Uncertainty of the Rate Coefficients

Theoretically one should not calculate the logarithm of a quantity with a physical dimension therefore, the original quantity has to be converted to a dimensionless value. According to accepted notation (JCGM 2008), a curly bracket indicates the specific value of a physical quantity having a given unit. 

Equation (5.54) means that the rate coefficient is uncertain according to a multiplication factor u = 10. Typical values of the uncertainty parameter/are 0.3, 0.5 and 0.7, which means that the extreme values differ from the recommended value by multiplication factors of 2.00, 3.16 and 5.01, respectively (see Table 5.1). This xmcertainty parameter/has been defined for a range of gas-phase systems by a series of researchers including Wamatz (1984), Tsang and Hampson (1986), Tsang (1992), Baulch et al. (1992, 1994, 2(X)5) and Konnov (2008). The specification of /allows the calculation of uncertainty ranges which may be used within the context of the global sensitivity methods described in the previous section. 

Uncertainty parameter/ Multiplication factor u r(logio i ) r(ln [k]) Multiplication factor corresponding to 1 T Multiplication factor corresponding to 2 t 

Reaction kinetic data collections in atmospheric chemistry define the uncertainty of the rate coefficient in a different way. The top of the troposphere is 7 km to 20 km from the Earth surface (Clarke and Tomlin 1999), depending on the season and the latitude. In the troposphere the temperature of the air is typically between —53 C and +47 °C (220 K 320 K). The stratosphere is located above the troposphere having a width of about 50 km with temperature increasing with altitude from about —53 °C to 3 °C (220 K — 270 K). This means that all chemical reactions in the troposphere and stratosphere occur between 220 K and 320 K. Thus, the temperature interval of atmospheric chemical reactions is much narrower than for combustion reactions (300 K 2,500 K). The rate coefficient of most atmospheric chemical reactions has been measured at room temperature, and therefore the uncertainty of the rate coefficient is expected to be lowest near 298 K. At higher and lower temperatures usually fewer measurements were carried out, and therefore the uncertainty of the rate coefficients is usually higher both above and below 298 K. For this reason atmospheric reaction kinetics data collections define the uncertainty of the rate coefficients so that its minimum is at 298 K. 

The lUPAC collections of atmospheric kinetic data (lUPAC 2014 Atkinson et al. 2004,2006,2007,2008) define the uncertainty of the rate coefficient as follows  

---

Table 5.1 Various representations of the uncertainty of the rate coefficient, assuming that the logio fe and logioil values correspond to 3

Table 5.1 shows the conversion of the uncertainty parameter/to other representations of the uncertainty of the rate coefficient The second colunrn shows, e.g., that an/value of 0.3 means that the rate coefficient is uncertain according to a factor of 2, that is, up to 200 % and down to 50 % of the recommended value is also possible. The table also shows that/= 0.1 and/= 0.3 (frequently adopted values of the uncertainty parameter used for the characterisation of well-known rate coefficients) approximately correspond to 8 % and 26 % uncertainty of the rate coefficient at the lo level but multiplication factors of 1.26 and 2.00 at the 3o level, respectively. 

In most uncertainty studies published so far (see e.g. Brown et al. (1999), Turanyi et al. (2002), Zsely et al. (2005), Zador et al. (2005a, b, 2006a) and Zsely et al. (2008)), where the uncertainties of the rate coefficients were utilised, the uncertainty of k was considered to be equal to the xmcertainty of the pre-exponential factor A. This implies that the uncertainty of parameters E and n is zero, which is an unrealistic assumption. In a global sensitivity analysis study of a turbulent reacting atmospheric plume, Ziehn et al. (2009a) demonstrated the importance of uncertainties in EIR for the reaction N0 + 03 = N02 + 02. In this case for the prediction of mean plume centre line O3 concentratiOTis, the sensitivity to the assumed value for EIR was almost a factor of 20 higher than that of the A-factor, based on input parameter uncertainty factors provided by the evaluation of Androulakis (2004, 2004). However, in this case the parameters of the Anhenius expression for the chemical reactions considered were allowed to vary independently. In fact, the characterisation of the joint uncertainty of the Arrhenius parameters is important for the reahstic calculatiOTi of the uncertainty of chemical kinetic simulation results as will be discussed in the next section. 

In this section, the relationship between the temperature dependence of the uncertainty of the rate coefficient and the joint pdf of the Arrhenius parameters is discussed based primarily on Nagy and Turanyi (2011, 2012). As was mentioned in Sect. 2.2.1, the temperature dependence of the rate coefficient k can be described by the modified Arrhenius equation A =AT exp( // 7). Introducing the transformed parameters x = ln, a = ln A and e = EIR, the linearised form of the modified Arrhenius equation is... 

Usually gas kinetics data collections suggest not only the Arrhenius parameters but also parameters that characterise the uncertainty of the rate coefficients at given temperatures as discussed in the previous section. The temperature interval [Tj, T2],... 

The covariance matrix of the three parameters of the modified Arrhenius equation contains six parameters. To determine these parameters, the uncertainty of the rate coefficient has to be known for at least six different temperatures. Fitting these parameters requires not only Eq. (5.73) but also expressions (5.69) and (5.70)... 

Kramp, F., and S. E. Paulson, On the Uncertainties in the Rate Coefficients for OH Reactions with Hydrocarbons, and the Rate Coefficients of the 1,3,5-Trimethylbenzene and m-Xylene Reactions with OH Radicals in the Gas Phase, J. Phys. Chem. A., 102, 2685-2690 (1998). 

Yet unless very detailed information is available to describe the initial distribution of separations, p(r, 0), it will not be possible to use measured time-dependent survival probabilities to probe details of dynamic liquid structure. Currently, experimental uncertainties at 30% are so large that such a probe is not possible, since the effects of the short-range caging region are only 30%, at the most, of the rate coefficient or escape probability. 

There is sufficient and convincing experimental evidence available already to support the need to consider activation and diffusion processes simultaneously. For instance, in Chap. 2, Sects. 5.2 and 5.4, mention was made of other instances where the reaction rate had been measured and found to be slower than anticipated from the Smoluchowski rate coefficient [eqn. (19)]. Using the Collins and Kimball expression enabled the workers to obtain reasonable estimates of the rate coefficients of encounter pair reactions. There is still some degree of uncertainty that the slower than expected reaction rate might not be attributable to partial... 

DMSO is a product of the IO/DMS reaction 126.271. The magnitude of the rate coefficient 126.271 indicates that an IO concentration in excess of 10s molecule cm 3 would make a significant contribution to DMS removal. IO concentrations as high as 108-10s molecule cm 3 have been predicted (281 but these are subject to considerable uncertainty as a consequence of the chemical scheme employed. Furthermore, high concentrations of IO are only likely near to strong sources of CH3I. As these tend to be located at coastal sites their contribution to the chemistry of an air parcel advecting at a moderate velocity will only be small. Consequently, IO chemistry has not been included in the current model. 

Enormous uncertainties remain in the theoretical models that have been proposed and major changes in them may yet be demanded by new observations. A major obstacle to further progress is the inadequacy of our knowledge of the rate coefficients of reactions involving the large molecules that are significant in the photochemistry of the planets. 

Estimation of model error bars and sensitivity analyses are based rai the same principle. AU rate coefficients (or other model parameters) of a system are randomly varied within a certain range. The chemical evolution is then computed for each set of rate coefficients. For a network containing 4,000 reactions, the model is typically run 2,000 times with different sets of rate coefficients. The distribution of the rate coefficients can be either log-normal or log-uniform (see Fig. 4.5). The first choice implies that the mean value ko is a preferred value. This is usually the case for rate coefficients, which are measured with an uncertainty defined by statistical errors. The factor Fq, which defines the range of variation, can be a fixed factor for aU reactiOTis for a sensitivity analysis or specific to each reaction for an uncertainty propagation study. Use of the same Fq for all reactions, in the case of a sensitivity analysis, assures the modeller that an underestimated uncertainty factor will not bias the analysis. The results of thousands of runs are used differently to identify important reactions and to estimate model error bars. 

The uncertainty parameter defined this way is also a piecewise linear functiiMi of and this uncertainty has a minimum at temperature To = 298 K. The upper and lower limits belonging to the standard deviation (lrecommended value of the rate coefficient by the parameter... 

Figure 5.19 shows the uncertainty values provided in the database and the uncertainty—temperature function of the rate coefficient, calculated from the uncertainties of the Anhenius parameters. The calculated uncertainty function passes through the points and has realistic values at other temperatures. Figure 5.20 shows the joint normal pc of the transformed Arrhenius parameters, whilst Fig. 5.21 presents the temperature dependence of the normal p of transformed rate coefficient k. The uncertainty range of the rate coefficient is narrower at intermediate temperatures therefore, the pdf of In (A ) is narrower at intermediate temperatures, which is easily seen in the upper projection of the pdf m Fig. 5.21. Since the integral of the p( of In A is of unit value at each temperature, a narrower pdf also means a higher maximum. This is the reason why the temperature-dependent p h s a hump at intermediate temperatures.

Following the estimation of predicted output uncertainties, sensitivity studies can then be used to identify the kinetic and thermodynamic data that cause the highest uncertainty in the model simulation result. The contribution of the uncertainty of the parameters can be assessed using Sobol indices as discussed in Sect. 5.5.3. For example, as Fig. 5.22 shows, at stoichiometric equivalence ratio, in a premixed laminar methane-air flame, the uncertainties in the rate coefficients of reactions O2 -1- H = OH -1- O and H -1- CH3 = CH4 cause the highest uncertainty in the calculated laminar flame velocity. Knowing these rate coefficients with lower... 

In the study, the reactions were treated as reversible, with reverse rates calculated from the appropriate equilibrium constants based on enthalpies of formatiOTi calculated using NASA polynomials (see Sect. 2.2.3). Because so many of the input parameters were estimated, derived from a low number of measurements or from single theoretical studies, the input distributions were cmisidered to be uniform between predefined minimum and maximum values (Tomlin 2006). Uncertainties in the rate coefficients were expressed using A-factors only, since for most reactions, there was insufficient information to determine the joint of the Arrhenius parameters. 

The four measurements of the rate coefficient for reaction of Cl-atoms with DEE (Wallington et al., 1988g Nelson et al., 1990a McLoughlin et al., 1993 Notario et al., 2000b), all conducted near 298 K, are in reasonable agreement see table ni-B-9. The average of the four determinations, k = 2.8 x 10" cm molecule" s", is recommended with an uncertainty of 20%. 

The three measurements (Nelson et al 1990a Harry et al., 1999 Notario et al 2000b) of the rate coefficient for reaction of Cl with DnBE are in good agreement see table in-B-28. The unweighted average of the three determinations, = 4.7 X 10 cm molecule" s is recommended, with an estimated uncertainty of 15%. 

There exists a single room-temperature determination of the rate coefficient for reaction of OH with each of the four compounds listed above (Dagaut et al., 1989a Platz et al., 1999 Aschmann et al., 2001b). The data are summarized in table ni-C-28. Given the lack of corroborative studies, these rate coefficients are recommended with uncertainties of 35%. 

The only study relevant to the atmospheric behavior of this species is a measure of the rate coefficient for its reaction with OH (Wallington et al., 1988a), k= 1.9 x 10 cm molecule" s See table III-E-4. In the absence of corroborative data, an uncertainty of 35% is assigned. 

---