Introduction

ACP

Atmospheric Chemistry and Physics

ACP

Atmos. Chem. Phys.

1680-7324

Copernicus GmbH

Göttingen, Germany

10.5194/acp-14-12815-2014

Henry's law constants of polyols

Compernolle

steven.compernolle@aeronomie.be

https://orcid.org/0000-0003-0872-0961

Müller

J.-F.

Belgian Institute for Space-Aeronomy, 1180 Brussels, Belgium

S. Compernolle (steven.compernolle@aeronomie.be)

8December2014

14 23 1281512837 8May2014 23May2014 24October2014 12November2014

This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/

This article is available from https://acp.copernicus.org/articles/.html

The full text article is available as a PDF file from https://acp.copernicus.org/articles/.pdf

Henry's law constants (HLC) are derived for several polyols bearing between 2 and 6 hydroxyl groups, based on literature data for water activity, vapour pressure and/or solubility. While deriving HLC and depending on the case, also infinite dilution activity coefficients (IDACs), solid state vapour pressures or activity coefficient ratios are obtained as intermediate results. An error analysis on the intermediate quantities and the obtained HLC is included. For most compounds, these are the first values reported, while others compare favourably with literature data in most cases. Using these values and those from a previous work , an assessment is made on the partitioning of polyols, diacids and hydroxy acids to droplet and aqueous aerosol.

Introduction

Henry's law constant (HLC) describes the partitioning of a compound between the gas phase and a liquid, highly dilute solution. In the atmosphere, such dilute solutions, with water as the solvent, can be reached in cloud droplets. Aqueous aerosols are another example where liquid water is important, but in this case the solvent must be regarded as multicomponent, with significant inorganic and/or organic contributions. In a previous work we determined additional HLC data for diacids and hydroxy polyacids, from water activities, solubilities and vapour pressures, employing thermodynamic relationships. We follow the same approach in this work, but with a focus on polyols: compounds with two or more hydroxyl groups, but no other functional group. Polyols such as 2-methyl tetrols were identified as important secondary organic aerosol (SOA) constituents . HLC data on polyols are limited. Data are available e.g. in the often-cited compilation of . In this compilation however, most values are estimated by a group-contribution method, while only for three molecules are experimental values included, and some of the data were evaluated as unreliable.

In Sect. , we first briefly review the thermodynamic relationships employed to derive HLC (kh) and the associated enthalpy of dissolution of a gas phase species (ΔHg→aq∞). In Sect. we derive intermediate results, namely infinite dilution activity coefficients (IDACs), solid state vapour pressures and activity coefficient ratios, which are necessary components in deriving HLC but not as such available in the literature for all compounds. Next (Sect. ), we present kh and ΔHg→aq∞ using the data from the previous section and other literature data. An error analysis is presented in Appendix . In Sect. , we discuss the implications for partitioning to cloud droplets and aqueous ammonium sulfate aerosol, making use of the activity coefficient model AIOMFAC (Aerosol Inorganic–Organic Mixtures Functional groups Activity Coefficients) . We extend this also to dicarboxylic acids and hydroxy polyacids, making use of a model presented at the site E-AIM (Extended AIM Aerosol Thermodynamics Model), available at http://www.aim.env.uea.ac.uk/aim/aim.php, see e.g. to calculate the acid dissociation.

Note that in this work, non-IUPAC names are used for polyols with three or more hydroxyl groups. Their structures are presented in Table .

Thermodynamic relationships between HLC and <?xmltex \hack{\\}?>other quantities

For HLC, several definitions exist. We will follow here the convention used by : kh≡lim⁡cs,ps→0csps with cs the solute molar concentration of the solution and ps its partial pressure above it. Note that we assume ideal gas behaviour for the solute. With this convention, a larger kh means a higher partitioning of the solute to the solution. At some point, we will employ also another definition of HLC, following again the notation of , khpx≡lim⁡xs,ps→0xsps with xs the solute mole fraction. The px superscript specifies that vapour pressure p is used for the gas phase concentration, and mole fraction x for the aqueous phase concentration.

Molecular structures of polyols with three or more hydroxyl groups, discussed in this work using non-IUPAC names.

name # OH structure name # OH structure glycerol 3

arabinitolb,c 5

erythritol 4

sorbitol 6

pentaerythritol 4

mannitole 6

xylitol 5

dulcitold,e 6

adonitola,c 5

a also named ribitol, b also named arabitol, c stereo isomer of xylitol, d also named galacticol, e stereo isomer of sorbitol.

The relation between both quantities is kh/khpx=csol with csol the total molar concentration of the solution (or solvent, since we assume that the solute is infinitely diluted). In the case that the solvent is pure water, we write csol=cw (equal to 55.5 molL-1 at 298.15 K).

If the solute is a liquid at the temperature of interest and the solvent is water, kh can be obtained by kh=cwγ∞pL0 with γ∞ the infinite dilution activity coefficient (IDAC) of the solute (mole fraction based and with the symmetric convention γ∞=1 for pure liquid solute) and pL0 its pure liquid state vapour pressure. The corresponding enthalpy of dissolution of an infinitesimal amount of gas phase species (ΔHg→aq∞) can be derived from the van 't Hoff equation see e.g. and (neglecting the small temperature dependence of cw) related to the enthalpy of vaporization (ΔHvap) and the enthalpy of solution of the liquid solute at infinite dilution (ΔHL→aq∞) using Hess's law , -Rdln⁡khd(1/T)=ΔHg→aq∞=ΔHL→aq∞-ΔHvap since the dissolution of a gas in a solvent can be considered as first a condensation of the gas to the pure liquid (corresponding to -ΔHvap) followed by a dissolution of this liquid in the solvent (corresponding to ΔHL→aq∞). ΔHvap and ΔHL→aq∞ can themselves be derived from the van 't Hoff equation or, for Eq. (), from the Clausius–Clapeyron equation : ΔHL→aq∞=Rdln⁡γ∞d1/TΔHvap=-Rdln⁡pL0d1/T.

If the solute is a solid at room temperature, as is generally the case for polyols with more than three hydroxyl groups, the following equation can be applied instead of Eq. () seefor the derivation: kh=γsatγ∞cwxsatpCr0 with xsat the solute mole fraction at the solubility limit and γsat the corresponding activity coefficient. In the case that the solubility is small, γ∞/γsat≈1 and Eq. () reduces to kh≈cwxsatpCr0. The corresponding enthalpy of dissolution of gas phase species, derived from the van 't Hoff equation, can be related to (again neglecting the temperature dependence of cw) the sublimation enthalpy (ΔHsub) and the enthalpy of solution of the solid at infinite dilution (ΔHCr→aq∞) using Hess's law , ΔHg→aq∞=-Rdln⁡khd(1/T)=ΔHCr→aq∞-ΔHsub with ΔHsub=-Rdln⁡pCr0d1/TΔHCr→aq∞=-Rdln⁡γsatxsatγ∞d1/T=ΔHfus+ΔHL→aq∞. Equation () is the Clausius–Clapeyron equation for sublimation . Equation () can be derived by combining Eqs. () and () and the van 't Hoff relation for the activity of the solid asCr see e.g.: dln⁡γssatxssatd(1/T)=dln⁡asCrd(1/T)=-ΔHfusR. The data required for Eqs. (), (), () and () is not always available in the literature. In Sect. , data for IDACs, solid state pressures and activity coefficient ratios are derived.

Intermediate results Infinite dilution activity coefficients

In the case that the compound is a liquid at room temperature, Eq. () applies and the IDAC is required. Estimation methods to calculate activity coefficients exist e.g. but experimental data are preferred. provide IDAC data for diols, but only for compounds with up to four carbon atoms. For many other polyols, the IDAC of the solute is not reported, but instead data are available on the water activity aw as a function of mixture composition. In that case, γs∞ can be obtained by the Gibbs–Duhem relation (here stated in its integral form) , ln⁡γs∞=∫01ln⁡γw(xw)1-xw2dxw with γw=aw/xw the activity coefficient of water and xw the water mole fraction. Note that we added the subscript “s” to γ∞ to distinguish clearly the activity coefficient of the solute and the activity coefficient of water.

If sufficiently precise, fine-grained aw data over the whole composition range would be available, and numeric integration of the integral in Eq. () would be the most straightforward. If this is not the case, an alternative is to fit the ln⁡γw data with an activity coefficient expression f, e.g. Margules, Van Laar, Wilson or UNIQUAC – see Appendix for the expressions.

provide aw data for 14 diols and two triols over a broad composition range (xw typically between 0.1 and 0.95.), but the data are rather coarse grained. This is especially critical in the dilute region; from Eq. () it can be concluded that a small change in ln⁡γw leads to a comparatively large change in ln⁡γs. Therefore, where possible, we included also more fine-grained data in the dilute region xw typically between 0.93 and 0.996. In Table the resulting γs∞ are presented. For eight diols we present also the γs∞ estimation without taking the available dilute region data into account, i.e. based on the data only. This can then be compared with the γs∞ based on all data. In most cases, the difference in the resulting IDAC is rather small (see Appendix ). This indicates that even in those cases where only the coarse-grained data of are available, the derived γs∞ are still quite reliable.

Often, we were able to fit an activity expression to all ln⁡γw data. In other cases, where the broad-ranged but coarse-grained data of was combined with the more fine-grained data in the dilute region, we had to split the integral in Eq. () into two parts. More details on the procedure to derive γs∞ are provided in Appendix . Also included in Appendix is an uncertainty analysis in the derived γs∞ values by applying systematic and random shifts to the aw data, by testing alternative activity expressions and by comparing γs∞ derived with and without dilute region data.

Infinite dilution activity coefficients derived in this work, the sources of water activity they are based on, and comparison with the literature.

molecule

γs∞

aw source γs∞ (lit.)f 1,2-ethane diol 0.75, 0.69g

a,b

0.8 1,2-propane diol 1.25, 1.08g

a,b

1.0 1,3-propane diol 1.23, 1.25g

a,b

1.2 1,2-butane diol 3.00, 3.74g

a,b,c

2.0 1,3-butane diol 2.14, 1.97g

a,b,c

2.2 1,4-butane diol 2.27, 2.12g

a,b,c

2.8 2,3-butane diol 2.10, 1.77g

a,b,c

1.6 1,5-pentane diol 5.99, 5.26g

a,b

1,2-pentane diol 11.9

1,4-pentane diol 3.8

2,4-pentane diol 2.8

1,2-hexane diol 26.3

2,5-hexane diol 5.7

1,7-heptane diol 27.9

glycerol 0.52

a,d,e

1,2,4-butane triol 0.45

a,d

a . b . c . d . e . f . g The second value is obtained by applying Eq. () to the data of only.

The γs∞ for pentane and hexane diols, derived from surface tension data, are reported by and . These are considerably higher than the data presented in Table . However, as explained by , very accurate surface tension data are a prerequisite to derive γs∞; the γs∞ of diols, derived from surface tension data, are all overestimated compared to the literature data in their analysis. Moreover, it is not clear to us whether the applied approximation the Volmer surface equation of state, see is valid in this case.

Solid state vapour pressures

The liquid state vapour pressure of organic compounds can be estimated e.g., but for polyfunctional compounds the result is often not accurate. Solid state vapour pressure is even more difficult to estimate, as this depends on the molecular arrangement in the crystal structure which is compound specific. Therefore, experimental data are preferred. Solid state and/or liquid state vapour pressure data for polyols with four or more hydroxyl groups is available , but obtained at temperatures considerably above room temperature. The solid state vapour pressure at Tref=298.15K is given by ln⁡pCr0Tref=1/RΔSsubTref-1TrefΔHsubTref. If at the temperature of measurement Tmeas, the compound is a solid, the following temperature correction is applied Kirchhoff's law, see e.g.: ΔSsubTref=ΔSsubTmeas+∫TmeasTrefCp,g-Cp,CrTdTΔHsubTref=ΔHsubTmeas+∫TmeasTrefCp,g-Cp,CrdT with ΔHsub and ΔSsub weak functions of temperature. If at Tmeas the compound is a liquid, the fusion point must be taken into account, and the temperature correction is ΔSsubTref=ΔSvapTmeas+∫TmeasTfusCp,g-Cp,LTdT+ΔSfusTfus+∫TfusTrefCp,g-Cp,CrTdTΔHsubTref=ΔHvapTmeas+∫TmeasTfusCp,g-Cp,LdT+ΔHfusTfus+∫TfusTrefCp,g-Cp,CrdT with Cp,g,Cp,L,Cp,Cr the constant pressure heat capacities for respectively gas, liquid and crystalline phase, Tfus the melting temperature,ΔSfus,ΔHfus the entropy and enthalpy of fusion, and ΔSvap,ΔHvap the entropy and enthalpy of vaporization. In most cases, the high-temperature p0 data are not measured at one temperature but in a temperature interval. Tmeas then corresponds to the centre of this interval.

Solid state pressure and enthalpy of sublimation of polyols, obtained from Eqs. (), () and/or (), at Tref=298.15K, and reference to the data used to obtain them. Cp,g is estimated by the method of .

Molecule

pCr0(Tref)¯

ΔHsub(Tref)¯

Data source for Eqs. (), () and/or () Pa kJ mol-1 pL0,ΔHvap (L)a

Tinterval¯

Tfus,

Cp,L (L)a or pCr0,ΔHsub (Cr) K

ΔHfus

or Cp,Cr (Cr) nonane-

1.4×10-3

148 Lb 323

Crk, Li diol

2.4×10-3

148 Lc 347–373

Crk, Lo decane-

1.4×10-4

151 Crb 342 Crk diol

1.7×10-4

161 Lc 351–377

Crk, Lo erythritol

4.7×10-7

155 Ld 397–428

L, Crh

3.6×10-6

136 Cre 379–392 Crh penta-

7.2×10-9

166 Crd 416–456 Crj erythritol

3.5×10-7

135 Cre 397–410 Crj

9.3×10-8

147 Crf 380–408 Crj xylitol

7.5×10-8

162 Ld 406–460

L, Crh adonitol

2.7×10-8

166 Ld 418–465

L, Crh arabinitol

2.1×10-8

166 Ld 414–466

L, Crl sorbitol

3.6×10-11

198 Ld 452–502

L, Crh mannitol

6.7×10-13

206 Ld 458–501

Crh, Lm dulcitol

1.9×10-13

210 Ld 464–500

Crn, Lm

a “L” (liquid) and “Cr” (crystalline) are used as shorthand to describe the phase state of the non-gaseous phase. b , c , d , e , f , g , h , i , j , k . l No Cp,Cr,Cp,L data were found for arabinitol. The data for adonitol were taken instead. m No Cp,L data were found for mannitol and dulcitol in the literature; the Cp,L data for sorbitol were taken instead. n No Cp,Cr data were found for dulcitol in the desired temperature range. Low-temperature (≤292.8K) data are comparable to those of mannitol, therefore, Cp,Cr of mannitol was taken instead. o No Cp,L data for decane diol were found in the literature. It was estimated by the method of .

Fusion data were taken from , and . Experimental heat capacity data for solid and liquid were taken from , and , while for the gas it was calculated by the method of , available from the NIST Chemistry WebBook (). In Table the derived solid state vapour pressures and sublimation enthalpies at room temperature are presented.

Fusion data of on one hand and of on the other hand could be compared for erythritol, xylitol, adonitol, sorbitol and mannitol. There is generally a good agreement between both data sets: Tfus was always within 6 K and ΔHfus within 4 %, with the exception of xylitol where the deviation is 11 %. Interchanging both data sets had an impact of a factor 1.3 on pCr0 at room temperature at most. Where available, the more recent data set of Tong et al. was preferred over that of .

Cp,L and Cp,Cr were not always available. In that case, the data of a stereo-isomer were taken instead. In Appendix , it is shown that liquid phase heat capacities of polyol stereo-isomers are very close. This is in agreement with the similar ΔSvap and ΔHvap of stereo-isomers reported by and indicates similar thermodynamic properties of the liquid phase. Regarding Cp,Cr, differences between stereo-isomers are larger (see Appendix ) but still the approximation of using a stereo-isomer seems reasonable.

Neglecting the integrals involving the heat capacity differences in Eqs. () and () can lead to serious error: while for the tetrols the change is only minor, there is a factor of 5 to 7 increase in pCr0(298.15K) for the pentols and most hexols, and a factor 40 increase for sorbitol. Estimating Cp,g with the method of instead of the method of led to changes in pCr0(298.15K) smaller than a factor of 2. Note that these two methods do not take the intramolecular hydrogen bonding into account. The group contribution Cp,g estimation method of and , based on quantum chemical data, does include corrections for intramolecular hydrogen bonds. However, it is not clear how to apply these correction terms for species with three or more hydroxyl groups. Using one HOCCO term (NNI5 in the terminology of ) per hydroxyl group for the linear polyols, one obtains a factor of 2 to 3 higher pCr0(298.15K) for the pentols and hexols, compared to the case when this term is neglected.

In most cases the high-temperature pL0 or pCr0 data are obtained from a single reference ; only for erythritol and pentaerythritol is a comparison possible between different data sources. The high-temperature pL0 or pCr0 data of erythritol and pentaerythritol are roughly comparable among the different data sources ; if the p0 parameterizations presented in these works are evaluated at mid-points between their respective Tmeas, differences ranging from 4 % up to 40 % are obtained. However, due to differences in ΔHvap or ΔHsub, the extrapolated pCr0(298.15K) is a factor of 7 to 50 higher if the older data of are used, compared to when the more recent data of are used. In the older studies the enthalpy was determined using far fewer data points (6–11, compared to 25–30 for the data of ), and specifically for the data of , over a quite narrow temperature interval (∼ 12 K, compared to 30–40 K for the other studies). Therefore, we consider the pCr0 derived from the high-temperature data of as more reliable.

Mole fraction solubilities xssat and activity coefficient ratios γs∞/γssat derived from aw data. Estimations of γs∞/γssat by UNIFAC-MP are also given.

Molecule

xssat

γs∞/γssat

aw data source γs∞/γssat MP nonane diol

1×10-3

1.0p – 1.06 decane diol

7.6×10-5

1.0p – 1.01 erythritol 0.074b 0.84

h,k,i

0.75 pentaerythritol 0.00946c 1.0q – 0.97 xylitol 0.18e 0.56

h,j

0.32 adonitol 0.15d 0.66l

0.37 arabinitol 0.20d 0.60l

0.30 sorbitol 0.196g 0.45o

h,j,m

0.18 mannitol 0.0209f 0.96n

h,m

0.80 dulcitol 0.0031d 1.0p – 0.97

Solubilities: a Merck Millipore (http://www.merckmillipore.com/), at 20 ∘C, b , c , d , e , f , g .Water activities: h , i , j , k , l (one-parameter fittings), m .n For mannitol, only the aw data of were used, as the data of led to γw>1, which is probably wrong. o Due to the extrapolation involved (no aw data at xsat), this value is more uncertain. p No aw data were found. γs∞/γssat=1 was assumed because of the low solubility, i.e. Eq. () is considered valid.

Uncertainties in the derivation of pCr0(Tref) are analysed in Appendix . The largest uncertainties are encountered for the polyols with four or more hydroxyl groups; due to the large difference between Tmeas and Tref, relatively small changes in ΔHvap or ΔHsub lead to large changes in pCr0(Tref). Uncertainty in heat capacity becomes important for the hexols. Uncertainty in fusion data is relatively unimportant.

Activity coefficient ratios

The ratio γs∞/γssat can be obtained from water activity data in the subsaturation range : ln⁡γs∞γssat=1-xssatxssatln⁡γwx̃w+∫x̃w1ln⁡γw(xw)1-xw2dxwx̃w=1-xssat. The polyols with more than three hydroxyl groups considered here are solid at room temperature. Their water activity is only measured up to the solubility limit, if measured at all. Similarly to our previous work , activity coefficient expressions – Margules, Van Laar, Wilson see e.g. – were fitted to aw data in the subsaturation range, and the fitting parameters were used to obtain the solute activity coefficient ratio γs∞/γssat.

The ln⁡γw data for erythritol, xylitol, mannitol and sorbitol, compared with the fitting using the Margules formula, and UNIFAC-MP results. Note that the data of are at 20 ∘C rather than 25 ∘C, but from their data at 35 ∘C, it can be deduced that the temperature dependence of ln⁡γw is small.

The precise procedure is described in Appendix A of , and the resulting parameters are shown in Fig. .

This was done for erythritol, xylitol, sorbitol and mannitol (Fig. ). The UNIFAC (UNIQUAC Functional-Group Activity Coefficient) method of (UNIFAC-MP, identical to AIOMFAC for polyol-water systems) underestimates γw of these polyol/water mixtures.

For adonitol and arabinitol, we calculated γs∞/γssat from the one-parameter Margules fittings of (see Appendix ). The results are presented in Table . For nonane diol, decane diol, pentaerythritol and dulcitol, no aw data were found, but reasonable assumptions for γs∞/γssat could be made (see Table ). As expected, the polyols with a lower solubility (erythritol, mannitol) have γs∞/γssat close to unity. We included estimations of the activity coefficient ratio by UNIFAC-MP. This method gave lower γs∞/γssat as compared to our results.

Henry's law constants and gas dissolution enthalpies at 298.15 K for the polyols considered in this work, from Eqs. (, , , ). γs∞ is taken from Table , pCr0,ΔHsub from Table and γs∞/γssat from Table .

Molecule

ΔHg→aq∞¯

Data source for Eqs. () and ().

Matm-1‾

kJ mol-1

pL0,

ΔHL→aq∞ (L)

Matm-1‾

ΔHvap

or ΔHCr→aq∞ (Cr) (lit.) 1,2-ethane diol

6.6×105

-72.9

4.1×105

1,2-propane diol

2.7×105

-78.8

1×105

1,3-propane diol

1.6×106

-79.1

9.2×105

1,2-butane diol

2.1×105

-82.1

Lg 1,3-butane diol

7.1×105

-84.5

Lg 1,4-butane diol

3.5×106

-89.6

1×105

2,3-butane diol

1.1×105

-82.2

4×104

1,2-pentane diol

1.4×105

1,4-pentane diol

2.3×106

1,5-pentane diol

7.1×106

-103.5

Le 2,4-pentane diol

3.9×105

1,2-hexane diol

1.7×105

2,5-hexane diol

1.4×106

1,7-heptane diol 4.6×106–8.4×106

1,9-nonane diol 2.4×106–4.0×106 1,10-decane diol 2.5×106–3.0×106 glycerol

4.8×108

-92.6

6×108

erythritol

1.1×1012

-133 Cri pentaerythritol

7.4×1012

-133 Cri xylitol

4.0×1013

-140 Cri adonitol

4.7×1013

-147 Cri arabinitol

6.8×1013

-147 Cri sorbitol

6.7×1016

-181 Cri mannitol

1.8×1017

-184 Cri dulcitol

9.1×1016

-181 Cri

a , , . For 1,2-propane diol: . b . c pL0 from either (giving rise to the lower kh) or (giving rise to the higher kh). pL0 is corrected to 298.15 K using Cp,L data from and Cp,g estimated with the method of , but the effect is small. d . e , f , g , h , i . j , value at 293 K. k , value obtained from vapour-liquid equilibrium data.

Henry's law constants and enthalpies of gas <?xmltex \hack{\\}?>dissolution

Using Eqs. (, , , ), the data provided in the previous tables, as well as literature data, the Henry's law constants and gas dissolution enthalpies can be derived. The values are given in Table .

The expected order hexols > pentols > tetrols > triol > diols in kh is generally followed. Diols have kh between 105 and 107 Matm-1. The diols with longer hydrophobic chains have considerably lower kh than their α,ω counterparts (e.g. an order of magnitude difference between 1,2- and 1,5-pentane diol). For the linear polyols, kh and ΔHg→aq∞ are roughly comparable among the different stereo-isomers. Clearly, the large differences in xsat and pCr0 of the hexols are mainly due to their different crystal structure, which does not affect kh. Although pentaerythritol has the same number of hydroxyl groups as erythritol, its kh is seven times larger. Probably the tetragonal arrangement of the hydroxyl groups of pentaerythritol facilitates bonding with the water molecules.

Impact on gas-particle partitioning

Similarly as for the diacids and hydroxy polyacids , we assess the importance of the partitioning to the particulate phase for polyols in clouds and aqueous aerosol, approximating the liquid phase as a dilute aqueous solvent. Moreover, we also perform a sensitivity test, aiming at determining the particulate fraction of polyols, diacids and hydroxy polyacids in the case of an aqueous ammonium sulfate aerosol.

We note that this discussion, based on Henry's law, is only applicable if the organic compound is present in a tiny amount, and this may not be justified for all situations encountered in the lower atmosphere.

Pure water as the solvent

For clouds, the liquid water content (LWC) varies between typically 0.1 and 1 gm-3, and for aqueous aerosols between 10-6 and 10-4 gm-3 . The particulate fraction of organic solute is equal to fp,s≡np,snp,s+ng,s with np,s,ng,s the moles of solute in particulate and gas phase respectively. Using the ideal gas law, Eq. () can be transformed into fp,s=11+ng,s/np,s=11+psVairRT1csVp with Vair a unit volume of air and Vp the particle volume. If partitioning between gas and aqueous phase is governed solely by Henry's law (Eq. ), and the solvent is considered pure water, Eq. () becomes fp,s=1k*/kh+1, with k*=VairVs1RT=ρwLWC1RT with ρw the density of pure water. From the LWC range given above, and fixing T to the reference temperature of 298.15 K, it follows that for clouds, k* varies between 4×104 and 4×105 Matm-1. From Table , it can be deduced that all polyols with three or more hydroxyl groups will be almost completely partitioned to the aqueous phase. Diols will be completely or partially in the aqueous phase, depending on the case. For aqueous aerosol, if one (falsely) assumes the aerosol phase to be pure water, it is obtained from the LWC range given above that at the reference temperature k* varies between 4×108 and 4×1010 Matm-1. With this assumption, diols will not partition appreciably to the aqueous phase, glycerol will partition to some extent, and only at the highest water content, while all polyols with four or more hydroxy groups should reside almost completely in the particulate phase.

Aqueous ammonium sulfate aerosol as the solvent

An aqueous aerosol is not a dilute aqueous solution, but is instead a concentrated mixture of organics and/or inorganics. HLC determined for a pure water solvent are less applicable to such mixtures. We present here a sensitivity test for a simple aerosol mixture of water and ammonium sulfate ((NH4)2SO4, AS).

Note however that this test is only applicable to a situation with a tiny amount of organics. If e.g. a separate organic phase is present, less hydrophilic compounds may partition substantially to the particles, even if the HLC-based analysis suggests otherwise.

It is more convenient to use here the alternative HLC definition khpx instead (Eq. ). Let us define ng,s and np,s as the number of moles of solute in gas and particulate phase respectively, and np,tot as the total number of moles of the solution. As the particulate solute mole fraction is equal to np,s/np,tot, and using the ideal gas law, Eq. () can be transformed into khpx=lim⁡ng,s,np,s→0np,s/np,totng,sVairRT=fp,s1-fp,sVairnp,totRT. In the last step Eq. () was used. Eq. () can be rearranged to fp,s=11+1khpxRTVairnp,tot Note that in the particular case of the AS–water system np,tot=np,w+3np,AS+np,s=np,w+3np,AS as each molecule of AS dissociates into three ions, and the amount of organic solute is infinitesimally small.

The organic solutes considered are the polyols discussed in this work, and the diacids and hydroxy polyacids treated in our previous work . The khpx of a solute for a solvent consisting of water and a mole fraction xAS of dissolved AS can be calculated from khpx(xAS)=khpx(xAS=0)γs∞(xAS=0)γs∞(xAS)=khpx(xAS=0)1γs* with γs* the activity coefficient of the organic solute using the asymmetric convention (i.e. γ*=1 if the solute is infinitely diluted in pure water). The khpx(xAS=0) was taken from Table or Table 3 of (recommended values only), after the appropriate conversion khpx=cwkh. The γs∞(xAS) and γs∞(xAS=0) were calculated with the model AIOMFAC , available online (http://www.aiomfac.caltech.edu). This model calculates activity coefficients taking interactions between water, organics and inorganics into account. A very small organic mole fraction (xorg=10-10) was chosen to ensure that γs∞(xAS) and γs∞(xAS=0) represent IDACs. As a consequence, the impact of the organic solute on the activities of water and the ions is negligible. Although the activities are estimated and not measured, we note that activity data sets of several AS–water–organic mixtures (organic being a polyol, diacid or hydroxy polyacid) are used to determine AIOMFAC's parameters .

Given a particular xAS, the water activity and hence the relative humidity (RH) are fixed by the AIOMFAC model. xAS was varied between 0.43 and 0, corresponding to a RH range between 30 and 100 %. Note that pure AS particles have a deliquescence RH (DRH) of 79.5 % and an efflorescence RH (ERH) of ∼ 35 % . The DRH is the equilibrium point below which solid AS is the thermodynamically stable phase and this corresponds to the solubility limit of AS in water. However, depending on the RH history of the particle, a metastable supersaturated solution may instead be present below the DRH. Below the ERH, only solid AS is present in the particulate phase.

The particulate fraction fp,s of the organic solute depends on the amount of solvent (water + AS) per volume of air. A fixed AS mass concentration of 4 µgm-3 was chosen, typical for inorganic aerosols at mid-latitudes over continents (http://vista.cira.colostate.edu/improve/). As a consequence, upon increasing RH from the ERH to 90 %, the LWC varies between 10-6 and 10-5 gm-3, a typical range for aqueous aerosol.

Activity coefficient methods that best fit the data of . For seven compounds, an alternative fitting is presented.

Molecule Fitting method

A12,A21

ln⁡γs∞

SDln⁡γw

SD aw

Δsyst

σrand

1,2-ethane diol Margules -0.13, -0.38 -0.38 0.0094 0.0033

±0.02

0.07 1,2-propane diol Margules 0.038,0.078 0.078 0.0098 0.0053

±0.03

0.05 1,3-propane diol UNIQUAC -144.83, 207.56 0.22 0.0064 0.0033

±0.02

0.06 1,2-butane diol Van Laar 0.40, 1.32 1.32 0.0085 0.0039

±0.04

0.08 UNIQUAC -75.45, 217.72 1.43 0.0087 0.0044 1,3-butane diol Wilson 1.84, 0.22 0.68 0.0101 0.0053

±0.05

0.13 Margules 0.12, 0.48 0.48 0.0107 0.0053 1,4-butane diol Wilson 2.00, 0.18 0.75 0.0083 0.0035

±0.04

0.11 UNIQUAC -103.07, 199.15 0.86 0.0090 0.0045 2,3-butane diol Margules 0.26, 0.57 0.57 0.0061 0.0030

±0.03

0.03 1,5-pentane diol Van Laar 0.37, 1.66 1.66 0.0126 0.0046

±0.02

0.14 UNIQUAC -60.77, 201.91 1.73 0.0127 0.0050 1,2-pentane diol UNIQUAC 9.40, 152.56 2.48 0.0125 0.0053

±0.03

0.08 1,4-pentane diol Wilson 1.70, 0.13 1.33 0.0107 0.0053

±0.06

0.098 Van Laar 0.35, 1.29 1.29 0.0108 0.0051 2,4-pentane diol Margules 0.20, 1.04 1.04 0.0111 0.0073

±0.03

0.02 1,2-hexane diol Van Laar 0.63, 3.27 3.27 0.0041 0.0033

±0.05

0.09 UNIQUAC 13.44, 185.39 3.19 0.0063 0.0050 2,5-hexane diol Van Laar 0.37, 1.74 1.74 0.0153 0.0080

±0.06

0.10 Wilson 1.75, 0.07 1.90 0.0158 0.0089 1,7-heptane diol UNIQUAC -4.99, 206.56 3.34 0.0158 0.0072

±0.04

0.06

aOptimized parameters to use in the activity coefficient expression for ln⁡γw. b Standard deviation between modelled and observed ln⁡γw. c Standard deviation between modelled and observed aw. d Range for ln⁡γs∞, obtained by applying systematic shifts of ±0.005 to the aw data. e Standard deviation on ln⁡γs∞, obtained by applying random shifts from a normal distribution with σ=0.0075 to the aw data.

(a) Activity coefficient correction (log⁡10γs*) of the organic solute vs. RH for an AS–water particle, as calculated by AIOMFAC. (b) Particulate fraction of the organic solute vs. RH. Acid dissociation is not taken into account. Tartaric and citric acid have fp=1 over the entire presented range. (c), as (b), but taking acid dissociation into account using the on-line model of E-AIM (see text for more details). Oxalic, malonic, tartaric and citric acid have fp=1 over the entire presented range.

Polyols. Due to their low kh, diols do not partition significantly to aqueous aerosol and hence are not included in this analysis. Stereo isomers of xylitol and sorbitol were also not included, given their similar kh and the fact that AIOMFAC does not distinguish between stereo isomers. For the polyols, AIOMFAC predicts an activity increase with lowering RH (or equivalently increasing the salt concentration) (Fig. a). The effect increases with the number of hydroxyl groups. However, this is more than compensated by the concomitant increase in kh(xAS=0) (Table ). The particulate fraction of polyols decreases with decreasing RH both due to the increase in solute activity, and the decrease of total absorbing mass. At RH = 90 % glycerol, with three hydroxyl groups, is 95 % in the gas phase while sorbitol, with six hydroxyl groups, is still 50 % in the particulate phase at RH = 44 %. This is due to the large difference (8 orders of magnitude) of their kh values.

Linear diacids. Diacids with two (oxalic) up to 10 (sebacic) carbon atoms are considered. Let us neglect acid dissociation for the moment. AIOMFAC's interaction parameters are negative (stabilizing) between the carboxylic acid group COOH on the one hand, and the ions NH4+ and SO42- on the other hand. For the group CH2, these interaction parameters are positive (destabilizing). As a consequence, the activity of the linear diacids with four carbon atoms or more increases with decreasing RH. The activity of oxalic acid, on the other hand, decreases with decreasing RH, while the activity of malonic acid stays roughly constant. Even without taking acid dissociation into account, it is clear that these diacids partition appreciably to the particulate phase (Fig. b). Note that for malonic acid, we chose the lower of the recommended kh values from Table 3 of ; the higher value would lead to fp near unity even without acid dissociation.

Hydroxy polyacids. Citric and tartaric acid exhibit a modest activity increase upon decrease of the RH. On the other hand, they have extremely high kh(xAS=0) values . Therefore, they will reside almost completely in the particulate phase from RH = 100 % to the ERH.

Impact of acid dissociation. The effective HLC, khpx,eff, of a polyacid is larger than khpx due to acid dissociation. For a diacid one has khpx,eff=lim⁡xH2A,xHA-,xA2-,pH2A→0xH2A+xHA-+xA2-pH2A with xH2A,xHA- and xA2- the mole fractions of the undissociated acid, monodissociated acid and twice dissociated acid respectively. Acid dissociation is governed by the acid dissociation constants Ka,1=γH+*xH+γHA-*xHA-γH2A*xH2AKa,2=γH+*xH+γA2-*xA2-γHA-*xHA- with Ka,i mole fraction based acid dissociation constants, and γ* mole fraction based activity coefficients, with the asymmetric convention (i.e. becoming unity at infinite dilution in pure water). Combining Eqs. () and () leads to khpx,eff=khpx1+Ka,1γH2A*γHA-*1γH+*xH++Ka,1Ka,2γH2A*γA2-*1γH+*xH+2. AIOMFAC does not calculate activity coefficients of ionized organic acids. To describe the ionization in the water–AS–diacid system, we used the models provided at the site of E-AIM (http://www.aim.env.uea.ac.uk/aim/aim.php). Specifically, model IV was used, which is an implementation of the parameterizations of and (for the diacids) of . Solid formation was prevented, and the dissociation equilibria H2O/OH-, NH4+/NH3, HSO4-/SO42- were taken into account. At the vanishingly small acid concentration used, E-AIM calculates the same γH2A*,γHA-* and γA2-* regardless of the identity of the diacid. This is not realistic; one expects a larger γ* value for a diacid with more CH2 groups. Therefore, γH2A*, equal to γs* in Eq. (), is still calculated by AIOMFAC, to take into account the destabilizing CH2–ion interaction. The γH+*xH+, γH2A*γHA-* and γH2A*γA2-* in Eq. () are determined by the E-AIM calculation. Note that due to the vanishingly small acid concentration, γH+*xH+ is determined by the amounts of water and AS only. Acid dissociation constants were taken from E-AIM or . Oxalic and malonic acid are predicted to be completely in the aqueous phase from RH = 100 % to the ERH, while the particulate fraction of the other diacids are clearly enhanced (Fig. c), compared to the calculation without acid dissociation (Fig. b).

Discussion and conclusions Assessing main uncertainties

HLC of polyols with 2–6 hydroxy groups are derived in this work, using experimental data and thermodynamic relationships. This study complements a previous work where the focus was on diacids and hydroxy polyacids.

An error analysis is performed in Appendix . The compounds that are liquid at room temperature (most diols and glycerol) have a relatively low uncertainty in kh (relative standard error 6–28 %, see Table ). For some compounds, further improvement is possible with more precise pL0 data and/or more fine-grained and precise aw data in the dilute region.

Estimated relative standard errors on pL0, γs∞ and the resulting relative standard error on kh. See Sects. and .

Molecule

SEpL0pL0

SEγs∞γs∞

SEkhkh

1,2-ethane diol 0.03 0.05 0.06 1,2-propane diol 0.03 0.05 0.06 1,3-propane diol 0.03 0.05 0.06 1,2-butane diol 0.03 0.05 0.06 1,3-butane diol 0.03 0.05 0.06 1,4-butane diol 0.03 0.05 0.06 2,3-butane diol 0.25 0.05 0.25 1,2-pentane diol 0.03 0.13 0.13 1,4-pentane diol 0.25 0.13 0.28 1,5-pentane diol 0.25 0.05 0.25 2,4-pentane diol 0.03 0.13 0.13 1,2-hexane diol 0.03 0.13 0.13 2,5-hexane diol 0.25 0.13 0.28 1,7-heptane diol 0.25 0.13 0.28 glycerol 0.03 0.05 0.06

Estimated relative standard errors on xssat, γs∞γssat and the various parts of pCr0, and the derived relative standard error on kh. See Sects. , and .

SExssatxssat

SEγs∞γssatγs∞γssat

TfusK

SEC

TmeasK

SEA

SEB

SEkhkh

1,9-nonane diol 0.16 0 318.7g 0.13 323b 0.27 0.01 0.34 360c 0.35 0.06 0.41 1,10-decane diol 0.16 0 345.8g 0.18 342b 0.31 0.03 0.39 364c 0.36 0.07 0.44 erythritol 0.05 0 390.25h 0.23 412d 0.46 0.2 0.55

386e 0.41 0.1 0.42 pentaerythritol 0.05 0

436d 0.51 0.28 0.58 404e 0.45 0.17 0.48 394f 0.42 0.14 0.44 xylitol 0.05 0.05 369.04h 0.19 433d 0.51 0.27 0.61 adonitol 0.05 0.16 369.08h 0.20 443d 0.52 0.31 0.66 arabinitol 0.05 0.16 379.4 d 0.21 440d 0.52 0.29 0.65 sorbitol 0.05 0.05 366.5h 0.18 477d 0.59 0.44 0.76 mannitol 0.05 0 437.25h 0.30 477d 0.59 0.44 0.80 dulcitol 0.05 0 460.3 d 0.33 482d 0.59 0.46 0.82

a Fusion point and SEC not relevant here as Tfus>Tmeas. b-h The same references as for Table apply.

The compounds that are solid at room temperature, especially the polyols with four or more hydroxy groups, bear a much larger uncertainty in kh (relative standard error 34–82 %, see Table ). This is mainly due to the use of high-temperature liquid or solid state vapour pressures. More specifically, it is due to the uncertainty in ΔHvap or ΔHsub in combination with the extrapolation over a large temperature interval. For the hexols, also the uncertainty in heat capacity becomes important, although we note that the error in Cp,g is speculative as this property is estimated. Measuring the (solid or liquid state) vapour pressure closer to room temperature will lower these uncertainties.

As noted above, the Cp,g values are estimated. Improvement here is possible by using Cp,g derived from experiment or from ab initio calculations rather than using a group contribution method. For nonane diol and decane diol, only solubilities from a secondary reference (Merck Millipore) could be retrieved, for which it is difficult to estimate the reliability. New solubility measurements are desirable to obtain a more reliable kh estimate.

Comparison with literature

HLC compilations of polyols are provided by e.g. and . However, most values in these studies are estimated. provide HLC measurements for 1,2-ethane diol and 1,3-propane diol (Table ). Their values are lower but reasonably close (within a factor of 2) to ours. While the majority of HLC values of polyols provided by are estimated, a few are derived from vapour–liquid equilibrium data. For 1,2-propane diol, 2,3-butane diol and glycerol, their HLC values are within a factor 3, but for 1,4-butane diol the difference is more than an order of magnitude. In conclusion, for five out of six HLC values, we have a reasonable agreement with the literature values.

The estimated values presented by are obtained by a group-contribution method (values not reproduced in Table ). For the diols, overestimations by ∼ 1 order of magnitude compared to our values are common. For the compounds with three or more hydroxyl groups, the overestimation ranges between 3 (glycerol) and 8 (mannitol) orders of magnitude, showing the limitations of such an estimation method.

pL0 calculations for erythritol, xylitol and sorbitol, with three vapour pressure estimation methods available at http://www.aim.env.uea.ac.uk/aim/aim.php.

method erythritol xylitol sorbitol Nannoolal Tb/K adj.a 612 634 674

pL0(Tref)/Pa

2.2×10-5

3.0×10-7

2.5×10-10

Moller Tb/K adj.a 649 705 787

pL0(Tref)/Pa

1.5×10-5

2.6×10-7

1.8×10-10

Myrdal–Yalkowsky Tb/K adj.a 656 703 772

pL0(Tref)/Pa

9.7×10-5

5.2×10-6

5.9×10-8

a Hypothetical boiling point, adjusted such that the experimental pL0 at Tmeas is reproduced. These experimental (Tmeas/K,pL0/Pa) data points are (412,8.6),(433,5.5) and (477,6.8) for erythritol, xylitol and sorbitol respectively.

Atmospheric implications

According to the HLC derived in this and the previous work , diols will be partially (e.g. 1,2 hexane diol, depending on the droplet size) or completely (e.g. 1,4-butane diol) in the aqueous phase in clouds, while polyols with three or more hydroxyl groups, diacids and hydroxy polyacids will be completely in the aqueous phase. Regarding aqueous aerosol, the sensitivity test performed here using aqueous AS aerosol indicates that polyols with four or more hydroxyl groups are significantly or totally in the particulate phase, depending on the RH. The same holds for the longer linear diacids (succinic and higher). The shorter linear diacids (oxalic and malonic) and the hydroxy polyacids (citric and tartaric) are completely in the particulate phase both at lower and higher RH, due to (i) their relatively high acid dissociation constants and/or (ii) stabilizing or only mildly destabilizing interactions with AS and/or (iii) very high kh values.

Note that this analysis is only applicable for aqueous AS aerosol in the limiting case of a small concentration of organics. If e.g. a separate organic phase is present in the aerosol, partitioning to this phase should be taken into account as well.

measured gas particle partitioning of diacids at a site in Japan in different seasons. According to this study, both particulate and gaseous fractions are significant, and RH influences the partitioning. measured gas particle partitioning of 2-methyl tetrols at a site in Denver and found about equal particulate and gaseous fractions. Our sensitivity test, based on a simple AS–water aerosol system, cannot be quantitatively compared with these studies, but does show that partitioning to the particulate phase is important for diacids and tetrols.

Error analysis

In this section we identify the main uncertainties contributing to the kh values, as well as the overall uncertainty. From the error propagation rule , and Eqs. () and (), the standard error (SE) on kh can be derived: SEkhkh≈SEpL0pL02+SEγs∞γs∞2SEkhkh≈SEpCr0pCr02+SExssatxssat2+SEγs∞γssatγs∞γssat2. In Eq. (), the covariance between xssat and γs∞/γssat is neglected. Such covariance exists in principle, as xssat enters the formula for deriving γs∞/γssat (Eq. ). However, due to the low error on xssat (see Sect. ), this can be neglected. Note that it also follows from the error propagation rule that SEuu=SEln⁡u.

We tried to obtain the uncertainties from the original studies. This is hindered by the fact that these errors are not always reported, or it is not always made clear what they exactly represent (e.g. once or twice the standard deviation). Discrepancies between results of different research groups are often larger than the reported errors of individual studies. Our error analysis is mostly based on this inter-laboratory error.

From Eqs. () and (), it is clear that relative standard errors (i.e. SEu/u) are relevant. They are cited in the text as percentages. To estimate how much data of two data sources 1 and 2 disagree, we calculated the root mean squared relative difference (RMSRD) RMSRD=1N∑i=1N2u1,i-u2,iu1,i+u2,i2 with u representing a physical quantity and i running over N data points (obtained by varying e.g. the temperature, the water content, or the molecule type). If we can consider the data of source 1 as more reliable than of source 2, the RMSRD is used to assign a relative standard error to source 2. Otherwise the RMSRD is assigned as relative standard error to both data sources.

Sometimes the absolute error is more relevant. To quantify the difference between two data sources, we use therefore the mean deviation (MD) and the root mean squared difference (RMSD): MD=1N∑i=1Nu1,i-u2,iRMSD=1N∑i=1Nu1,i-u2,i2. An example where the absolute error is more relevant is for water activity data aw in function of water content, which is used as input to calculate γs∞ and γs∞/γssat (Sects. and ). Of course aw of both data sources was evaluated at the same water content, with interpolation of data points if necessary.

Liquid vapour pressure

provide the bulk of pL0 data for diols used in Table (eight compounds in total). They report that their measurements are “reliable within 1–3 %” and point to a good consistency with other literature data. An additional advantage is that the measurements are performed at or near room temperature. We consider the data of this group the most reliable for diols – based on our experience with other vapour pressure data of this group, e.g. on aldehydes and mono-alcohols – and will use the data as a benchmark for other pL0 data of diols. Regarding the triol glycerol, reports an accuracy of “somewhat less” than 0.6 % and the authors state consistency with other work. Their measurements also include room temperature. We consider this work as reliable as well. We assign to all pL0 data with label “a” and “d” in Table a relative standard error of 2 %.

The pL0 data of four diols used in Table are from the compilation . Comparing pL0 data of a set of eight diols with measurements of Verevkin and co-workers (which we consider more reliable), we find a RMSRD of 25 %. This is therefore assigned as relative standard error to all diols in Table with label “b”.

For 1,7-heptane diol, we have two data sources: and . Comparing room-temperature data of six diols from Knauth and Sabbah on the one hand and from Verevkin and co-workers on the other hand , we obtain again a RMSRD of 25 % and assign this as the relative standard error to pL0. The data of could not be directly compared with those of Verevkin and co-workers because the two groups did not measure the same diols. We assign the same relative standard error of 25 % to their pL0 data due to lack of alternative.

Infinite dilution activity coefficient

The infinite dilution activity coefficient is calculated from aw data using Eq. (). If these data were sufficiently fine-grained, precise and cover the entire xw range from 0 to 1, a numerical integration of the integral in Eq. () would be appropriate. However, if this is not the case, it might be better to fit the aw data with a reasonable model. The following models for ln⁡γw were considered in this work: Margules, Van Laar, Wilson and UNIQUAC , as they were derived from physical considerations. These models are reproduced below: Margules:fx1,A12,A21=A12+2(A21-A12)x1x22Van Laar:fx1,A12,A21=A12A21x2A12x1+A21x22Wilson:fx1,A12,A21=-ln⁡x1+A12x2+x2A12x1+A12x2-A21A21x1+x2UNIQUAC:fx1,A12,A21=ln⁡Φ1x1+z2q1ln⁡θ1Φ1+Φ2l1-r1r2l2-q1ln⁡θ1+θ2e-A21/T+θ2e-A21/Tθ1+θ2e-A21/T-e-A12/Tθ2+θ1e-A12/T with x1 the mole fraction of the first component (here water), x2=1-x1 the mole fraction of the second component (here the solute), and A12,A21 the parameters to fit. To determine A12,A21, a least square regression was performed, minimizing the difference between experimental and modelled ln⁡γw. For a detailed overview of the quantities used in the UNIQUAC equation, see . Errors in γs∞ arise from uncertainty and sparsity in the aw data, but also from a suboptimal model choice. This is especially important if no data in the highly dilute concentration range are available.

All the γs∞ derived in this work are based – partially or totally – on the aw from . This work presents aw data, relatively coarse-grained, with a xw range typically between 0.1 and 0.95. The main limitation of this data source is that no data in the highly dilute range are available. We present here first an error analysis for all the 14 diols from Table , but using the data of only. The four activity coefficient expressions were all tested, and the one that fitted the data best was chosen. The results are presented in Table .

report that the aw data are “accurate within ±0.015”, as specified by the employed instrument, but this does not make clear whether the error is random and/or systematic, or whether this range corresponds to e.g. one or two standard deviations. If we assume that the selected fitting model is correct and any systematic error in aw is absorbed in the fitting parameters, then any remaining deviation between modelled and experimental values should be due to the random error in the experiment . Standard deviations between modelled and observed aw vary between 0.003 and 0.009 (Table ). We drew repeatedly random errors from a normal distribution with a standard deviation of 0.0075 (corresponding to 2σ=0.015), using the python scipy.stats module and applied these to the aw data. This resulted in a distribution of γs∞ values. A relative standard error between 2 % and 14 % on γs∞ is obtained in this way. We note however, that this error depends also on the selected model, i.e. γs∞ based on Margules models tend to be the least sensitive to the random shifts, while γs∞ based on Van Laar and Wilson models are the most sensitive.

Systematic errors in aw can be estimated by comparing with other data sources . The aw data of are typically higher than from the other data sources, with a MD of 0.005. We applied systematic shifts of ±0.005 on the aw data of . This had a minor impact on γs∞, between 2 and 6 %.

We note that we had expected that the largest γs∞ (from 1,7-heptane diol and 1,2-hexane diol) would exhibit the largest sensitivity on these random and systematic shifts. This turned out not to be true, however.

Errors due to a suboptimal model choice are more difficult to quantify. Per compound, the other three activity coefficient models (apart from the “best” one) were also considered. If the fitting was considerably worse, the model was rejected because it was probably not appropriate. If the fitting gave essentially the same γs∞ result, it was also not retained, because in that particular case the models were not truly different. In this way, for seven diols γs∞ from an alternative model were selected (presented in Table ). The RMSRD between γs∞ of the “best” model and “alternative” model was 12 %. The most important reason for the discrepancy between the activity coefficient models was their behaviour at the highly dilute region, where no data were used to constrain them.

For eight diols in Table , also data from two other sources were available. Note that do not provide the data points but only fitted expressions. These two data sources are very precise; e.g. report a relative uncertainty in aw of 0.005 % or less. They are also in good agreement with each other; the difference in aw is typically 0.001 or less. These data cover the highly dilute region (xw typically in the range between 0.93 and 0.996) and therefore enable us to largely correct for the error due to a suboptimal model choice. Using these data as well, more accurate estimations of γs∞ were made (Table ). Comparing those with the γs∞ derived only from the data, a RMSRD of 13 % was obtained, in good agreement with our above estimate of the error due to a suboptimal model choice. Therefore, we assign to all γs∞ where only data from are used, a standard error of 13 %.

Practically, the derivation of γs∞ when multiple data sources are available was done in the following way. The integral in Eq. () was split into two parts: ln⁡γs∞=∫t=0x̃wln⁡γw(xw)1-xw2dxw+∫t=x̃w1ln⁡γw(xw)1-xw2dxw where x̃w marks the start of the highly dilute region for which data of and are available. For the region [0,x̃w] we employed for ln⁡γw the corresponding activity coefficient expression from Table . For the region [x̃w,1], the data from and were integrated. This was attained either by a numeric integration, or by fitting an activity coefficient expression over this small region, or – if only data of were available – the analytic expression given in this study was used. We assign to these more accurate estimations of γs∞ a small standard error of 5 %.

For the two triols in Table , the data of and of and were combined. There is a good agreement between the data; e.g. report that for glycerol, there is only 0.1 % relative deviation between their aw data and those of . Also here we assign a small standard error of 5 % on γs∞.

Solubility

As some solubility data of the polyols are from literature compilations, uncertainties are not always available. But when reported, they are typically very low. For xylitol, reported an error of no more than 0.5 % on xssat. For pentaerythritol, a 0.3 % error can be derived from the study of and 0.15 % from the study of . Higher errors are found when comparing different studies. For erythritol, report a 4 % uncertainty by comparing with literature data. For pentaerythritol, we found by comparing data of , and that the RMSRD on room temperature xssat is 5 %. Compared to the estimated errors on solid state pressure (see Sect. ), these errors are very minor. We assign a standard error of 5 % to all solubility values, except for those of 1,9-nonane diol and 1,10-decane diol. For these latter two compounds, there are two issues. First, the data are from a secondary reference (Merck Millipore; http://www.merckmillipore.com/) from which it is difficult to assess the reliability. Second, the solubility is not reported at the reference temperature of 298.15 K but at 293.15 K instead. The temperature dependence of solubility can be described by a van 't Hoff relation : dln⁡xssatd1/T=-ΔHCr→aqvHR. The ΔHCr→aqvH is not known for these two diols. For mono-alcohols it can be either positive or negative and is (in absolute value) in the range 0–30 kJ mol-1 based on data from. Based on this, we estimate roughly the standard error on xssat at 298.15 K as 15 %.

Activity coefficient ratio

The activity coefficient ratio γs∞/γssat is calculated from aw data. The error on γs∞/γssat depends on the solubility. If the solubility is very low, γs∞/γssat will necessarily be very close to unity and the uncertainty on γs∞/γssat will vanish (see Eq. ). For all compounds with xssat<0.1 we neglect the error. Xylitol, adonitol, arabinitol and sorbitol have relatively high solubilities (xssat≈0.2) and γs∞/γssat considerably deviating from 1 (Table ). For xylitol and sorbitol, precise aw data are available ; e.g. report a reproducibility of ±0.001 aw units. There is a good agreement between the data sources. For sorbitol, there is a near perfect agreement between the data sets of and (no significant MD, and a RMSD of 0.0005). The RMSD between the aw data of on the one hand, and the data of and on the other hand, for both xylitol and sorbitol, is 0.003. This is similar to the standard deviation between modelled aw and experimental aw (0.0015 for sorbitol, 0.003 for xylitol). Therefore, we applied random shifts from a normal distribution with a standard deviation of 0.003 to the aw data. This resulted in a 5 % uncertainty on γs∞/γssat for sorbitol and a 2.5 % uncertainty for xylitol. We adopted 5 % as a relative standard error for the γs∞/γssat of xylitol and sorbitol (Table ).

presented one-parameter fittings of the form ln⁡γw=exp⁡-Axs, from which γ∞/γsat can be derived. Using Eq. (), one has ln⁡γs∞γssat=-Axssat2-xssat. For adonitol and arabinitol, these are the only data available. γs∞/γssat values derived in this way are likely more uncertain than the values given above for sorbitol and xylitol. For these two compounds, we compared our own γs∞/γssat values (in Table ) with the values obtained using the one-parameter fittings of and found a RMSRD of 16 %. Therefore, we assign a relative standard error of 16 % to the γs∞/γssat of adonitol and arabinitol (Table ).

Derivation of sublimation data

Starting from high-temperature solid state pressure data, and heat capacity data, the solid state pressure pCr0 at Tref=298.15 K is given by ln⁡pCr0Tref=A+B, withA=ln⁡pCr0Tmeas-1R1Tref-1TmeasΔHsubTmeasB=1R∫TmeasTrefCp,g-Cp,Cr1T-1TrefdT. In the case that the high-temperature data corresponds to liquid vapour pressure pL0, the relation is ln⁡pCr0Tref=A+B+C, with A=ln⁡pL0Tmeas-1R1Tref-1TmeasΔHvapTmeasB=1R∫TmeasTfusCp,g-Cp,L1T-1TrefdT+1R∫TfusTrefCp,g-Cp,Cr1T-1TrefdTC=-1R1Tref-1TfusΔHfusTfus. Uncertainties on terms A (involving high temperature vapour pressure data), B (involving heat capacity data) and, depending on the case, C (involving fusion data) will contribute to the error on ln⁡pCr0Tref: SEpCr0TrefpCr0Tref=SEln⁡pCr0Tref=SEA2+SEB2+SEC2. In the next sections these three uncertainties are analysed.

High-temperature <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mtext>Cr</mml:mtext><mml:mn mathvariant="normal">0</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mtext>L</mml:mtext><mml:mn mathvariant="normal">0</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> data

To simplify the discussion, in this section, p0,ΔS and ΔH stand for either pCr0,ΔSsub and ΔHsub or pL0,ΔSvap and ΔHvap respectively, depending on whether the compound is a solid or a liquid at Tmeas. Term A is then given by A=ln⁡p0Tmeas-1R1Tref-1TmeasΔHTmeas. The standard error of A, from the error propagation rule (neglecting the uncertainty in Tmeas), is equal to

SE(A)=SEp0Tmeasp0Tmeas2+SEΔHTmeasR1Tref-1Tmeas2. In the study of , p0Tmeas and ΔHTmeas are measured separately and at one fixed temperature. In the other studies, however , ΔH was derived from a linear regression of ln⁡p0 vs. 1/T. Over the relatively small temperature interval, the temperature dependence of ΔH can be neglected Clausius–Clapeyron relation, see e.g.. Eq. () still applies, if pL0 is taken at the centre of the measurement interval . Tmeas then corresponds to the centre of the temperature interval (see Table ) and is presented in Table . Most high-temperature p0 data are from Piacente and co-workers, covering both diols and polyols with four or more hydroxyl groups .

As before (see Sect. ), we estimate SEp0Tmeasp0Tmeas as 0.25 for the data of and . By comparing ΔH data on diols from and , we find a RMSD of 3.5 kJ mol-1, which we adopt as SEΔHTmeas for both data sources. As is often the case when comparing different data sources, this error is higher than the errors reported in the individual studies themselves (typically 1–2 kJ mol-1). For large differences between Tref and Tmeas, SE(A) is determined mainly by uncertainty in the enthalpy term. This is the case for the polyols with four or more hydroxyl groups. The largest SE(A), about 0.6, is obtained for the hexols (Table ).

Most data on polyols with four or more hydroxyl groups is from Piacente and co-workers . There are a few other, much older studies on erythritol and pentaerythritol, with ΔH values strongly deviating (15–30 kJ mol-1 lower) from the data of . These large discrepancies cannot be understood from the reported precisions, or from the error of 3.5 kJ mol-1 adopted above. We think that this points to problems with one or more of the experimental studies, rather than to a typical experimental uncertainty. We judge the data of Piacente and co-workers as the more reliable, for the following reasons. (i) A relatively good agreement in ΔH of diols with . (ii) Piacente and co-workers also present pL0 measurements on diols using another technique, and with comparable results . (iii) Consistency, in the sense that the high-temperature pL0 data of stereo-isomers of polyols are comparable, as one would expect . (iv) Compared to the older studies, those of Piacente and co-workers employ more data points and a larger temperature interval. Obviously, more p0 measurements on polyols with four or more hydroxyl groups are desirable.

Heat capacity data

Assuming that the error on heat capacity is temperature independent, one derives from Eq. () for the standard error on term B (if no fusion point is involved) SEB=1Rln⁡TrefTmeas-1TrefTref-TmeasSECp,g2+SECp,Cr2. If there is a fusion point, starting from Eq. (), the standard error on term B becomes SEB=1Rln⁡TfusTmeas-1TrefTfus-TmeasSECp,g2+SECp,L2+1Rln⁡TrefTfus-1TrefTref-TfusSECp,g2+SECp,Cr2. Most Cp,L and Cp,Cr data we use is from Tong and co-workers , as it covers a wide range of polyols and temperatures. Quoted errors on the experimental values of Cp,L and Cp,Cr are very low, e.g. 0.2 % for those of Tong and co-workers. This corresponds to about 0.5 J/(Kmol) error on the heat capacity value. However, the discrepancy between data of different groups is much larger. We compared the data of Tong and co-workers with other data where possible: for erythritol , sorbitol and mannitol . Based on the RMSD of the data, we assign SECp,L=SECp,Cr=20 J/(Kmol), much larger than the reported errors. By taking identical errors for Cp,L and Cp,Cr (there are not enough data to treat them separately), Eq. () simplifies to Eq. ().

In a few cases (arabinitol, dulcitol, liquid state mannitol) we did not found heat capacity data in the appropriate temperature range and we took heat capacity data of a stereo-isomer instead (see Table ). To test the validity of this assumption, we compared heat capacity data at or above room temperature of stereo-isomers from the same research group. Regarding the liquid state, Cp,L data of the stereo-isomers erythritol and threitol , and of the stereo-isomers xylitol and adonitol differ by less than 2 J/(Kmol). This justifies the approximations Cp,Larabinitol≈Cp,Lxylitol, Cp,Ldulcitol≈Cp,Lsorbitol and Cp,Lmannitol≈Cp,Lsorbitol applied in this work. Regarding the solid state, we compared Cp,Cr of stereo-isomers erythritol and threitol , xylitol and adonitol , mannitol and sorbitol , only taking into account the data at T≥Tref (as this is the most relevant for our work) and found a RMSD of 9 J/(Kmol). This is larger than for the liquid state but still smaller than the discrepancies between different works on the same molecule. Moreover, for the low-temperature (88–291 K) Cp,Cr data for the stereo-isomers dulcitol and mannitol , the RMSD was only 2 J/(Kmol). Therefore, also the approximations Cp,Crarabinitol≈Cp,Crxylitol and Cp,Crdulcitol≈Cp,Crmannitol seem to be justified.

Regarding Cp,g, the error is difficult to quantify as this value is not measured but obtained by the Benson group contribution method. quote a relative error of 1 % for this method, but as polyols are under-represented in the fitting set of this method, this is probably not realistic. The method of is a very close relative to the Benson group contribution method (although limited to room temperature), yielding very similar results for Cp,g. Importantly, this method also estimates Cp,L and Cp,Cr. Comparing Cp,Cr for erythritol, xylitol and sorbitol from Tong and co-workers with estimations by the method of yields a RMSD of 20 J/(Kmol). We make now the assumption that this error is applicable to Cp,g as well. Hence we assign SECp,g=20 J/(Kmol).

For the diols, where Tmeas is relatively close to Tref, SEB is small (1–7 %), but for the compounds with four or more hydroxyl groups it becomes more important – over 40 % for the hexols (see Table ).

Fusion data

In this section the error due to the term C=-1R1Tref-1TfusΔHfusTfus in Eq. () is estimated. From the error propagation rule the standard error can be derived as SEC=SETfus2ΔHfusTfus2R2+SEΔHfus21R1Tref-1Tfus2. Fusion data were collected from , , and . The selected Tfus are presented in Table . Reported errors in these works are typically 0.1 K for Tfus and 0.1–0.3 kJ mol-1 for ΔHfus. However, comparing the data between the different references reveals larger uncertainties: a standard deviation of 2.8 K for Tfus and 2.2 kJ mol-1 for ΔHfus. Using these errors, SE(C) ranges from 0.13 for nonane diol to 0.33 for dulcitol (see Table ).

Overall discussion of errors

Table gives an overview of the estimated relative standard errors on pL0, γs∞ and the derived relative standard error on kh. This applies to the compounds which are liquid at room temperature, i.e. most diols discussed in this work and glycerol. The relative standard error on kh ranges between 6 % and 30 %.

Table gives the different error contributions when the compound is solid at room temperature, and the resulting relative standard error on kh. In all cases, SEA is the dominant error contribution. For all polyols with four or more hydroxyl groups, this is caused by the large difference between Tmeas and Tref and the uncertainty on ΔHfus. The estimated relative standard error on kh ranges between 30 and 80 %.

Alternative method to estimate <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mtext>Cr</mml:mtext><mml:mn mathvariant="normal">0</mml:mn></mml:msubsup><mml:mo>(</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mtext>ref</mml:mtext></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> <?xmltex \hack{\\}?>data from high-temperature <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mtext>L</mml:mtext><mml:mn mathvariant="normal">0</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> data

As is clear from Sect. , using high-temperature p0 data at Tmeas far above Tref contributes the largest uncertainty to the derived room temperature pCr0 and kh data. One of the reviewers suggested an alternative approach. It can be described as follows:

Select a vapour pressure estimation method that uses a boiling point as input e.g..

Adjust the boiling point such that the experimental high-temperature pL0 data at Tmeas are reproduced.

With this setting, estimate the subcooled pL0 at Tref.

Use triple point or fusion point data to calculate pCr0 at Tref Eq. (1).

We applied this procedure up to step 3. The selected vapour pressure estimation methods are those of , and , available at the site E-AIM (http://www.aim.env.uea.ac.uk/aim/aim.php), and applied to the compounds erythritol, xylitol and sorbitol. The boiling point was adjusted until the pL0 data at Tmeas of could be reproduced, and pL0 at Tref calculated. The results are presented in Table . The method of gives much higher pL0(Tref) estimations than the other two methods; for sorbitol it is more than two orders of magnitude higher. The pL0(Tref) estimations using the methods of and of are within a factor of 2. However, the adjusted boiling point varies strongly between both methods (almost 100 K for sorbitol), indicating also problems here.

From the theses describing these two methods it can be deduced that to derive these methods, only compounds were selected where also a normal boiling point is available. This excludes the polyols with four or more hydroxyl groups. Therefore, these methods may not be well suited to calculate the vapour pressure of these kind of compounds. In fact, the original version of the method of contained a bug that we pointed out , showing up only for highly polyfunctional compounds, and giving very unrealistic pL0 values. While this bug has been corrected since then, it does indicate that the method was not devised for highly polyfunctional compounds.

In conclusion, we do not think that this approach is a good alternative to obtain pCr0 at Tref.

Acknowledgements

This work was supported by the project BIOSOA (SD/CS/05A, 2011–2014) funded by the Belgian Science Policy Office. Edited by: D. Topping

References Apelblat(2002)

Apelblat, A.: Dissociation constants and limiting conductances of organic acids in water, J. Mol. Liq., 95, 99–145, 2002.

Atkins and de Paula(2006)

Atkins, P. and de Paula, J.: Physical Chemistry, Oxford University Pres, 2006.

Badea et al.(2014)Badea, Nowicka, and Della Gatta

Badea, E., Nowicka, B., and Della Gatta, G.: Thermodynamics of fusion and sublimation for a homologous series of eleven alkane-α, ω-diols HO-(CH2)n-OH: structure-related odd-even effect, J. Chem. Thermodyn., 68, 90–97, 2014.

Bao et al.(2012)Bao, Matsumoto, Kubota, Sekiguchi, Wang, and Sakamoto

Bao, L., Matsumoto, M., Kubota, T., Sekiguchi, K., Wang, Q., and Sakamoto, K.: Gas/particle partitioning of low-molecular-weight dicarboxylic acids at a suburban site in Saitama, Japan, Atmos. Environ., 47, 546–553, 2012.

Barone et al.(1990)Barone, Gatta, Ferro, and Piacente

Barone, G., Gatta, G. D., Ferro, D., and Piacente, V.: Enthalpies and entropies of sublimation, vaporization and fusion of nine polyhydric alcohols, J. Chem. Soc. Faraday T., 86, 75–79, 1990.

Bastos et al.(1988)Bastos, Nilsson, Ribeiro da Silva, Ribeiro da Silva, and Wadsö

Bastos, M., Nilsson, S.-O., Ribeiro da Silva, M. D., Ribeiro da Silva, M. A., and Wadsö, I.: Thermodynamic properties of glycerol enthalpies of combustion and vaporization and the heat capacity at 298.15 K. Enthalpies of solution in water at 288.15, 298.15, and 308.15 K, J. Chem. Thermodyn., 20, 1353–1359, 1988.

Benson(1976)

Benson, S. W.: Thermochemical Kinetics: Methods for the Estimation of Thermochemical Data and Rate Parameters, John Wiley & Sons, New York, 1976.

Bevington(2003)

Bevington, P. R.: Data reduction and error analysis for the physical sciences, New York: McGraw-Hill, third edn., 2003.

Bone et al.(1983)Bone, Cullis, and Wolfenden

Bone, R., Cullis, P., and Wolfenden, R.: Solvent effects on equilibria of addition of nucleophiles to acetaldehyde and the hydrophilic character of diols, J. Am. Chem. Soc., 105, 1339–1343, 10.1021/ja00343a044, 1983.

Bonner and Breazeale(1965)

Bonner, O. D. and Breazeale, W. H.: Osmotic and activity coefficients of some nonelectrolytes, J. Chem. Eng. Data, 10, 325–327, 10.1021/je60027a007, 1965.

Borghesani et al.(1989)Borghesani, Pedriali, and Pulidori

Borghesani, G., Pedriali, R., and Pulidori, F.: Solute-solute-solvent interactions in dilute aqueous solutions of aliphatic diols. Excess enthalpies and gibbs free energies, J. Solution Chem., 18, 289–300, 1989.

Bower and Robinson(1963)

Bower, V. E. and Robinson, R. A.: Isopiestic vapor pressure measurements of the ternary system: sorbitol-sodium chloride-water at 25 ∘C, J. Phys. Chem., 67, 1540–1541, 10.1021/j100801a033, 1963.

Bradley and Cotson(1953)

Bradley, R. S. and Cotson, S.: The vapour pressure and lattice energy of hydrogen-bonded crystals, Part II. α- and β-anhydrous oxalic acid and tetragonal pentaerythritol, J. Chem. Soc., 1684–1688, 10.1039/JR9530001684, 1953.

Brocos et al.(2007)Brocos, Piñeiro, Amigo, and Gracia-Fadrique

Brocos, P., Piñeiro, Á., Amigo, A., and Gracia-Fadrique, J.: A proposal for the estimation of binary mixture activity coefficients from surface tension measurements throughout the entire concentration range, Fluid Phase Equilibr., 260, 343–353, 2007.

Cammenga et al.(1977)Cammenga, Schulze, and Theuerl

Cammenga, H. K., Schulze, F. W., and Theuerl, W.: Vapor pressure and evaporation coefficient of glycerol, J. Chem. Eng. Data, 22, 131–134, 1977.

Carlson and Colburn(1942)

Carlson, H. C. and Colburn, A. P.: Vapor-liquid equilibria of nonideal solutions, Ind. Eng. Chem., 34, 581–589, 10.1021/ie50389a013, 1942.

Cheon et al.(2005)Cheon, Kim, and Kim

Cheon, Y.-H., Kim, K.-J., and Kim, S.-H.: A study on crystallization kinetics of pentaerythritol in a batch cooling crystallizer, Chem. Eng. Sci., 60, 4791–4802, 10.1016/j.ces.2005.03.035, 2005.

Chianese et al.(1995)Chianese, Karel, and Mazzarotta

Chianese, A., Karel, M., and Mazzarotta, B.: Nucleation kinetics of pentaerythritol, Chem. Eng. J. Bioch. Eng., 58, 209–214, 10.1016/0923-0467(94)02896-6, 1995.

Chirife et al.(1984)Chirife, Favetto, and Fontán

Chirife, J., Favetto, G., and Fontán, C. F.: Microbial growth at reduced water activities: some physicochemical properties of compatible solutes, J. Appl. Bacteriol., 56, 259–268, 10.1111/j.1365-2672.1984.tb01346.x, 1984.

Claeys et al.(2004)Claeys, Graham, Vas, Wang, Vermeylen, Pashynska, Cafmeyer, Guyon, Andreae, Artaxo, and Maenhaut

Claeys, M., Graham, B., Vas, G., Wang, W., Vermeylen, R., Pashynska, V., Cafmeyer, J., Guyon, P., Andreae, M. O., Artaxo, P., and Maenhaut, W.: Formation of secondary organic aerosols through photooxidation of isoprene, Science, 303, 1173–1176, 10.1126/science.1092805, 2004.

Clegg and Seinfeld(2006)

Clegg, S. L. and Seinfeld, J. H.: Thermodynamic models of aqueous solutions containing inorganic electrolytes and dicarboxylic acids at 298.15 K. 2. Systems including dissociation equilibria, J. Phys. Chem. A, 110, 5718–5734, 10.1021/jp056150j, 2006.

Cohen et al.(1993)Cohen, Marcus, Migron, Dikstein, and Shafran

Cohen, S., Marcus, Y., Migron, Y., Dikstein, S., and Shafran, A.: Water sorption, binding and solubility of polyols, J. Chem. Soc. Faraday T., 89, 3271, 10.1039/ft9938903271, 1993.

Comesaña et al.(2001)Comesaña, Correa, and Sereno

Comesaña, J. F., Correa, A., and Sereno, A. M.: Water activity in sorbitol or xylitol + water and sorbitol or xylitol + sodium chloride + water systems at 20 ∘C and 35 ∘C, J. Chem. Eng. Data, 46, 716–719, 10.1021/je0003187, 2001.

Compernolle et al.(2009)Compernolle, Ceulemans, and Müller

Compernolle, S., Ceulemans, K., and Müller, J.-F.: Influence of non-ideality on condensation to aerosol, Atmos. Chem. Phys., 9, 1325–1337, 10.5194/acp-9-1325-2009, 2009.

Compernolle et al.(2010)Compernolle, Ceulemans, and Müller

Compernolle, S., Ceulemans, K., and Müller, J.-F.: Technical Note: Vapor pressure estimation methods applied to secondary organic aerosol constituents from α-pinene oxidation: an intercomparison study, Atmos. Chem. Phys., 10, 6271–6282, 10.5194/acp-10-6271-2010, 2010.

Compernolle et al.(2011)Compernolle, Ceulemans, and Müller

Compernolle, S., Ceulemans, K., and Müller, J.-F.: EVAPORATION: a new vapour pressure estimation method for organic molecules including non-additivity and intramolecular interactions, Atmos. Chem. Phys., 11, 9431–9450, 10.5194/acp-11-9431-2011, 2011.

Compernolle and Müller(2014)

Compernolle, S. and Müller, J.-F.: Henry's law constants of diacids and hydroxy polyacids: recommended values, Atmos. Chem. Phys., 14, 2699–2712, 10.5194/acp-14-2699-2014, 2014.

Della Gatta et al.(1999)Della Gatta, Jóźwiak, and Ferloni

Della Gatta, G., Jóźwiak, M., and Ferloni, P.: Heat capacities near room temperature of ten solid alkane-α, ω-diols HO-(CH2)n-OH where n=6 and 8–16, J. Chem. Thermodyn., 31, 537–546, 1999.

Domalski and Hearing(1993)

Domalski, E. S. and Hearing, E. D.: Estimation of the Thermodynamic Properties of C-H-N-O-S-Halogen Compounds at 298.15 K, J. Phys. Chem. Ref. Data, 22, 805–1159, 1993.

Engineering Sciences Data Unit Ltd(1995)

Engineering Sciences Data Unit Ltd: Vapour Pressures and Critical Points of Liquids. Glycols and Cyclic Diols, Item No 95002, vol. 2d(ii), ESDU International, London, 1995.

Ervens et al.(2011)Ervens, Turpin, and Weber

Ervens, B., Turpin, B. J., and Weber, R. J.: Secondary organic aerosol formation in cloud droplets and aqueous particles (aqSOA): a review of laboratory, field and model studies, Atmos. Chem. Phys., 11, 11069–11102, 10.5194/acp-11-11069-2011, 2011.

Fredenslund et al.(1975)Fredenslund, Jones, and Prausnitz

Fredenslund, A., Jones, R. L., and Prausnitz, J. M.: Group-Contribution Estimation Of Activity-Coefficients In Nonideal Liquid-Mixtures, AIChE J., 21, 1086–1099, 1975.

Friese and Ebel(2010)

Friese, E. and Ebel, A.: Temperature dependent thermodynamic model of the system H+-NH4+-Na+-SO42--NO3--Cl--H2O, J. Phys. Chem. A, 114, 11595–11631, 10.1021/jp101041j, 2010.

Góralski and Tkaczyk(2008)

Góralski, P. and Tkaczyk, M.: Heat capacities of some liquid α, ω-Alkanediols within the Temperature Range between (293.15 and 353.15) K, J. Chem. Eng. Data, 53, 1932–1934, 10.1021/je800356x, 2008.

Gracia-Fadrique et al.(2002)Gracia-Fadrique, Brocos, Piñeiro, and Amigo

Gracia-Fadrique, J., Brocos, P., Piñeiro, A., and Amigo, A.: Activity coefficients at infinite dilution from surface tension data, Langmuir, 18, 3604–3608, 10.1021/la011761y, 2002.

Hao et al.(2005)Hao, Hou, Wang, and Zhang

Hao, H.-X., Hou, B.-H., Wang, J.-K., and Zhang, M.-J.: Solubility of erythritol in different solvents, J. Chem. Eng. Data, 50, 1454–1456, 10.1021/je0501033, 2005.

Jasra and Ahluwalia(1982)

Jasra, R. and Ahluwalia, J.: Enthalpies of solution, partial molal heat capacities and apparent molal volumes of sugars and polyols in water, J. Solution Chem., 11, 325–338, 10.1007/2BF00649291, 1982.

Joback and Reid(1987)

Joback, K. and Reid, R.: Estimation of pure-component properties from group-contributions, Chem. Eng. Commun., 57, 233–243, 1987.

Jones et al.(2001)Jones, Oliphant, Peterson et al.

Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: Open source scientific tools for Python, http://www.scipy.org/ (last access: October 2014), 2001.

Knauth and Sabbah(1990a)

Knauth, P. and Sabbah, R.: Energetics of intra- and intermolecular bonds in o-alkanediols. III. Thermochemical study of 1,6-hexanediol, 1,7-heptanediol, 1,8-octanediol, 1,9-nonanediol, and 1,10-decanediol at 298.15 K, Can. J. Chem., 68, 731–734, 1990a.

Knauth and Sabbah(1990b)

Knauth, P. and Sabbah, R.: Energetics of inter. and intramolecular bonds in alkanediols. IV. the thermochemical study of 1,2-alkanediols at 298.15 K, Thermochim. Acta, 164, 145–152, 1990b.

Knauth and Sabbah(1990c)

Knauth, P. and Sabbah, R.: Energetics of intra- and intermolecular bonds in ω-alkanediols, Struct. Chem., 1, 43–46, 10.1007/BF00675783, 1990c.

Kulikov et al.(2001)Kulikov, Verevkin, and Heintz

Kulikov, D., Verevkin, S. P., and Heintz, A.: Enthalpies of vaporization of a series of aliphatic alcohols: Experimental results and values predicted by the ERAS-model, Fluid Phase Equilib., 192, 187–207, 2001.

Lian et al.(1982)Lian, Chen, Suurkuusk, Wadsö, Sokolov, Spiridonov, and Strand

Lian, Y.-N., Chen, A.-T., Suurkuusk, J., Wadsö, I., Sokolov, V. B., Spiridonov, V. P., and Strand, T. G.: Polyol–Water Interactions as Reflected by Aqueous Heat Capacity Values., Acta Chem. Scand., 36a, 735–739, 10.3891/acta.chem.scand.36a-0735, 1982.

Lopes Jesus et al.(2000)Lopes Jesus, Ermelinda Eusébio, Redinha, and Leitão

Lopes Jesus, A., Ermelinda Eusébio, M., Redinha, J., and Leitão, M.: Enthalpy of solvation of butanediols in different solvents, Thermochim. Ac., 344, 3–8, 2000.

Lopes Jesus et al.(2005)Lopes Jesus, Tomé, Eusébio, and Redinha

Lopes Jesus, A. J., Tomé, L. I. N., Eusébio, M. E., and Redinha, J. S.: Enthalpy of Sublimation in the Study of the Solid State of Organic Compounds. Application to Erythritol and Threitol, J. Phys. Chem. B, 109, 18 055–18 060, 2005.

Mackay et al.(2006)Mackay, Shiu, Ma, and Lee

Mackay, D., Shiu, W. Y., Ma, K.-C., and Lee, S. C.: Handbook of Physical-Chemical Properties and Environmental Fate for Organic Chemicals, CRC Press, 2006.

Mansoori(1980)

Mansoori, G. A.: Classical thermodynamic basis of activity coefficients: predictive and consistency rules for binary and ternary mixtures based on the relation between excess Gibbs free energies of (c)- and (c–1)-component mixtures, Fluid Phase Equilib., 4, 197–209, 1980.

Marcolli and Peter(2005)

Marcolli, C. and Peter, Th.: Water activity in polyol/water systems: new UNIFAC parameterization, Atmos. Chem. Phys., 5, 1545–1555, 10.5194/acp-5-1545-2005, 2005.

Martin(2000)

Martin, S. T.: Phase transitions of aqueous atmospheric particles, Chem. Rev., 100, 3403–3454, 2000.

Matsumoto et al.(1977)Matsumoto, Touhara, Nakanishi, and Watanabe

Matsumoto, Y., Touhara, H., Nakanishi, K., and Watanabe, N.: Molar excess enthalpies for water + ethanediol, + 1,2-propanediol, and + 1,3-propanediol at 298.15 K, J. Chem. Thermodyn., 9, 801–805, 1977.

Moller(2007)

Moller, B.: Development of an Improved Group Contribution Method for the Prediction of Vapour Pressures of Organic Compounds, Ph.D. thesis, University of KwaZulu-Natal Durban, 2007.

Moller et al.(2008)Moller, Rarey, and Ramjugernath

Moller, B., Rarey, J., and Ramjugernath, D.: Estimation of the vapour pressure of non-electrolyte organic compounds via group contributions and group interactions, J. Mol. Liq., 143, 52–63, 2008.

Mullin(2001)

Mullin, J. W.: Crystallization, 4th edn., Butterworth-Heinemann, Linacre House, Jordan Hill, Oxford OX2 8DP, 2001.

Myrdal and Yalkowsky(1997)

Myrdal, P. B. and Yalkowsky, S. H.: Estimating Pure Component Vapor Pressures of Complex Organic Molecules, Ind. Eng. Chem. Res., 36, 2494–2499, 10.1021/ie950242l, 1997.

Nannoolal(2006)

Nannoolal, Y.: Development and Critical Evaluation of Group Contribution Methods for the Estimation of Critical Properties, Liquid Vapour Pressure and Liquid Viscosity of Organic Compounds, Ph.D. thesis, University of Kwazulu-Natal, Durban Campus, 2006.

Nannoolal et al.(2008)Nannoolal, Rarey, and Ramjugernath

Nannoolal, Y., Rarey, J., and Ramjugernath, D.: Estimation of pure component properties: Part 3. Estimation of the vapor pressure of non-electrolyte organic compounds via group contributions and group interactions, Fluid Phase Equilibr., 269, 117–133, 10.1016/j.fluid.2006.11.014, 2008.

Nichols et al.(1976)Nichols, Sköld, Spink, and Wadsö

Nichols, N., Sköld, R., Spink, C., and Wadsö, I.: Thermochemistry of solutions of biochemical model compounds 6. α, ω-dicarboxylic acids, -diamines, and -diols in aqueous solution, J. Chem. Thermodyn., 8, 993–999, 1976.

Ninni et al.(2000)Ninni, Camargo, and Meirelles

Ninni, L., Camargo, M. S., and Meirelles, A. J. A.: Water activity in polyol systems, J. Chem. Eng. Data, 45, 654–660, 10.1021/je990303c, 2000.

Nitta et al.(1950)Nitta, Seki, Momotani, Suzuki, and Nakagawa

Nitta, I., Seki, S., Momotani, M., Suzuki, K., and Nakagawa, S.: On the phase transition in pentaerythritol (II), P. Jpn Acad., 26, 11–18, 1950.

Nitta et al.(1951)Nitta, Seki, and Suzuki

Nitta, I., Seki, S., and Suzuki, K.: Energy of hydrogen bond in tetragonal pentaerythritol, B. Chem. Soc. Jpn., 24, 63–69, 1951.

Nordström and Rasmuson(2008)

Nordström, F. L. and Rasmuson, Å. C.: Determination of the activity of a molecular solute in saturated solution, J. Chem. Thermodyn., 40, 1684–1692, http://dx.doi.org/10.1016/j.jct.2008.06.016, 2008.

Páez et al.(2011)Páez, Alvis, and Arrazola

Páez, M. S., Alvis, A., and Arrazola, G.: Comportamiento superficial e interfacial de soluciones acuosas diluidas de isómeros de pentanodiol a 288.15 K, Inf. Tecnol., 22, 65–70, 10.4067/s0718-07642011000100009, 2011.

Pankow and Asher(2008)

Pankow, J. F. and Asher, W. E.: SIMPOL.1: a simple group contribution method for predicting vapor pressures and enthalpies of vaporization of multifunctional organic compounds, Atmos. Chem. Phys., 8, 2773–2796, 10.5194/acp-8-2773-2008, 2008.

Paraskevas et al.(2013)Paraskevas, Sabbe, Reyniers, Papayannakos, and Marin

Paraskevas, P. D., Sabbe, M. K., Reyniers, M.-F., Papayannakos, N., and Marin, G. B.: Group additive values for the gas-phase standard enthalpy of formation, entropy and heat capacity of oxygenates, Chem. Eur. J., 19, 16431–16452, 10.1002/chem.201301381, 2013.

Parks and Anderson(1926)

Parks, G. S. and Anderson, C. T.: Thermal data on organic compounds. III. The heat capacities, entropies and free energies of tertiary butyl alcohol, mannitol, erythritol and normal butyric acid, J. Am. Chem. Soc., 48, 1506–1512, 10.1021/ja01417a009, 1926.

Parks and Huffman(1926)

Parks, G. S. and Huffman, H. M.: Thermal data on organic compounds. IV. The heat capacities, entropies and free energies of normal propyl alcohol, ethyl ether and dulcitol, J. Am. Chem. Soc., 48, 2788–2793, 10.1021/ja01690a004, 1926.

Peng et al.(2001)Peng, Chan, and Chan

Peng, C., Chan, M. N., and Chan, C. K.: The hygroscopic properties of dicarboxylic and multifunctional acids: measurements and UNIFAC predictions, Environ. Sci. Technol., 35, 4495–4501, 2001.

Piacente et al.(1993)Piacente, Ferro, and Della Gatta

Piacente, V., Ferro, D., and Della Gatta, G.: Vaporization enthalpies of a series of α, ω-alkanediols from vapour pressure measurements, Thermochim. Ac., 223, 65–73, 1993.

Piacente et al.(1994)Piacente, Ferro, and Della Gatta

Piacente, V., Ferro, D., and Della Gatta, G.: Vaporization enthalpies of five odd-numbered (C7 to C15) α, ω-alkanediols, Thermochim. Ac., 232, 317–321, 1994.

Poling et al.(2001)Poling, Prausnitz, and O'Connell

Poling, B. E., Prausnitz, J. M., and O'Connell, J. P.: The properties of gases and liquids, McGraw-Hill, New York, USA, 5 edn., 2001.

Prausnitz et al.(1999)Prausnitz, Lichtenthaler, and de Azevedo

Prausnitz, J. M., Lichtenthaler, R. N., and de Azevedo, E. G.: Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd edn., Prentice-Hall PTR, Upper Saddle River, New Jersey 07458, 1999.

Ramsey and Shafer(1997)

Ramsey, F. L. and Shafer, D. W.: The Statistical Sleuth: A Course in Methods of Data Analysis, Duxbury Press, 1997.

Robinson and Stokes(1961)

Robinson, R. A. and Stokes, R. H.: Activity coefficients in aqueous solutions of sucrose, mannitol and their mixtures at 25 ∘C, J. Phys. Chem., 65, 1954–1958, 10.1021/j100828a010, 1961.

Romero and Páez(2006)

Romero, C. M. and Páez, M. S.: Isopiestic determination of osmotic and activity coefficients of aqueous solutions of aliphatic polyols at 298.15 K, Fluid Phase Equilibr., 240, 140–143, 2006.

Romero et al.(2007)Romero, Páez, Miranda, Hernández, and Oviedo

Romero, C. M., Páez, M. S., Miranda, J. A., Hernández, D. J., and Oviedo, L. E.: Effect of temperature on the surface tension of diluted aqueous solutions of 1,2-hexanediol, 1,5-hexanediol, 1,6-hexanediol and 2,5-hexanediol, Fluid Phase Equilibr., 258, 67–72, 2007.

Sabbe et al.(2008)Sabbe, De Vleeschouwer, Reyniers, Waroquier, and Marin

Sabbe, M. K., De Vleeschouwer, F., Reyniers, M.-F., Waroquier, M., and Marin, G. B.: First principles based group additive values for the gas phase standard entropy and heat capacity of hydrocarbons and hydrocarbon radicals, J. Phys. Chem. A, 112, 12235–12251, 10.1021/jp807526n, 2008.

Sander(1999)

Sander, R.: Compilation of Henry's Law Constants for Inorganic and Organic Species of Potential Importance in Environmental Chemistry (Version 3), Public Report, available at: http://www.henrys-law.org/ (last access: 1 May 2014), 1999.

Saxena and Hildemann(1996)

Saxena, P. and Hildemann, L. M.: Water-soluble organics in atmospheric particles: a critical review of the literature and application of thermodynamics to identify candidate compounds, J. Atmos. Chem., 24, 57–109, 1996.

Scatchard et al.(1938)Scatchard, Hamer, and Wood

Scatchard, G., Hamer, W. J., and Wood, S. E.: Isotonic solutions. I. The chemical potential of water in aqueous solutions of sodium chloride, potassium chloride, sulfuric acid, sucrose, urea and glycerol at 25 ∘C, J. Am. Chem. Soc., 60, 3061–3070, 10.1021/ja01279a066, 1938.

Seidell(1941)

Seidell, A.: Solubilities of Organic Compounds, D. Van Nostrand Co., 3rd Edn., 250 Fourth Avenue, New York, 1941.

Spaght et al.(1931)Spaght, Thomas, and Parks

Spaght, M. E., Thomas, S. B., and Parks, G. S.: Some Heat-Capacity Data on Organic Compounds obtained with a Radiation Calorimeter, J. Phys. Chem., 36, 882–888, 10.1021/j150333a009, 1931.

Stein and Brown(2014)

Stein, S. and Brown, R.: NIST Chemistry WebBook, NIST Standard Reference Database Number 69, chap. Structures and Properties Group Additivity Model, National Institute of Standards and Technology, Gaithersburg MD, 20899, available at: http://webbook.nist.gov, last access: 1 May 2014.

Suleiman and Eckert(1994)

Suleiman, D. and Eckert, C. A.: Limiting activity coefficients of diols in water by a dew point technique, J. Chem. Eng. Data, 39, 692–696, 10.1021/je00016a011, 1994.

Suzuki et al.(1992)Suzuki, Ohtaguchi, and Koide

Suzuki, T., Ohtaguchi, K., and Koide, K.: Application of principal components analysis to calculate Henry's constant from molecular structure, Comput. Chem., 16, 41–52, 1992.

Toktonov(2009)

Toktonov, A. V.: Thermochemical insight into “green chemistry” processes: experiment and ab initio calculations, Ph.D. thesis, Rostock University, available at: http://rosdok.uni-rostock.de/resolve/id/rosdok_disshab_000000000219 (last access: 1 May 2014), 2009.

Tong et al.(2007)Tong, Tan, Shi, Li, Yue, and Wang

Tong, B., Tan, Z.-C., Shi, Q., Li, Y.-S., Yue, D.-T., and Wang, S.-X.: Thermodynamic investigation of several natural polyols (I): heat capacities and thermodynamic properties of xylitol, Thermochim. Ac., 457, 20–26, 10.1016/j.tca.2007.02.022, 2007.

Tong et al.(2008)Tong, Tan, Shi, Li, and Wang

Tong, B., Tan, Z., Shi, Q., Li, Y., and Wang, S.: Thermodynamic investigation of several natural polyols (II), J. Therm. Anal. Calorim., 91, 463–469, 10.1007/s10973-007-8361-8, 2008.

Tong et al.(2009)Tong, Tan, Zhang, and Wang

Tong, B., Tan, Z., Zhang, J., and Wang, S.: Thermodynamic investigation of several natural polyols, J. Therm. Anal. Calorim., 95, 469–475, 10.1007/s10973-008-9268-8, 2009.

Tong et al.(2010a)Tong, Liu, Meng, Yu, Ji, and Tan

Tong, B., Liu, R.-B., Meng, C.-G., Yu, F.-Y., Ji, S.-H., and Tan, Z.-C.: Heat capacities and nonisothermal thermal decomposition reaction kinetics of d-mannitol, J. Chem. Eng. Data, 55, 119–124, 10.1021/je900285w, 2010a.

Tong et al.(2010b)Tong, Yu, Tan, Meng, Cui, Xiao, and Liu

Tong, B., Yu, Y., Tan, Z.-C., Meng, C.-G., Cui, L.-J., Xiao, G., and Liu, R.-B.: Thermodynamic investigation of several natural polyols (IV): heat capacities and thermodynamic properties of adonitol, Thermochim. Ac., 499, 117–122, 10.1016/j.tca.2009.11.011, 2010b.

Verevkin(2004)

Verevkin, S. P.: Determination of vapor pressures and enthalpies of vaporization of 1,2-alkanediols, Fluid Phase Equilibr., 224, 23–29, 2004.

Verevkin(2007)

Verevkin, S. P.: Vapor pressures and enthalpies of vaporization of a series of the 1,3-alkanediols, J. Chem. Eng. Data, 52, 301–308, 10.1021/je060419q, 2007.

Verevkin and Schick(2007)

Verevkin, S. P. and Schick, C.: Vapour pressures and heat capacity measurements on the C7-C9 secondary aliphatic alcohols, J. Chem. Thermodyn., 39, 758–766, 2007.

Verevkin et al.(2003)Verevkin, Krasnykh, Vasiltsova, Koutek, Doubsky, and Heintz

Verevkin, S. P., Krasnykh, E. L., Vasiltsova, T. V., Koutek, B., Doubsky, J., and Heintz, A.: Vapor pressures and enthalpies of vaporization of a series of the linear aliphatic aldehydes, Fluid Phase Equilib., 206, 331–339, 2003.

Verevkin et al.(2009)Verevkin, Emel'yanenko, and Nell

Verevkin, S. P., Emel'yanenko, V. N., and Nell, G.: 1,2-propanediol. Comprehensive experimental and theoretical study, J. Chem. Thermodyn., 41, 1125–1131, 2009.

Wang et al.(2013)Wang, Wang, Liu, Fang, Li, and Xiao

Wang, Z., Wang, Q., Liu, X., Fang, W., Li, Y., and Xiao, H.: Measurement and correlation of solubility of xylitol in binary water+ethanol solvent mixtures between 278.00 K and 323.00 K, Korean J. Chem. Eng., 30, 931–936, 10.1007/s11814-012-0225-7, 2013.

Xie et al.(2014)Xie, Hannigan, and Barsanti

Xie, M., Hannigan, M. P., and Barsanti, K. C.: Gas/particle partitioning of 2-methyltetrols and levoglucosan at an urban site in Denver, Environ. Sci. Technol., 48, 2835–2842, 10.1021/es405356n, 2014.

Zhang and Yang(1989)

Zhang, Z. Y. and Yang, M. L.: Heat capacities and phase transitions of 1,1,1-trihydroxymethylpropane and pentaerythritol over the superambient temperature range, Thermochim. Ac., 156, 157–161, 1989.

Zuend et al.(2011)Zuend, Marcolli, Booth, Lienhard, Soonsin, Krieger, Topping, McFiggans, Peter, and Seinfeld

Zuend, A., Marcolli, C., Booth, A. M., Lienhard, D. M., Soonsin, V., Krieger, U. K., Topping, D. O., McFiggans, G., Peter, T., and Seinfeld, J. H.: New and extended parameterization of the thermodynamic model AIOMFAC: calculation of activity coefficients for organic-inorganic mixtures containing carboxyl, hydroxyl, carbonyl, ether, ester, alkenyl, alkyl, and aromatic functional groups, Atmos. Chem. Phys., 11, 9155–9206, 10.5194/acp-11-9155-2011, 2011.

</app></app-group></back> </article>