Research Article | Open Access
A. J. Komkoua Mbienda, C. Tchawoua, D. A. Vondou, F. Mkankam Kamga, "Evaluation of Vapor Pressure Estimation Methods for Use in Simulating the Dynamic of Atmospheric Organic Aerosols", International Journal of Geophysics, vol. 2013, Article ID 612375, 13 pages, 2013. https://doi.org/10.1155/2013/612375
Evaluation of Vapor Pressure Estimation Methods for Use in Simulating the Dynamic of Atmospheric Organic Aerosols
The modified Mackay (mM), the Grain-Watson (GW), Myrdal and Yalkovsky (MY), Lee and Kesler (LK), and Ambrose-Walton (AW) methods for estimating vapor pressures () are tested against experimental data for a set of volatile organic compounds (VOC). required to determine gas-particle partitioning of such organic compounds is used as a parameter for simulating the dynamic of atmospheric aerosols. Here, we use the structure-property relationships of VOC to estimate . The accuracy of each of the aforementioned methods is also assessed for each class of compounds (hydrocarbons, monofunctionalized, difunctionalized, and tri- and more functionalized volatile organic species). It is found that the best method for each VOC depends on its functionality.
Atmospheric aerosols (AA) have a strong influence on the earth’s energy balance  and a great importance in the understanding of climate change and human health (respiratory and cardiac diseases, cancer). They are complex mixtures of inorganic and organic compounds, with composition varying over the size range from a few nanometers to several micrometers. Given this complexity and the desire to control AA concentration, models that accurately describe the important processes that affect size distribution are crucial. Therefore, the representation of particle size distribution is of interest in aerosol dynamics modeling. However, in spite of the impressive advances in the recent years, our knowledge of AA and physical and chemical processes in which they participate is still very limited, compared to the gas phase . Several models have been developed that include a very thorough treatment of AA processes such as in Adams and Seinfeld , Gons et al. , and Whitby and McMurry . Indeed, the evolution of size distribution of AA is made by a mathematical formulation of processes called the general dynamic equation (GDE). It is well known that the first step in developing a numerical aerosol model is to assemble expressions for the relevant physical processes. The second step is to approximate the particle size distribution with a mathematical size distribution function. Thus, the time evolution of the particle size distribution of aerosols undergoing coagulation, deposition, nucleation, and condensation/evaporation phenomena is finally governed by GDE . This latter phenomenon is characterized by the mass flux for volatile species between gas phase and particle which is computed using the following expression :
describes the noncontinuous effects . When (), there is condensation (evaporation). is assumed to be at local thermodynamic equilibrium with the particle composition  and can be obtained from the vapour pressure () of each volatile compound with average molar mass of the atmospheric aerosol, mole fraction, and activity coefficient, through the following equation: where is given by the Clausius-Clapeyron law:
In (3), is equal to 156 kJ/mol as stated by Derby et al. . Consequently, the vapour pressures of all aerosol compounds are needed to calculate the mass flux of condensing and evaporating compounds. The knowledge of for organic compound at the atmospheric temperature is required whenever phase equilibrium between gas phase and particle is of interest. Often, most of the compounds able to condense have experimental vapor pressures unavailable, and because of that, their estimation becomes necessary. To solve this problem of estimation, many methods have been developed. For example, in Tong et al. , a method based on atomic simulation is applied only for compounds bearing acid moieties. Quantum-mechanical calculations are making steady effort in vapor pressure prediction (Banerjee et al. ; Diedenhofen et al. ). Furthermore, current models describing gas-particle partitioning use semiempirical methods for vapor pressure estimation based on molecular structure, often in the form of a group contribution approach. Therefore, these methods require in most cases molecular structures (e.g., boiling point , critical temperature , and critical pressure ), which usually have themselves to be estimated. For example, The MY method  was used by Griffin et al.  and Pun et al.  for modeling the formation of secondary organic aerosol. Jenkin , in a gas particle partitioning model, used the modified form of the Mackay method . Some methods (Pankow and Asher , Capouet and Müller ) assume a linear logarithmic dependence () on several functional groups, but this consideration fails when multiple hydrogen bonding groups are present. Furthermore, more investigations are needed to clarify which method can give values closer to the experimental data. Camredon and Aumont , Compernolle et al. , and Barley and McFiggans  have made an assessment of different vapor pressure estimation methods with experimental data for compounds of relatively higher volatility. For this latter reason, large differences in the estimated vapor pressure have been reported.
In this paper, our focus will be on the (i) evaluation of a number of vapor pressure estimation methods against experimental data using all volatile organic compounds present in our database and (ii) assessment of the accuracy of each of these methods on the base of each class of compounds.
2. Data and Vapor Pressure Estimation Methods
2.1. Experimental Data
The molecules selected in this study have been identified during in situ campaigns  and during chamber experiments . In fact, they are hydrocarbons, monofunctionalized and multifunctionalized species, and bearing alcohol, aldehyde, ketone, carboxylic acid, ester, ether, and alkyl nitrate functions. The experimental vapor pressures are taken from NIST chemistry website (http://www.nist.gov/chemestry) and from Myrdal and Yalkowsky , Asher et al. , Lide , Yams , and Boulik et al. , and they have a range from atm to 1 atm. Molecular properties (boiling point, critical temperature, and critical pressure) are also taken from the NIST chemistry website. All vapor pressure estimation methods used in this study take into account these properties. In most cases, these properties have also to be estimated.
2.2. Estimation of Molecular Properties
2.2.1. Boiling Temperature
Using the boiling temperature of Joback  and its extension, the group contribution technique denoted by is written as where is the contribution of group , and is the occurrence of this group in the molecule. As in Camredon and Aumont , the extension is made by adding some other group contribution to take into account molecules bearing hydroperoxide moiety (–OOH), alkyl nitrate moiety (ONO2), and peroxyacyl nitrate moiety (–C(=O)OONO2). Thus, the first group is divided into the existing Joback groups –O– and –OH. The second group value is provided by the NIST chemistry website, and the last group value is provided by Camredon and Aumont  using boiling point value from Bruckmann and Willner .
2.2.2. Critical Temperature
New group contributions have been added by Camredon and Aumont  for hydroperoxide, alkyl nitrate, and peroxyacyl nitrate moieties:
2.2.3. Critical Pressure
The two techniques listed in the previous section have been used to estimate critical pressure . Here, it is also assumed that is the sum of group contributions. Therefore, denoting by and the Joback and Lydersen critical pressures, respectively, we can write where is the molar mass, is the number of atoms in the molecule, and is the critical pressure contribution of group . The new group contributions described in the previous subsection are taken into account here.
2.3. Vapor Pressure Estimation Methods
As said earlier, many methods for vapor pressure estimation have been developed and are based on the Antoine or on the extended form of the Clausius-Clapeyron equation. Let us present in what follows each of the five methods used.
2.3.1. The Myrdal and Yalkowsky (MY) Method
The MY method  starts from the extended form of the Clausius-Clapeyron equation obtained by using Euler’s cyclic relation. Here, the expression of is given by where is the vaporization entropy at the boiling temperature, is the gas-liquid heat capacity, and is the gas constant. used in this method is an empirical expression given by Myrdal et al. :
In (9), the parameters and which characterize the molecular structure represent the torsional bond (see Vidal ) and the hydrogen bonding number (see [19, Section 3.1.2]), respectively. is a linear dependence of :
Thus, in the MY method, vapor pressure is estimated by the relatively simplified formula
2.3.2. The Modified Mackay (mM) Method
Often called simplified expression of Baum , this method is also based on the extended form of the Clausius-Clapeyron equation. The simplifying assumption here is to consider the ratio to be constant :
In (12), the vaporization entropy, , takes into account the van der Waals interactions and is based upon the Trouton’s rule. For its calculation, Lyman  has suggested the following expression: where is a structural factor of Fishtine  which corrects many polar interactions. It has different values as follows: (i) for nonpolar and monopolar compounds; (ii) for compounds with a weak bipolar character; (iii) for primary amines; (iv) for aliphatic alcohols. Finally, the mM method is reduced to the following equation:
2.3.3. The Grain-Watson (GW) Method
The GW method is based on the following equation : where and is inversely proportional to (). has the same form like that used in the mM method.
2.3.4. The Lee and Kesler (LK) Method
Like the methods described previously, the LK method required critical temperature, critical pressure, and boiling temperature. Here, the vapor pressure is estimated on the base of Pitzer expansion : where is reduced vapor pressure, is reduced temperature, and is the Pitzer’s acentric factor which accounts for the nonsphericity of molecules: with . In (16) and (17), and are the Pitzer’s functions which are polynomials in . Lee and Kesler have suggested the following equations :
2.3.5. The Ambrose-Walton (AW) Method
AW method  is also based on the Pitzer expansion. They have reported their analytical expressions of Pitzer’s functions in the form of a Wagner type of vapor pressure equation:
In (19), .
3.1. Molecular Properties
We present in this section the results obtained for the Joback and Lydersen techniques described previously. The accuracy of each of the five vapor pressure estimation methods used in this study is assessed taking into account the reliability of pure substance property estimates. The reliability of the two techniques presented in Section 2.2 is therefore crucial.
In Figure 1, where the results of Joback technique are displayed, is plotted against experimental values for a set of 253 volatile organic compounds. The scatter tends to be larger for boiling temperature higher than 500 K. The correlation coefficient () shows that estimated values match very well experimental . The root mean square error (RMSE) and the mean absolute error (MAE) are, respectively, 17.60 K and 12.65 K. This MAE agrees with 12.9 K and 12.1 K calculated in Reid et al.  and Camredon and Aumont  for a set of 252 and 438 volatile organic compounds, respectively. Hence, these results show that can be used in vapor pressure estimation methods. Moreover, Joback reevaluated Lydersen’s group contribution scheme. He added several new functional groups and deducted new contribution values.
The two techniques for the estimation of are compared to experimental data of 138 compounds in Figure 2.
Figures 2(a) and 2(b) show that the Joback and Lydersen techniques give similar results for values lower than 700 K. Joback technique shows a negative bias for higher than 700 K. This technique gives for the overall compounds an RMSE of 24.98 K. This value is higher than the 19.81 K provided by the Lydersen technique. The mean bias error (MBE) for and is −4.9 and −1.9, respectively. These results and Figures 2(a) and 2(b) show clearly that Joback technique underpredicts critical temperature, mostly for compounds which can be condensed onto particle phase, with high boiling temperature. The experimental group contributions provided by Lydersen are therefore more accurate than those provided by Joback. Thus, the Lydersen technique is more reliable than the Joback technique to estimate .
Figure 3 shows and versus experimental values for a set of 117 compounds. The RMSE is 4.5 atm and 2.6 atm for Joback and Lydersen, respectively. According to Figures 3(a) and 3(b), estimated by Lydersen matches fairly better () experimental data than estimated by Joback (). Joback technique considerably overpredicts critical pressure with MBE of 1,95 K higher than 0,39 K obtained with the Lydersen technique. In fact, besides group contribution, Lydersen technique takes into account molecular weight. Therefore, the Lydersen technique is retained to the critical pressure estimation in this paper. This is in agreement with Poling et al.  who found that the Lydersen technique is one of the best techniques for estimating critical properties. For the five vapor pressure estimation methods described previously, we will use estimated , , and because there is in general a lack of experimental data.
Some of the five methods described in Section 2.3 need , , and , while others need only and . This last pure substance property is estimated by the Joback technique, and the two critical properties are estimated by the Lydersen technique.
3.2. Evaluation of Vapor Pressure Estimation Methods
The accuracy of each method is assessed in terms of the mean absolute error (MAE), the main bias error (MBE), and the root mean square error (RMSE) (Table 3). The MBE measures the average difference between the estimated and experimental values, while the MAE measures the average magnitude of the error. The RMSE also measures the error magnitude, but gives some greater weight to the larger errors. Their expressions are given by
In (20), and are the estimated and experimental values of VOC , respectively, and is the total number of VOC. We have also used linear correlation coefficient which measures the degree of correspondence between the estimated and experimental distributions.
The logarithms of vapor pressures estimated at K for different methods are compared in the scatter plots shown in Figure 4 for a set of 262 VOC. Corresponding MAE, MBE, and RMSE are given in Table 1. In this figure, it is clear that all the five methods give similar scatter for vapor pressures higher than atm. The species concerned are hydrocarbons (Figure 5). For the set of 28 tri- and more functionalized species used in this study, vapor pressures are lower than atm (Figure 8).
Figure 4(d) shows that MY method is well correlated with experimental values. For this method, we have one of the best correlation coefficients . This method shows no systematic bias for vapor pressure lower than atm, while it is not the case for other methods. The MBE found here is 0.027. This value is one of the lowest ones of the total VOC (see Table 1). Thus, the MY method does not show any systematic bias. This method also provides the smallest values of MAE and RMSE (0.265 and 0.381, resp.). These results are in agreement with those found by Camredon and Aumont  using Ambrose technique to estimate critical properties. It is also found that this method provides the smallest values of these errors for a set of 74 hydrocarbons (see Table 2). Those are compounds with vapor pressure higher than atm. This result is not the same for mono- and difunctionalized species whose errors are some of the largest ones. The vapor pressures higher than atm are fairly well estimated. Using a set of 45 multifunctional compounds, Barley and McFiggans  found that MY method tends to overpredict vapor pressure of lower volatility compounds. Furthermore, it is important to note that MY method provides one of the poor results for difunctionalized VOC (see Table 2). Thus, for a set of 32 difunctionalized VOC, estimation values fit the experimental ones with a coefficient (Figure 7) which is the smallest value obtained for this class of species. Vapor pressures are overpredicted with a bias of 0.24, while the RMSE = 0.54 is of the same order of magnitude as those obtained by other methods. In contrast, Figure 8 shows that MY method has the best correlation for tri- and more functionalized species and is therefore the best method to estimate vapor pressure for this class of species. The systematic errors reported in Table 2 allow us to conclude that assumption. Indeed, the peculiarity of the MY method is that it takes into account the molecular structure.
Figure 4(c) displays the results for the mM method. This method gives one of the lowest scatterings with a coefficient and agrees with other methods for the highest vapor pressures. It shows a positive bias (MBE = 0.026) for the total set of 262 VOC. As the GW method, the mM method tends to overpredict vapor pressures lower than atm. Furthermore, these methods describe vaporisation entropy by taking into account van der Waals interactions. The root mean square error (RMSE = 0.442) is close to those provided by GW and AW methods. The mM method is then less appropriate than the four others for all classes of VOC.
Figure 5(c) shows that the predicted values match the experimental values with a coefficient for 32 difunctionalized species. For this class of species, estimates are provided with a positive bias (MBE = 0.208) and an RMSE of 0.512. These are the best values obtained from all the five methods (see Table 2) for difunctionalized species. Thus, mM method is more accurate than others to estimate vapor pressure for difunctionalized species, but does not provide good results for monofunctionalized (Figure 3) and tri- and more functionalized species (Figure 5).
It can be seen in Figure 7 that estimated values provided by GW method are strongly correlated with experimental values () for difunctionalized species. This method tends to overpredict vapor pressure for this class of species (MBE = 0.288), but does not show any bias for other classes of species (Table 2). Figure 8 shows that we have very acceptable results for tri- and more functionalized species with RMSE and MAE equal to 0.549 and 0.440, respectively.
Except for tri- and more functionalized species (see Figure 8), it is clear from Figures 4 to 7 that the LK method gives accurate values, based upon best correlation coefficient values. This method is the second best one of the five methods, but it has the greatest systematic bias (RMSE = 0.490, MAE = 0.32) for the total set of VOC and for difunctionalized species (RMSE = 0.70, MAE = 0.56). The MAE of hydrocarbons and monofunctionalized species are 0.142 and 0.36, respectively. For these species, Figures 5 and 6 give the best correlations.
The AW and LK methods are both based on Antoine’s equation. According to all figures plotted, it is clear that these two methods give very similar results. The peculiarity of AW is that, for the monofunctionalized compounds, the predicted and experimental values are strongly correlated with a coefficient .
Vapor pressures for a set of 74 hydrocarbons are higher than atm. It is found for the five methods that estimated values for this class of species are well correlated (Figure 5). Furthermore, vapor pressures of tri- and more functionalized species are below atm (Figure 8). For this class of species, LK method yields a weak correlation and has the largest positive bias. Therefore, this method is the least reliable to estimate vapor pressure for tri- and more functionalized species.
We have evaluated in this study five vapor pressure estimation methods useful for simulating the dynamics of atmospheric organic aerosols. These are the Myrdal and Yalkovsky (MY), the Lee and Kesler (LK), the Grain-Watson (GW), the modified Mackay (mM), and the Ambrose-Walton (AW) methods. Some of them are based on the Antoine equation, while others are based on the extended form of the Clausius-Clapeyron equation. But all of them take into account boiling temperature and (or) critical temperature . Therefore, Joback technique has been used to estimate , while the Lydersen technique was found to be better for estimation.
When using Joback to provide the values, LK, AW, and MY are the best three methods for all classes of species. Moreover, for a set of 262 volatile organic compounds and as illustrated in the scatter plots and errors computed, the MY method which appears to be the best one fails for difunctionalized species. For these latter species, the mM method provides good results, according to the correlation coefficient and the least errors reported in Table 2. GW method is the least reliable, which provides the lowest results for all VOC and also for each class of species. Predictions made with the AW method for monofunctionalized species are more reliable than those made with the other four methods employing the Joback technique to provide the . For vapor pressure higher than atm, all the five methods give similar results.
This work highlights that the choice of a method to predict vapor pressure of volatile organic compounds depends on the number of functional groups existing in the species.
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