Technical Note : Estimating fusion properties for polyacids

Introduction Conclusions References


Introduction
Diacids can be a significant part of OA, according to both field measurements and smog chamber experiments (Limbeck et al., 2001;Baboukas et al., 2000;Claeys et al., 2007;Yu et al., 1999).Due to the multicomponent nature of OA it is often glassy or liquid-like at ambient temperature even if the individual components are crystalline solids when in pure state, as was recently demonstrated for a mixture of diacids (Cappa et al., 2008b).To describe the partitioning of a compound to the aerosol, its liquid vapor pressure is required.Vapor pressures of polyacids have been measured since decades (Bradley and Cotson, 1953;Arshadi, 1974) but recently work in this area has intensified, with several publications in only this year (Booth et al., 2010(Booth et al., , 2011;;Frosch et al., 2010;Soonsin et al., 2010;Pope et al., 2010).
Unfortunately, pure diacids are solid at ambient temperature.To obtain the liquid vapor pressure, one could extrapolate from measurements above the melting point T fus , Introduction

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Full but as T fus can be a few hundred Kelvin above the temperature of interest, this approach is very prone to error.Some groups have measured the vapor pressure of the liquid diacid in a mixture with water (Zardini et al., 2006;Riipinen et al., 2007;Koponen et al., 2007;Pope et al., 2010;Soonsin et al., 2010).In that case, also the activity coefficient is needed in order to determine the vapor pressure of the pure diacid, which can be calculated using empirical methods (Peng et al., 2001;Hansen et al., 1991).
Other groups have measured the solid vapor pressure (Cappa et al., 2007(Cappa et al., , 2008a;;Booth et al., 2010Booth et al., , 2011;;Frosch et al., 2010;Ribeiro da Silva et al., 2000, 2001;Salo et al., 2010;Bilde and Pandis, 2001;Bilde et al., 2003;Soonsin et al., 2010).However, there can be orders of magnitude difference between measurements of different groups on the same compound (e.g. for sebacic acid, 3 orders of magnitude between Cappa et al., 2007 andSalo et al., 2010), way above the reported experimental errors (typically 30-50%).It has been speculated that this might be due to the experimental technique employed (Cappa et al., 2007;Pope et al., 2010) or to the physical nature of the diacids (Zardini et al., 2006;Soonsin et al., 2010;Salo et al., 2010) (presence of defects; partially or completely liquid/amorphous character).Soonsin et al. (2010) have measured supercooled liquid vapor pressures with only a very small water content, and vapor pressures of the saturated solution, which allows the derivation of the vapor pressure of the pure liquid and solid, respectively.Even if a given experimental sublimation pressure can be considered accurate, one still needs fusion data to obtain a subcooled liquid vapor pressure (Prausnitz et al., 1999): with p 0 l , p 0 s the vapor pressures of the liquid and solid state, respectively, R the ideal gas constant, ∆S fus the entropy of fusion and ∆C p,sl the difference between solid and liquid heat capacity.The fusion temperature, enthalpy of fusion ∆H fus and entropy of fusion are related by Introduction

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Full Although the second term in Eq. ( 1) can be significant for large values of the difference T fus −T , it will generally be much less important than the first term.Moreover, as ∆C p,sl is frequently unavailable experimentally, it is often estimated from ∆S fus (e.g., Booth et al., 2010).Therefore, the availability of measured or estimated fusion data appears critical.However, it occurs often that fusion data is unavailable, or only the fusion temperature is known (Ribeiro da Silva et al., 2000;Monster et al., 2004;Frosch et al., 2010).General estimation methods for T fus , ∆H fus (Joback and Reid, 1987;Marrero and Gani, 2001;Zhao and Yalkowsky, 1999) or ∆S fus (Myrdal and Yalkowsky, 1997;Jain et al., 2004a) give significant errors for diacids, as we will show below.Therefore, a new simple estimation method is developed specifically for this class of compounds.

Literature data
In Table 1 we present experimental fusion data for polyacids and keto or hydroxy polyacids, taken from Roux et al. (2005); Booth et al. (2011Booth et al. ( , 2010)).In case also solid-solid transitions were present, always the sum over all fusion data was taken: The standard deviations (STD) were obtained by comparing overlapping data for linear diacids from Cingolani and Berchiesi (1974); Hansen and Beyer (2004) evaluated at 298.15 K is given in Table 1.Roughly speaking and excluding oxalic acid, ω is about 1 for odd-numbered linear diacids and 2 for even-numbered linear diacids.
For oxalic acid, T fus reported by Booth et al. (2010) conflicts with the transition data reported by Linstrom and Mallard and Thalladi et al. (2000).It is possible that Booth et al. (2010) found a solid-solid transition point rather than a fusion point (Booth 2010, personal communication).Furthermore, Soonsin et al. (2010) have measured the vapor pressure of both solid and supercooled liquid oxalic acid, and found two orders of magnitude difference.This would result in a ω of about 2, much more than the value of 0.12 calculated from the data of Booth et al. (2010), but corresponding satisfactorily with those of the other even-numbered linear diacids.However, interpretation of experimental data is hampered due to uncertainty regarding the precise structure of the solid oxalic acid (Soonsin et al., 2010).Therefore, we exclude oxalic acid in our comparison analysis of experimental with modeled data.

Testing existing estimation methods
The methods considered are presented in Table 3.Both the methods of Joback and Reid (1987) (JR) and Marrero and Gani (2001) (MG) are group contribution methods providing both T fus (JR(T), MG(T)) and ∆H fus (JR(H), MG(H)).While the former is relatively simple, the second is a detailed method involving first, second and third order groups.The method of Myrdal and Yalkowsky (1997) The JR(T) method gives a very large error on the fusion temperature.This can be ascribed to the fact that this method considers T fus as a sum of group contributions, and this gives large overestimations for larger molecules (up to 275 K for citric acid).
Clearly, the JR(T) method is not suitable to estimate fusion point of polyacids.We note that a similar failure occurs for the estimation of boiling points by the JR method (Stein and Brown, 1994).The ZY(T) and MG(T) methods perform best for fusion temperature, while the more recent version of ZY(T), JYY(T), actually performs worse.Notwithstanding its high detail, the MG(H) method performs worse than the JR(H) method in estimating fusion enthalpy, and has a relatively high bias.For fusion entropy estimation, the MY(S) and JYY(S) method have a similar precision, but the last one (Jain et al., 2004a) has the highest bias.Introduction

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Full For the calculation of ω, it is possible that no experimental fusion data are available and hence the estimation of two fusion properties is necessary.Combinations with JR(T) give a large positive bias for ω due to the large overestimation of T fus .This method will not be considered further.The combinations MG(T) + MY and MG(T) + JR(H) perform the best in terms of low bias and STD, with the first having the smallest bias.Better estimation is possible when one fusion property is already known.This is typically the fusion temperature.Best results are obtained when employing MY(S) in combination with the experimental fusion temperature.Note that the more recent version of this method, JYY(S), performs worse.
Even in the best case, the MAE and STD of ω remains quite substantial.The methods are general purpose and are apparently not well suited to polyacids.Therefore, we developed a simple specialized method, based on the experimental data from Table 1.

Development of a new estimation method
As is well known, the fusion data of linear diacids follow an even-odd alternation (Roux et al., 2005).Using the data from  where #CH 2 denotes the number of methylene groups.Using correlation Eq. ( 13), a T fus of 490 K would be predicted for oxalic acid, even higher than the fusion point reported by Linstrom and Mallard, Thalladi et al. (2000).This gives further argument that the fusion point reported by Booth et al. (2010) could be a solid-solid transition point.
No such clear correlation of fusion enthalpy or entropy with carbon number exists for the nonlinear polyacids.For example, for the cyclic diacids, the diacids with the highest carbon number have the lowest ∆H fus and ∆S fus , while the other three have very similar values.Instead, we use as independent variable the effective torsional bond number τ (Dannenfelser and Yalkowsky, 1996).Also the number of nonacid functional groups (keto and hydroxy groups), and the identification of the molecule as a linear evennumbered chain, are taken as independent variables.Our estimation method has then the following form with i even = 1 if the molecule is a linear even-numbered chain and 0 otherwise, and n OH and n CO the number of hydroxy and keto groups, respectively.In Table 5 the optimal parameters, obtained by linear regression, are given, as well as the STD and MAE, and the prediction sum of squares (PRESS).This last statistical diagnostic is based on the leave-one-out principle and is calculated by where is an experimental measurement and f est (i ) a model calculation, using parameters fitted to all experimental data except f exp i .In this way, PRESS is a measure of the predictive power of the model, while MAE and STD merely show how well the model can fit the observations.It is always higher than the STD.Cyclicity as an extra independent variable was tested but this did not improve the PRESS.Introduction

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Full Comparing the PRESS of Table 5 with the STD of Table 3, it is clear that this new model performs much better than the methods considered in Sect.3.This should of course not be a surprise, as these methods have a much wider scope, and most experimental data used to develop this model are more recent than these methods.If T exp fus is available, this leads to an important improvement of the estimation of ω.It makes thereby not much difference whether ∆S fus or ∆H fus is used in conjunction with T exp fus .When testing our method on the molecules of Table 2, an STD of 31 K is obtained, relatively close to the PRESS obtained from the data of Table 1.This confirms the robustness of our model, as the data of Table 2 was in no way included in the development of our method.

Predicting fusion data for some compounds
In Table 6 we present the fusion data estimations for the compounds in Table 2.The lowest ω is predicted for pinic acid, and the highest for 4-oxo pimelic acid.

Conclusions
To derive subcooled liquid vapor pressure from solid vapor pressure, knowledge of the fusion properties is necessary.Several fusion property estimation methods are tested for polyacids, having possibly keto-or hydroxy groups.Best results are obtained by combining the experimental fusion temperature with the method of Myrdal and Yalkowsky (1997), and estimating the fusion temperature with Marrero and Gani (2001) if it is not available.We have also developed a simple method to estimate the fusion properties for this kind of compounds, with a smaller error compared to the other methods.Introduction

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a
Booth et al. (2010) b Roux et al. (2005) c Booth et al. (2011) * For oxalic acid, Linstrom and Mallard and Thalladi et al. (2000) report a solid-solid transition point at 393.2 K and a fusion point at 463-464 K, which are both significantly higher than the fusion point reported by Booth et al. (2010).Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (Jain et al., 2004b)ent variant Introduction ofJain et al. (2004a)(JYY) estimate ∆S fus from the number of torsional bonds and rotational symmetry of the molecule.Although the method of Zhao and Yalkowsky (1999) (ZY), and its more recent variant(Jain et al., 2004b)(JYY), are formally ∆H fus group contribution methods, they are rather T fus estimating methods, as the group contributions are fitted to experimental fusion points, with the entropy of fusion fixed by the MY method.In Table4, the bias, the mean absolute error MAE and the standard deviation STD are presented.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Table 1 .
Experimental fusion data of several polyacids.

Table 5 .
Parameters and statistic diagnostics of the new estimation method.

Table 6 .
Estimated ∆H fus , ∆S fus and ω for the compounds of Table2.