Formaldehyde reacts with solvents that contain hydroxyl groups ( R-OH) in oligomer- ization reactions to oxymethylene oligomers ( R-(OCH 2)n-OH). The chemical equilibria of these reactions have been studied in the literature for water, for the mono-alcohols methanol, ethanol, and 1-butanol, as well as for the diols ethylene glycol and 1,4- butynediol. In the present work, the collective data were analyzed. It was found that the prolongation of the oxymethylene chains by the addition of formaldehyde can be described very well with a generalized chemical equilibrium constant KR-OH x,n≥2, which is independent of the substructure ( R) of the solvent. This holds for the oligomerization reactions leading to R-(OCH 2)n-OH with n≥2. The chemical equilibrium constant KR-OH x,1 of the reaction of formaldehyde with the solvent R-OH depends on the sol- vent, but simple trends are observed. The hypotheses of the existence of a generalized 1 chemical equilibrium constant KR-OH x,n≥2was tested for the reactions of formaldehyde with ethanol and 1-propanol, for which neither KR-OH x,1 norKR-OH x,nwere previously available. The corresponding equilibria were studied by13C NMR spectroscopy and the equilib- rium constants were determined. A novel method was developed and used in these studies to obtain data on KR-OH x,1 by NMR spectroscopy, which is dicult due to the low amount of molecular formaldehyde. It was found that the generalized equilibrium constant is even valid for the acid-catalyzed formation of poly(oxymethylene) dimethyl ethers (OME). Introduction Formaldehyde (FA) is one of the most important chemical intermediates.1It reacts with sol- vents that contain hydroxyl groups R-OH in oligomerization reactions, which leads to very high solubilities of formaldehyde in these solutions. As pure formaldehyde tends to polymer- ize, formaldehyde is used in industrial processes almost exclusively in such solutions. The most common solvent for formaldehyde is water (W), but also the reactions with methanol and many other solvents R-OH are of interest.2Due to the oligomerization reactions of formaldehyde with R-OH, these solutions are always complex multicomponent mixtures.3 For the poly(oxymethylene) oligomers of the type R-(OCH 2)n-OH the abbreviation Ol nis used in the present work, where ndenotes the number of CH 2O segments in the oligomer. The oligomers of formaldehyde with water are named poly(oxymethylene) glycols (MG n), those with alcohols are named poly(oxymethylene) hemiformals (HF n). Under acidic con- ditions also poly(oxymethylene) acetals R-(OCH 2)n-O-Rare formed. These oligomerization reactions strongly in uence the properties of the mixtures, e.g. they have to be considered for describing the vapor-liquid equilibrium.3{10 In the literature, information on the chemical equilibrium of mixtures of formaldehyde with water,11{14with the mono-alcohols methanol,12,15,16ethanol,17and 1-butanol18,19as well as with the diols ethylene glycol17and 1,4-butynediol20is available. In the present 2 work, the entire set of available data on the oligomerization reactions of formaldehyde was analyzed. Moreover, the experimental basis was extended to oligomerization reactions of formaldehyde with the mono-alcohols ethanol and 1-propanol by studying the species dis- tribution in the systems (formaldehyde + water + ethanol) and (formaldehyde + water + 1-propanol) by NMR spectroscopy, which is a well-established tool for the examination of reactive formaldehyde mixtures.16,18,19,21{24It may be astonishing that mixtures contain- ing water were investigated here, when only the reactions in the system (formaldehyde + ethanol) and (formaldehyde + 1-propanol) are of interest. This is related to the diculty in determining the equilibrium constant of reactions of monomeric formaldehyde with the alcohol by means of NMR spectroscopy. Monomeric formaldehyde is only present in very small amounts, such that its direct quanti cation by NMR spectroscopy is not possible. By using the competition between water and the alcohol in combination with the fact that the reactions of formaldehyde with water are well-known, this problem can be circumvented and the chemical equilibrium constant of the reaction of monomeric formaldehyde with the alcohol can be determined by NMR spectroscopy. Chemical equilibrium constants should preferably be reported in a thermodynamically consistent manner; i.e. for liquid phase reactions, activities should be used rather than concentrations.25,26Nonetheless, we use mole fraction-based equilibrium constants in the present work. The mole fraction-based equilibrium constant Kxis related to the activity- based equilibrium constant Kaby: Ka=Kx⋅K ; (1) where K is de ned analogous to Kx, but with the corresponding activity coecients . In general, both KaandKxdepend on the temperature T, but the thermodynamic theory yields that Kais independent from the composition, while Kxis in general composition-dependent. (The in uence of pressure is neglected here). 3 Why have we, despite the theoretical shortcomings, nevertheless preferred to work with Kxinstead of Ka? There are several reasons for this: The determination of activity coe- cients, which are required for the calculation of Ka, is far from being trivial in formaldehyde- containing systems.3,6{8,27,28In principle, o -the-shelf group contribution methods, such as UNIFAC,29,30could have been applied to calculate K , but it has been known for a long time that the standard methods yield poor results for formaldehyde-containing mixtures:3as a rule, at least some of the parameters of the group-contribution methods have to be revised to obtain a good description of thermodynamic data of formaldehyde-containing systems, see e.g. Maurer3for an early example, and Schmitz et al.31,32for recent examples. Establishing a reliable and consistent group-contribution model for the non-ideality of all mixtures studied in the present work is a formidable task that would have exceed the scope of the present research by far. Moreover, even if such a group-contribution model have been developed for all solvents studied here, it is by no means clear that it would yield good results when used for prediction of systems that were not included in the present study. Luckily, from a practical standpoint, and if the task is only to describe the chemical equi- libria in formaldehyde-containing systems in solvents like those studied here, it is completely sucient to use Kx; using Kahas no practical advantages: It is known that the numbers for Kxin solutions of formaldehyde in the polar and H-bonding solvents that are studied here do not depend strongly on the composition of the solution for a given solvent, see e.g. Berje et al.20Furthermore, Schmitz et al.9have recently compared modeling chemical equilibria in the system (formaldehyde + OME) with KxandKa. Their results for Ka, obtained from studies of solutions of di erent composition, even scatter more strongly than the correspond- ing results for Kx. We will show here, that these ndings can be extended to the di erent solvents: We nd a universal behavior using simply Kx, and cannot hope to obtain a more favorable results by using Ka. This implies that the K in the studied systems do not vary strongly for the di erent systems, which is not unexpected as all solvents that are considered here a strongly polar and H-bonding. We do not claim that the simple ndings presented 4 here hold for solvents from other classes. Last but not least: The results for Kxcan very easily be used to calculate Kaif an activity coecient model is available. In the present research, it was revealed that the prolongation of oxymethylene chains can be described very well with a generalized chemical equilibrium constant KR-OH x,n, which is independent of the solvent substructure ( R). This holds for the oligomerization reactions that yield R-(OCH 2)n-OH with n≥2. For solvents with two or multiple hydroxyl groups, however, symmetries have to be considered in the description of the chemical equilibria. Contrary to the generalized chemical equilibrium constant KR-OH x,nof the chain prolon- gation, the chemical equilibrium constant KR-OH x,1 of the rst addition of formaldehyde to the solvent depends on R, but simple trends were found. Furthermore, we found that the generalized equilibrium constant of the chain prolongation in hemiformals KR-OH x,nseems to be valid also for the chain prolongation in the formation of the corresponding acetals. This was shown for the poly(oxymethylene) dimethyl ethers (OME n, H3C-(OCH 2)n-O-CH 3). This paper is organized as follows: First, the chemical reaction system and the way, in which it is modeled here, are introduced. Then, the experiments that were carried out in the present work for the systems with ethanol and 1-propanol are described and the novel method for determining KR-OH x,1 is introduced. Finally, the results for the present work and those from the literature are analyzed and the general ndings are presented, before we conclude. We have chosen this standard organization of the paper to facilitate reading. In the actual research, the analysis of the literature came rst, then came the hypothesis of the generalization, and only then, the experiments were carried out for validation. 5 Oligomerization Reactions of Formaldehyde Mono-alcohols Formaldehyde reacts with mono-alcohols R-OH in oligomerization reactions to poly(oxymethylenes) Oln, which are hemiformals in the case of alcohols and glycols in the case of water. These reactions take place without any catalyst.12By the rst addition of formaldehyde, the small- est oligomer Ol 1is formed, the corresponding chemical equilibrium constant is denoted as KR-OH x,1. R-OH+CH2O /dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlymid/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyright FAKR-OH x,1/leftrightline/leftrightline/leftrightline⇀/uni21BD/leftrightline/leftrightline/leftrightline R-OCH 2-OH /dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlymid/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyright Ol1(2) The formation of higher poly(oxymethylene)s Ol n(n≥2), is described by the chain prolon- gation with the chemical equilibrium constant KR-OH x,n. R-(OCH 2)n−1-OH /dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlymid/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyright Oln−1+CH2O /dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlymid/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyright FAKR-OH x,n/leftrightline/leftrightline/leftrightline⇀/uni21BD/leftrightline/leftrightline/leftrightline R-(OCH 2)n-OH /dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlymid/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyright Oln;n≥2: (3) The mole fraction-based chemical equilibrium constants of Reactions (2) and (3) are: KR-OH x,1=xOl1 xR-OH⋅xFA(4) KR-OH x,n=xOln xOln−1⋅xFA;n≥2 (5) Due to Reactions (2) and (3), the amount of monomeric formaldehyde in solutions of formaldehyde in hydroxyl group-containing solvents is always very low, since the formalde- hyde is predominately bound in the oligomers. In the present work, we distinguish between overall concentrations and true concentra- tions. The latter are found taking into account Reactions (2) and (3), whereas in the former, it is assumed that all reaction products are split up into formaldehyde and the solvent. 6 Diols The reaction scheme presented in Equations (2) and (3) is too simplistic in cases, in which R contains additional hydroxyl groups. We discuss only the case of diols here, and denote them by HO- R′-OH. The reactions of formaldehyde with diols lead to di erent types of oligomers of the general structure HO-(CH 2O)l-R′-(OCH 2)r-OH. In the following, a short notation will be used in which a given oligomer is simply denoted by the pair of natural numbers ( l r) that indicate the number of formaldehyde segments attached to the hydroxyl groups of the diol (formally: lfor left and rfor right). This notation does not imply that the resulting products are distinguishable, i.e. (23) may be the same molecule as (32). The formation of these oligomers can be described by the pyramidal reaction scheme presented in Figure 1a. Figure 1: The panel (a) shows a reaction scheme for the formation of formaldehyde oligomers from diols. In the short notation in the gure, the oligomer HO-(CH 2O)l-R′-(OCH 2)r-OH is denoted by the two numbers for land r. This notation does not imply that the result- ing products are distinguishable. The arrows indicate forward and backward reactions and involve a monomeric formaldehyde molecule. For comparison, the panel (b) shows the corre- sponding scheme for mono-alcohols in a similar notation where the number simply stands for the number of CH 2O segments in the oligomer. The asterisk labels rst addition reactions in which formaldehyde adds directly to a hydroxyl group of the diol or mono-alcohol. The corresponding chemical equilibrium constants Kxare de ned in the same way as in Equations (4) and (5), where one educt is always monomeric formaldehyde. For comparison, 7 Figure 1b gives the corresponding scheme for mono-alcohols. In Figure 1 reactions in which formaldehyde adds directly to R′are labeled with an asterisk. The arrows in Figure 1 indicate the forward and backward reaction, as usual. Our hypothesis is that the values of these rate constants are the same for all reactions that are not labeled with an asterisk in Figure 1, i.e. they are not only the same for a given mono-alcohol or a given diol but also the same for all mono-alcohols and all diols and that, furthermore, there is also no di erence between the mono-alcohols and diols. In contrast, the rate constants of the reactions labeled with an asterisk depend on the alcohol, but for a given diol, they are all identical. Let us now assume that all species shown in the pyramidal reaction scheme in Figure 1a are distinguishable, i.e. that the structure of the diol is not symmetrical and the two OH- groups can be distinguished. In this case, it then follows from the above that all reactions shown in Figure 1a are distinguishable and that the equilibrium constants of all reactions that are not labeled with an asterisk in Figure 1a are identical. Also the reactions labeled with an asterisk in Figure 1a have the same equilibrium constant, which, however, generally has a di erent value than that of the remaining reactions. Things get more complicated when not all species can be distinguished. The most im- portant case is that of a symmetrical diol, in which the two hydroxyl groups cannot be discriminated, and, as a consequence, also the species ( l r) and the species ( r l) are indis- tinguishable. Let us consider the reaction scheme shown in Figure 1a for this case. It then contains pairs of reactions that cannot be discerned; we will call such a pair an apparent re- action . All distinguishable apparent reactions in the scheme shown in Figure 1a then belong to one of the following three categories: a) two indistinguishable species are formed from one distinguishable species, e.g. (00) → (01) or (10); b) one distinguishable species is formed from two indistinguishable species, e.g. (10) or (01)→(11); 8 c) two indistinguishable species are formed from two indistinguishable species, e.g. (20) or (02)→(21) or (12). Using the same assumptions on the equality of the rate constants of the di erent elemen- tary reactions that were described above leads to the following consequences for the relation between the equilibrium constant of the apparent reaction ~Kxand that of the elementary reaction Kxfor the three categories de ned above: a)~Kx= 2⋅Kx b)~Kx= 0.5⋅Kx c)~Kx=Kx Water is a special case that does not fall into any of the categories discussed above. In principle, also for water, a scheme similar to the one shown in Figure 1a can be set up, as water has two locations at which formaldehyde can be added. But for water, all species that have the same number of CH 2O segments are indistinguishable, e.g. (20) = (11) = (02). The apparent reactions then have the same simple structure as that for the mono-alcohols, c.f. Figure 1b, i.e. there is only the simple chain prolongation. All apparent reactions have two pathways for the prolongation and two for the cleavage, such that for water, there is no di erence between the equilibrium constant of the apparent reaction and that of the elementary reaction. The term distinguishable is used here in a mathematical sense. As a rule, not all mathematically distinguishable species can be distinguished in an analysis of formaldehyde- containing mixtures, as the resolution of the analytical instruments is limited. The inter- pretation of the results of such measurements based on the reaction scheme discussed above requires an assignment of species to signals, as usual. In the following, we report values for Kxand not for ~Kxfor the diols as they allow a simple direct comparison with the values of mono-alcohols and water. 9 Literature Data Table 1 gives an overview of the literature on the chemical equilibria of the liquid phase reactions of formaldehyde with hydroxyl group-containing solvents. Only