Gas Phase Reaction Networks

Ion-molecule and neutral exchange reactions are important contributors to the abundances of interstellar molecules, and networks containing thousands of such gas phase reactions (and other types) have been explored; they have had great success in describing the variety of observed molecules in different regions and their abundances, though some problems remain. These reaction schemes depend on the presence of an initial molecule, which is either H2 or a molecule formed from H2. The formation of this seminal species, that is, the molecule on which all of interstellar chemistry rests, will be discussed in the next chapter.

In studying reaction networks, we are hoping to describe the chemistry of interstellar clouds in sufficient completeness that we can calculate abundances of the molecule formed in the chemistry. Suppose we wish to calculate the number density n(M) of a molecule, M, formed in the reaction

which has a rate coefficient kt -m3 s-1 per molecule; the molecule M may be destroyed by photodissociation at a rate ДМ) per second and by reactions such as

which has a rate coefficient kd m3 s 1 per molecule. Then, we can write down a differential equation for /j(M), showing how it changes in time, t,

where the first term on the right-hand side of equation (3.23) is the rate of formation of molecule M by reaction (3.21) (and all such reactions must be included), and the second term is the rate of loss of molecule M by photodissociation and photoionization and by reaction (3.22) with species X (again, all such reactions must be included).

An equation similar to equation (3.23) must be written down for each species included in the chemistry, so rate coefficients and photodissociation rates must be known (or estimated) for a large number of reactions. The physical parameters (number density, temperature, elemental abundances, radiation intensity, etc.) need to be specified or an evolving scenario must be defined. Then the equations can be integrated numerically. The process is simple in conception but may be complicated and laborious in practice. Extensive databases for astrochemistry are maintained and updated (see end of chapter), and codes for setting up and integrating the rate equations are available.

Sometimes, if the chemistry proceeds rapidly compared to any other changes, it may be sufficient to calculate the steady state abundances, that is, with d/?(M)/ d/ = 0. To find the steady state /?(M) from equation (3.23), we see that we need to know the number densities of all other species. А, В, X, and so on, so we have to write down a set of algebraic equations of the form of equation (3.23) for each atom, molecule, and ion in chemistry, but with each derivative on the left-hand side set equal to zero. Such sets of equations are solved numerically.

We can occasionally obtain useful information about steady state chemistry without solving equation but by considering a simple enough set of chemical equations. For example, in the centre of a dense cloud of molecular hydrogen, say /;(H2) 5 Ю10 m-3, the level of ionization is probably dominated by the cosmic ray ionization of H2 at a rate ( per second, the formation of H,+ in reaction (3.11) with rate coefficient k and the dissociative recombination of H?+ with electrons in reaction (3.12) with a rate coefficient к m3 s-1 per molecule. From the rate equations, we find

This equation allows us to predict that /?(Hj) ^(Hj) so that ~ и(е),

and we find

Typically, С - 10 17 s 1 and к = 10 12 m3 s 1 per molecule, so that the fractional ionization is

which shows that the level of ionization to be expected in dense clouds is really very small. In clouds in which n(H2) = 10U) m-3, the ionization fraction predicted by equation (3.26) is only 3 x 10_8. Although this treatment is greatly simplified, and a proper discussion should include other processes affecting the ionization, this theoretical result does seem to be approximately correct. In addition, observations of various molecular ions in dense clouds are consistent with a low level of ionization.

Initial Steps in Interstellar Cloud Gas Phase Chemistry

As we’ll see in detail in Chapter 5, the radiation field from hot stars creates ionized regions (Hll regions) around those stars. These HI1 regions trap all the stellar photons with energies greater than the ionization potential of hydrogen atoms, that is, 13.6 eV. This means that the spectrum of mean starlight outside those regions is truncated in energy, and possesses only photons with energies less than this value. However, the ionization potential of carbon atoms is 11.3 eV and of sulfur atoms is 10.4 eV, so in interstellar regions that are penetrated by starlight, these two elements (and some others) are photoionized, and are present in dilfuse clouds as C+ and S+ ions. Molecular hydrogen is the most probable collision partner and at the low temperatures in diffuse clouds, C+ ions and H2 molecules radiatively associate to form CH2+, and successive reactions of this product with H2 molecules leads to CH3+ and CH5+. Dissociative recombination of these ions with electrons forms a variety of simple hydrocarbons: CH, CH2, CH3, and CH4. These species then undergo further reactions. For example, an exchange reaction of CH with an oxygen atom

forms the important molecule carbon monoxide, while greater molecular complexity can be achieved in reactions such as

leading to formaldehyde (a detected species in interstellar clouds). Evidently, a rich carbon chemistry may arise even in diffuse clouds. However, sulfur does not behave similarly to carbon, because the association of S+ with H2 is slow, so that sulfur chemistry in both diffuse and dark clouds grows not through sulfur hydride chemistry but mainly through exchange reactions with other species such as CH and OH, forming CS and SO.

In denser, dark clouds where starlight is excluded, carbon atoms are mainly neutral. Carbon chemistry proceeds through reaction with H3+ (formed in reaction (3.11)) to form CH+ and subsequent hydrogen abstraction reactions with H2 lead to CH2+ and CH3+, and a slower radiative association with H2 provides CH5+. As in diffuse clouds, recombination reactions of these ions with electrons provide CH, CH2, CH ,, and CH4, and these molecules generate further chemistry.

Nitrogen and oxygen atoms have ionization potentials of 14.5 and 13.6 eV, respectively, so that they cannot be photoionized by starlight in the interstellar radiation field, and these species are mainly neutral in both diffuse and dark clouds. Oxygen atoms do not react with hydrogen molecules at the low temperatures of dark interstellar clouds, as there is too great a barrier of the type illustrated in Figure 3.5. However, oxygen can be chemically converted to other molecules even at low temperatures by reactions with the H^ ion, produced through reaction (3.11). The H3 ion can readily donate a proton to many species, including О atoms,

and the OH+ ion can react in a sequence of hydrogen abstraction reactions with H2 molecules,

though the sequence stops there as 0+ has a valency of 3 and can bind no more H atoms. Dissociative recombination reactions with electrons produce OH and H20:

Both OH and H20 are detected in dark clouds, and can undergo further ion- molecule reactions to produce other oxygen-containing molecules, including HCO+ and CO:

However, the equivalent reactions of N with H3+ to form NH+ or NH2+ do not proceed. An alternative entry to nitrogen chemistry exists: cosmic rays colliding with N atoms create N+ ions, which react with H2 molecules to form NH+. Hydrogen abstraction reactions with H2 molecules lead to the formation of NH4+, and dissociative recombinations of these ions with electrons give the species NH. NH2, and NH3. These species are then available for reactions with other atoms and ions, for example, to form products containing the CN radical.

Although the problem of generating chemistry in interstellar clouds seems at first difficult, once entry routes can be identified it is evident that enormous chemical opportunities become available. The schemes adopted are all broadly similar. In dark clouds, cosmic rays drive the formation of H3+ which donates protons to many species. Since the abundance of molecular hydrogen is overwhelming in interstellar clouds, if reactions of these ions can occur with H2, they will do so; successive hydrogen abstraction reactions may occur (as in reactions (3.30)) followed by neutralization in reactions with electrons; this scheme creates hydrides and hydride ions which provide the feedstock for more complex reactions. The networks are essentially unrestricted, so that molecules containing more than just a few atoms can be readily produced in gas phase networks similar to those whose first steps are indicated here.

The molecular species that have been identified (by 2019) in many types of interstellar region (not only diffuse and dark clouds) are shown in Table 3.2. In fact, the actual number of interstellar molecular species must be much larger than the number included in this list of detected species (about 200 species), because the chemical networks to produce this variety of molecules include many species that have not yet been detected. The lack of detection of these species may be a result of low abundance or of the species not having a suitable transition for detection by current technology. Table 3.2 contains many molecules which have been detected in which the main isotopes have been replaced by minor isotopes. For example, H may be replaced by D, 12C by l3C, and l60 by l70. These versions of the molecules are called isotopologues. Many of the species in Table 3.2 are also detected as isotopologues; if these are counted separately, then the total number of detected species is several times larger.

It is remarkable that very many of these detected interstellar species can be formed in simple but extensive gas phase schemes of the type outlined in this section. However, a significant number of exceptions exist, including the seminal molecule, H2, and many of the largest molecules in the list. For these exceptions, it may be that appropriate gas phase chemical networks do not exist, or that those that do exist are incapable of providing particular molecules in the detected abundances. We shall describe an alternative to gas phase chemistry in the next chapter.

It is clear from this brief description that interstellar chemistry is dependent on the presence of molecular hydrogen. There are a number of ways in which interstellar molecular hydrogen can be formed in gas phase reactions but these ways are incapable of providing a fast enough formation rate, at least in the Milky Way galaxy. The direct association of two H atoms in the gas phase is very strongly forbidden by quantum mechanics and so does not contribute, and formation in exchange reactions such as XH + H —> X + H2 is a circular argument since XH almost certainly requires H2 for its formation. Two other possibilities remain. Hydrogen atoms may radiatively associate with electrons to form hydrogen negative ions, H~, which then react with H atoms to form H2 molecules, releasing the electrons back to the gas phase. The second possibility is a similar

Detected interstellar molecular species (by 2019).

TABLE 3.2

Cll (methylidyne) CN (cyanide) Cll+ (methylidyne cation) OH (hydroxyl) CO (carbon monoxide) 1I2 (molecular hydrogen) SiO (silicon monoxide) CS (carbon monosulfidc) SO (sulfur monoxide) SiS (silicon monosulfidc) NS (nitrogen sulfide) C2 (diatomic carbon) NO (nitric oxide) IICI (hydrogen chloride) NaCI (sodium chloride) A1CI (aluminium monchloridc) KCI (potassium chloride) AIK (aluminium monofluoride) PN (phosphorus mononitride SiC (silicon carbide) CP (carbon monophosphide) NH (nitrogen monohydride) SiN (silicon mononitride) SO+ (sulfur monoxide cation) CO+ (carbon monoxide cation) I IF (hydrogen fluoride) N2 (molecular nitrogen) CF+ (fluoromethylidynium cation) PO (phosphorus monoxide) 02 (molecular oxygen) AIO (aluminium monoxide) CN" (cyanogen anion) OH+ (hydroxyl cation) SH+ (sulfur monohydride cation) HCI+ (hydrogen chloride cation) SH (sulfur monohydridc cation) TiO (titanium oxide) ArH+ (argoniunt cation) NS+ (nitrogen sulfide cation) HeH+ (helium hydride cation)

1I20 (water) HCO+ (formylium cation) HCN (hydrogen cyanide) OCS (carbonyl sulfide) HNC (hydrogen isocyanidc) H2S (hydrogen sulfide) N2II+ (protonated nitrogen) C2H (ethynyl) S02 (sulfur dioxide) HCO (formyl) UNO (nitroxyl) HCS+ (thioformyl cation) HOC+ (hydroxymcthyliumidcnc cation) SiC2 (cylacyclopropynylidcnc) C2S (dicarbon sulfide) C, (tricarbon) C02 (carbon dioxide) CH2 (methylene) C20 (dicarbon monoxide) MgNC (magnesium isocyanidc) NH2 (amidogen) NaCN (sodium cyanide) N20 (nitrous oxide) MgCN (magnesium cyanide) Hj+ (protonated molecular hydrogen) SiCN (silicon ntonocyanidc) A1NC (aluminium isocyanidc) SiNC (silicon monoisocyanidc) HCP (phosphacthync) CCP (dicarbon phosphide) AIOH (aluminium hydroxide) H20+ (water cation) H2CI+ (chloronium cation) KCN (potassium cyanide) FeCN (iron cyanide) H02 (hydroperoxyl) ТЮ2 (titanium dioxide) CCN (cyanomethylidine) Si2C (disilicon carbide) S2H (hydrogen disulfide) IICS (thioformyl) IISC (isothioformyl) NCO (isocyanate)

NH, (ammonia) H2CO (formaldehyde) HNCO (isocyanic acid) H2CS (thioformaldehyde) C2II2 (acetylene) C3N (cyanoethynyl) HNCS (isothiocyanic acid) HOCO+ (protonated carbon dioxide) C.,0 (tricarbon monoxide) l-C.,H (propynylidyne) HCNH+ (protonated hydrogen cyanide) H.,0+ (protonated water) C.,S (tricarbon monosulfidc) c-C,H (cyclopropenylidcnc) HC2N (cyanocarbcne) 1I2CN (methylene amidogen) SiC, (silicon tricarbidc) CH, (methyl) C3N" (cyanoethynyl anion) PH., (phosphine) HCNO (fulminic acid) HOCN (cyanic acid) HSCN (thiocyanic acid) 1IOOII (hydrogen peroxide) 1-C,H+ (propynylidyne cation) HMg.NC (hydromagnesium isocyanidc) HCCO (ketenyl) CNCN (isocyanogen) MONO (nitrous acid)

IIC3N (cyanoacetylcne) IICOOII (formic acid) CH2NH (mcthaniminc) NF12CN (cyanamidc) ll2CCO (kctcnc) C,H (butadiynyl) SiH, (silane) e-C,H2 (cyclopropenylidcnc) CF12CN (cyanomethyl) C5 (pcntacarbon) SiC, (silicon tctracarbidc) ll2CCC (propadicnylidcnc) CH, (methane) HCCNC (isocyanoacctylcnc) HNCCC (?) H2COH+ (protonated formaldehyde) C,H" (butadiynyl anion) CNCHO (cyanoformaldehyde) HNCNH (carbodiimidc) CH,0 (methoxy) NH,D+ (deutcratcd ammonium cation) H2NCO+ (protonated isocyanic acid) NCCNH+ (protonated cyanogen) CH3CI (chloromcthanc)

CH3OH (methanol) CH.,CN (methyl cyanide) NH2CHO (formamide) CH3SH (methyl mercaptan) C2H, (ethylene) C5H (pentynylidyne) CH,NC (methyl isocyanidc) HC2CHO (propynal) C5S (pcntacarbon monosulfidc) HC,NII+ (protonated cyanoacetylcne) C5N (cyanobutadiynyl) IIC,H (diacetylene) HC,N (?) c-H2C,0 (cyclopropcnonc) CH2CNH (kcteniminc) CSN" (cyanobutadiynyl anion) HNCHCN (E-cyanomcthaniminc) SiH,CN (silyl cyanide)

(Continued)

TABLE 3.2 (Cont.)

CHjCHO (acetaldehyde) CH3CCH (methyl acetylene) CH3NH2 (mcthylaminc) CH2CIICN (vinyl cyanide) HC5N (cyanodiacctylenc) C6H (hcxatriynyl) c-C2H40 (ethylene oxide) CH2CII01I (vinyl alcohol) C6H" (hcxatriynyl anion) CHjNCO (methyl isocyanate) HC5O (butadiynylformyl)

HCOOCH3 (methyl formate) CH3C3N (mcthylcyanoacetylcnc) C71I (hcptatriynylidync) CH3 COOU (acetic acid) ll2C6 (hexpentaenylidene) CH2OHCHO (glycolaldehydc) HC6H (triacetylene) CH2CHCHO (propcnal) CH2CCHCN (cyanoallene) NH2CH2CN (aminoacetonitrile) CH3CHNH (cthanimine) CH3S1II3 (methyl silane)

CHjOCHj (dimethyl ether) CH3CH2OH (ethanol) CH,CH2CN (ethyl cyanide) IIC7N (cyanotriacctylcnc) CH3C4H (methyl diacctylene) C»II (octatriynyl) CH3CONH2 (acetamide) C8H' (octatriynyl anion) CH2CHCH3 (propylene) CH,CH2SH (ethyl mercaptan) HC7O (hexadiynylformyl)

(СНз)2СО (acetone) HO(CH2)2OH (ethylene glycol) СНзСН2СНО (propanal) CH3 C5N (mcthycyanodiacetylenc) CH3CHCH20 (propylene oxide) CH3OCH2OH (methoxymcthanol) IIC9N (cyanotetraacctylcnc) CH3C6H (methyltriacetylene) CH3CH2OCHO (ethyl formate) CH, COOCH3 (methyl acetate)

C6II6 (benzene) C3H7CN (/t- and /-propyl cyanide)

c-C6HsCN (benzonitrile)

Species arc grouped by the number of atoms they contain. All of them arc potential probes of the wide range of physical conditions in the interstellar medium. Many of these molecular species may be found not only in interstellar clouds but also in circumstcllar regions. Most of them arc detected by millimetre-wave and submillimetre-wave observations. Two other molecular species, the fullcr- cncs (cage molecules) Сбо+ and C7+ ions to form Hi+ ions, which then react with H atoms to form H2 molecules, releasing the H+ ions to the gas phase. These schemes are valid and operated to form the first molecules in the early Universe. However, under the physical conditions in the interstellar medium of Milky Way, the H2 formation rate is inadequate to compete with the photodissociation by starlight described in Section 3.3.2. Another method of forming interstellar molecular hydrogen in the interstellar medium of the Milky Way galaxy is required, and is described in Section 4.6.2.

Chemical networks containing hundreds of species interacting in thousands of reactions are routinely used in astrochemical studies, creating a huge demand for accurate reaction rate data. Websites such as Kinetic Database for Astrochemistry (KIDA) and UM1ST Database for Astrochemistry (UDfA) maintain extensive lists of assessed and updated data.

PROBLEMS

  • 1. We shall define a cooling time tc by - TI(dTldt). Consider a cloud in which /;(Н) = 108 m-3, n(e) - 105 nT3, and /?(C+) = 4 x 104 m~3, at a temperature of 200 К. Suppose that cooling occurs solely by the excitation of transitions in C+ by collisions with electrons. Calculate the cooling time, tc, in years.
  • 2. Find the abundance of H2 necessary in a cloud of density n = 10s m-3 at a temperature of 100 К to give a cooling rate due to H2 equal to that due to C+ and e, if я(е) = и(С+) = 104 m-3.
  • 3. Show that, in equilibrium, the heating rate due to photoionization of atom X is ««(e)/?(X+)£, where a is the recombination rate coefficient for X+ + e —* X + hv, and E is the excess energy released in the photoionization.What is the heating due to C atoms in the cloud of problem 1? (Use a = КГ17 m3 s-1.)

Show that at T = 15 К (approximately) the heating and cooling due to C+ balance in any cloud.

  • 4. Find the fraction,/, of hydrogen in molecular form (f = 2n(H2)/n) which will maintain a cloud temperature of 100 К by H2 cooling in a cloud heated by photoelectric emission from grains at a rate КГ28 J m-3 s-1.
  • 5. The molecule AB is made in the following ways: A + В —» AB + hv and A + BC —> AB + C and the rate coefficients are k{ and k2, respectively. AB is lost in the reaction AB + D —> A + BD (rate coefficient k2) and in photodissociation AB —» A + В (rate [}). Write down an expression for the equilibrium abundance of AB in terms of other abundances.
  • 6. Suppose that the chemistries of H2, Hj, and H}- are described by

H2 + cosmic ray —>H|+e, f=10_17s_l

H2 + Ht —> Hj" + H, ki = 10_l5m3s_l

Hj- + CO -» HCO+ + H2, k2 = 10_,5m3s_l

If //(CO) = 1(T4/7(H2) and /;(Н2) = Ю10 m-3 (constant), find the steady state abundances of and Н/

7. For the chemistry of problem 6, write down the differential equation that determines the time dependence of H2. Show that и(Н^) approaches steady state on a timescale 1/[A'!//(H2)]. If the typical lifetime of a cloud is 106 years, is steady state a good approximation for the abundance of Hj?

 
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