it is proved directly by the very fact of Titan’s existence. This implies that the value alow ¼ 0:72610 www.scilet.com

3

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should be taken as a lower theoretical limit with a reliable accuracy of + 100%, Titan’s conditions being fixed. Uncertainties in the packing index a and the local transfer rates between zones 2 and 3 are mutually dependent and closely interconnected. The essential difference is that the former can be found experimentally while there is no way to specify the latter. 3.2 Recycling of reconverted CH4 Primordial CH4 is not the only source of its replenishment in the atmosphere. To determine any parallel source, features of the atmospheric kinetics should be examined. They lie in the fact that Titan’s atmospheric photochemistry is imperfect thermodynamically because both material and energy resources are in acute shortage. Kinetically, the conversion of CH4 and N2 into final products proceeds via the formation of acetylene C2H2 and hydrogen cyanide HCN, which are the key intermediates in the general chemical process CH4 ?fC2 H2 g ?fPure hydrocarbonsg N2 þ CH4 ?fHCNg ?fTholins ¼ CyN

mixed hydrocarbons þ nitrilesg

Despite the fact that CH4 and N2 are the main background species, Titan’s photochemistry should actually be titled ‘‘acetylene chemistry’’ rather than ‘‘methane chemistry’’, because C2H2 formation acts as the bottleneck for the general chemical process. The material restriction relates exactly to C2H2 formation, i.e. the following complete conversion of this equiatomic (C=H : 1 : 1) key intermediate into saturated hydrocarbons with an increased atomic ratio (C=H 4 1 : 1) needs unavailable complementary sources of carbon. The energy (thermodynamic) restriction implies that the perfect termination of the process, i.e. the process proceeding right up to the formation of the most stable saturated hydrocarbons, needs an energy Eth ¼ 0.7867 W m 2, which is higher than the available energy Lch ¼ 0.6923 W m

2

. This is why the final

products consist not only of saturated species but of various unsaturated species of near-equiatomic ratio (C=H * 1 : 1) as well, most of them consisting of dienes. Thermodynamically, this means that the full termination of atmospheric chemical activity remains incomplete [33]. In physical terms, the final products are gaseous and condensed species, the latter appearing because of phase conversion. The condensation of gaseous species begins by generation of primary aerosol nuclei and occurs over a wide range of altitude because it depends equally on local physical conditions and www.scilet.com


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specific indices of any given component [31]. Thus, aromatic and diene aerosol nuclei first appear at Z * 800 km in very small initial concentration ( * 106 m 3), the speed of generation being maximal (dN=dt * 1012 m 3s 1) at Z * (150 – 300) km. Tholin and polyacetylene nuclei appear at Z * 350 – 200 and 250 – 100 km, respectively, while saturated and unsaturated pure hydrocarbons condense mainly at the troposphere, Z570 km. These products fall onto Titan’s surface as a deposit. Afterwards some part of this deposit re-evaporates into the atmosphere as an outgoing flux, some part is irreversibly accumulated on the surface as an involatile residue, while some part, namely – unsaturated hydrocarbons – gains the capacity to undergo further interaction with the actively oxidizing ammonia-water medium, the conditions being new. Once produced (due to the incompleteness of atmospheric activity), and then extracted from the sphere of gaseous reaction (due to phase conversion), these condensed unsaturated hydrocarbons complete their conversion into saturated forms including methane which is the most important constituent of the outgoing flux. The general unspecified scheme of the ground-based conversion is of the form: fUnsaturated hydrocarbons C2x H2x ; þ þ g þ fH2 O; NH4 OHg ? ?ðAldehydesyKetonesÞ* ?Alcohols ?Acids* ? ?fSaturated hydrocarbons Cx H2xþ2 g ; þ fCH4 þ HCN þ H2 g : Specifically, for CH4 as the simplest saturated hydrocarbon, this scheme is realised as follows: fC2 H4 þ H2 O ?ðHCHOÞ* ?CH3 OH ?ðCH3 COOHÞ* ?CH4 In such a way, any limited reservoir of condensed ethene serves as a continuous chemical parallel source of methane replenishment in ground conditions. Kinetically, this trend has a very low chemical efficiency and the final yield hardly exceeds (5 – 10) % [34]. Yet it is not prohibited by thermodynamics and should be considered within the framework of methane balance. Materially, chemical reconversion during one cycle enables the recycling of ðCH4 Þ0B * (ethene bulk)6(yield of chemical reconversion {C2H4 ? CH4}) ¼ (561014) 6(10 1) ¼ 561013 kg CH4. Considering both the past and following cycles Nmax ¼ NP þ Nf ¼ 200 þ 500 ¼ 700, the overall reconverted bulk along trend B is estimated as ðCH4 ÞB *561013 6700*3:561016 kg CH4. www.scilet.com

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4. SUMMARY In the global methane circulation, the pathway B of chemical reconversion is minor as compared to pathway A. At the same time, surface and internal chemistries needs detailed consideration due to their influence on the global physico-chemical processes. As to the features of methane retention, the upper limit a ¼ 5.0610

2

provides the overall admissible methane reserve for cases (A1 þ B) and (A2 þ B) as ðCH4 ÞS1 ¼ 15:361020 kg and ðCH4 ÞS2 ¼ 33:361020 kg, respectively. Meanwhile, the complete maintenance of current intensity of the atmospheric chemical process needs only ðCH4 Þcrit *ðTSun yT0 Þ6ðCH4 Þ0 *ð5:061017 y7:061014 Þ62:3361017 *1:6561020 kg

CH4.

Physically,

this

critical reserve (CH4)crit provides for the continuous activity of N * 700 cycles during the Sun’s lifetime, TSun. The foregoing lower theoretical limit alow ¼ 0.72610 3 is unable to support such a level of chemical activity independent of the availability of methane in zone 2. Indeed, for cases A1 and A2, the admissible methane feed ðCH4 Þfeed ¼ alow 6ðCH4 ÞS equates respectively to ðCH4 Þfeed1 ¼ 0:72610 3 615:361020 ¼ 1:161018 kg and ðCH4 Þfeed2 ¼ 0:72 610 3 633:361020 ¼ 2:461018 kg, both of these figures being far less compared with the required quantity (CH4)crit, i.e. {1.161018, 2.461018} 5 1.6561020. In order to ensure the continuous history, the cagefilling index for the cases A1 – A2 (methane in zone 2 is unavailable=available) should be equal, respectively, to A1 :

acrit1 ðTSun yT0 Þ6½ðCH4 Þ0 y½ðCH4 ÞS1 ¼ 5:45610 3 ;

A2 :

acrit2 ðTSun yT0 Þ6½ðCH4 Þ0 y½ðCH4 ÞS2 ¼ 2:51610 3 :

In other words, the critical cage-filling efficiency acrit lies in the range 2:51610

3

¼ acrit2 acrit acrit1 ¼ 5:45610

3

This range should be taken as the compulsory lower limit which governs Titan’s situation, see Figure 2. Hence the most relevant cases can be classified in the following way: (1)

a5alow ¼ 0:72610 3 . Since the quantity 0.72610 3 must be considered reliable to 100%, this case is irrelevant. Considering that the exact value of a for Titan’s conditions as yet remains unknown, this allows us to suggest www.scilet.com


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Figure 2 Dependence of availability of methane for chemical activity on cage-filling efficiency a and total methane stock (CH4)S. Crucial methane stock (CH4)crit ¼ 1.6561020 kg demarcates continuous and mosaic histories. 1. Total methane stock in zones 3 and 4 ðCH4 ÞS1 ¼ 15:361020 kg. 2. Total methane stock in zones 2 – 4 ðCH4 ÞS2 ¼ 33:361020 kg. acrit1 ¼ 5:45610 3 and acrit2 ¼ 2:51610 3 are critical values that provide continuous history depending on unavailability=availability methane stock in zone 2, respectively.

that the future experimental specification of a hardly results in the figure a5alow . Titan’s existence by itself is a direct validation of this assertion. (2)

a5acrit2 ¼ 2:51610 3 . A ‘mosaic history’ is the most probable scenario regardless of both the total inventory of methane and features of its allocation. In this scenario, Titan will inevitably lose its veil. www.scilet.com

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(3)

a5acrit1 ¼ 5:45610 3 . Titan’s history depends on both total methane reserve and features of its allocation (availability). The case acrit1 5a5acrit2 is the most intriguing because of the total number of problems increases considerably.

(4)

a4acrit1 . The higher the value of a, the higher is the probability of a ‘continuous history’.

5. CONCLUSIONS (1)

The global evolution of Titan’s chemical activity has been analysed in terms of the methane cycle and cage-filling efficiency. The absolute methane content CH40 ¼ 2.3361017 kg, the cycle duration T0 ¼ 7.061014 s, the number of past cycles NP ¼ 200, and the total number of cycles

(2)

Nmax ¼ 700 are the main quantitative indices of chemical activity. Methane is the only limiting factor of chemical activity. Its global circulation occurs due to outgassing from Titan’s interior and chemical reconversion of the deposited products of atmospheric chemical activity. The former presents the major source of methane replenishment while the latter is the

(3)

minor co-source. The cage-filling efficiency a is the most important index of global chemical activity. To a first approximation, it is restricted to 7:2610 4 5a55:0610 2 . The upper limit confines the overall admissible methane reserve for cases (A1 þ B) and (A2 þ B) as ðCH4 ÞS1 ¼ 15:361020 kg and ðCH4 ÞS2 ¼ 33:361020 kg, respectively. To predict reliably the fate of Titan’s aerosol veil, the direct specification of the packing index areal is necessary, and as much knowledge of the experimental conditions approaching Titan’s reality as possible.

(4)

The stock ðCH4 Þcrit ¼ 1:6561020 kg is crucial for Titan’s history. If the real reserve exceeds the critical one, i.e. ðCH4 Þreal ðCH4 Þcrit , the history will be continuous, otherwise if ðCH4 Þreal ðCH4 Þcrit then the history will be mosaic. In order for the critical stock to be ensured for cases A1 and A2, the critical values of cage-filling efficiency should be equal to acrit1 ¼ 5:45610

3

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6. ACKNOWLEDGMENTS This research was supported by the KAMEA Foundation of the Israel Ministry of Absorption.

7. REFERENCES [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13]

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