Journal of Molecular Liquids 275 (2019) 741–748 Contents lists available at ScienceDirect Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq Physically adequate approximations for abnormal temperature dependences of water characteristics Alexander Kholmanskiy ⁎, Nataliya Zaytseva Moscow State University of Medicine and Dentistry, Russia a r t i c l e i n f o a b s t r a c t Article history: Temperature dependences (TD) for thirteen basic characteristics of water in the range of −30 °C to 100 °C were Received 3 August 2018 approximated by function fa = Т±βexp(±Ea/RT). Signs and values of β and those of reaction heat (Ea) have been Received in revised form 11 November 2018 referred to the well-known relationships that link water characteristic with Van't Hoff and Arrhenius equations, Accepted 14 November 2018 as well as with Boltzmann and Maxwell distribution laws. Reactions of water internal structure transformation Available online 17 November 2018 responsible for TD characteristics abnormality have been classiﬁed as exothermic and endothermic ones while Keywords: characteristics themselves have been categorized as dynamic and conditionally static ones. Values of Ea for the Water former category were associated with hydrogen bond activation energy while those of the latter category were Characteristics attributed to the energies of translational and torsional molecular oscillations. Growth of Ea values for dynamic Abnormal characteristics at temperatures below 25 °C is associated with that of the share of hexagonal ice-like clusters in Temperature supramolecular structure of water. Extremal temperature values for TDs of static characteristics were derived Extremums from solutions of equation ∂fa/∂T = 0, and they were proportional to those of Ea, for corresponding Approximations approximations. © 2018 Elsevier B.V. All rights reserved. 1. Introduction like hexagonal clusters within the lattice of tetrahedral hydrogen bonds (Н-bonds) [5,7]. Advent of life on the Earth, in all its forms, has become possible Physical nature of liquid water abnormalities and that of TD extrema owing to abnormal physical-and-chemical properties of liquid water. of its characteristics has not been sufﬁciently explored, in spite that a In the process of evolution, thermodynamic properties of water were huge number of various studies of water properties carried out in the an instrumental in the development of living systems' mechanisms of frames of thermodynamic laws. Evidently, selection rules of quantum adaptation to the impact of global and changeable geocosmo-physical physics, as well as non-linear, cooperative and resonance-type effects factors. Abnormalities of liquid water, for normal atmospheric pressure, have to be regarded for understanding the nature of water properties become apparent, ﬁrst of all, in temperature dependencies (TD) of abnormality, as well as that of mechanisms of inﬂuence of external its characteristics, in a wide range of real temperatures (−30 °C to physical factors on water content of biosystem [9,10]. For example, se- 100 °C). For biogenesis, abnormal properties of cool and supercooled lection rules for transitions between the spin states in water molecule water remaining ﬂuid in form of water emulsions and bioorganic sys- are responsible for peculiarities of hemorheology near the point tE tems at 30 °C b t b 10 °C [1,2] are also of prime importance. ~ 36.6 °C and abnormality of TD [α] of sugar water solution . For TDs of volumetric density (ρ), speciﬁc heat capacity at constant pres- this case, energies of ground states and transitions between water sure (Ср), isothermal compressibility (γ) and sound velocity (V), for t N ortho- and para-isomers correlate with the values of activation energy 0 °C have their extremum points close to (tE) 4 °C, 36 °C, 46 °C and 75 °C, (Еа) in temperature dependences [α]. correspondently. Along with point of 0 °C, that of 25 °C was also in- Resonance effects may also be responsible for abnormality of TDs for cluded in this sequence since TDs of water dynamic characteristics dynamic properties of water close to point 25 °C. In , the following change in close proximity to this point [3,4]. Dependences of shearing formula was derived for the frequency of proton coherent oscillation z viscosity (ηs) and self-diffusion coefﬁcient (D) on pressure at t ≤ 25 °C (z ≥ 6), in helix chain of tetrahedral Н-bonds: have their extrema for ~1 kbar [5–7]. For t b 25 °C, growth of speciﬁc ro- tation ([α]) for sugar solution can be observed . TDs for infrared and νz ¼ 22z−1 ðz−1Þ1=2 :ðTHzÞ x-ray spectra [5,9] of ﬂuid water, at t b 40 °C, indicate presence of ice- From this formula, for z = 12, we obtain ν12 = 6.08 ТHz and oscilla- ⁎ Corresponding author. tion energy ~2.4 kJ/mol which corresponds to the value of thermal en- E-mail address: email@example.com (A. Kholmanskiy). ergy at T = 298 K (25 °C). Since one ice-like hexagonal cluster https://doi.org/10.1016/j.molliq.2018.11.059 0167-7322/© 2018 Elsevier B.V. All rights reserved.
742 A. Kholmanskiy, N. Zaytseva / Journal of Molecular Liquids 275 (2019) 741–748 bond breakage, as well as those of translational, rotation-vibrational and spin-spin transitions. Within the frames of equilibrium thermodynamics laws, each of these reactions and transitions will be characterized by its own values of reaction heat (ΔQ) and activation energy (Е) (see Fig. 1). These values apply to Arrhenius and Van't Hoff equations and deﬁne kinetics of equi- librium processes of water structure reconﬁguration responsible for ab- normalities of TDs of water characteristics. Dependence of restructuring kinetics on Т can be formally expressed via either TD of equilibrium constant (Кe) or that of the ratio of speciﬁc reaction rates for direct reaction (К1) and reverse reaction (К−1) pre- vailing in each type of structure transformation. Therefore, function, in Т, of the following form has to represent the essence of physically ade- quate approximation for TDs of water characteristics: Ea C 1 exp − К1 RT ΔQ f a ðТÞ∝Кe ¼ ¼ ¼ А exp Æ : ð1Þ К−1 Ea Æ ΔQ RT C −1 exp − RT Pre-exponential factor A in Eq. (1) does not normally depend on Т Fig. 1. Energy proﬁles of endo- (1) and exothermic (2) reactions of water structure rear- while values of ΔQ change within the range from quant energy for rota- rangement. E1, E2 - reaction barriers (activation energy); ΔQ - thermal effects. tional transition to the energy of Н-bond breakage. In empirical approx- imations for TDs of water characteristics, ΔQ plays the role of effective contains 12 protons assumption can be made that, in the vicinity of energy of activation. That is why its traditional designations Еа will be point 25 °C, transition of hexagonal cluster into a helix chain consisting used hereafter. of 6 water molecules takes place, in accordance with resonance-type In works dedicated to the research of water physics, various combi- mechanism. Therefore, ηs reduces while D grows. Probably, a similar ef- nations of functions of type fa, are normally applied as approximations fect high pressure produces on water structure at temperatures t ≤ 25 for TDs of water characteristics in which pre-exponential factor A may °C. have the form of power function in Т. Below are typical examples of fa Taking various theoretical models of structure and dynamics of liq- functions for dynamic characteristics of water: uid water into consideration, the assumption was made in [3,4] that Ea structure conversion takes place, in points tE, and supramolecular struc- f η ¼ Cexp ; ð2Þ RT ture of water splits into dynamic quasi-phases that differ in their level and type of molecular clusterization. Exactly such supramolecular struc- Ea tures may feature cooperative properties and add to high sensitivity of f 1 ¼ CT 1:5 E−1 exp a ½12; RT water-containing biosystems to external factors of various origins. As a result of water dynamic structure conversion, energy state E1 E2 f 2 ¼ С1 exp þ С2 exp ½13; of separate molecule changes, and hydrogen bonds get redistributed. RT RT Therefore, intermolecular links may get stronger or weaker. Such restructuring occurs on micro- and macro-levels involving Ea f 3 ¼ Cexp ½12 intercorrelated states of both separate molecules and various cluster- RðT−T o Þ type molecular complexes. The assumption can be made that physics α α2 of abnormalities of each TD of water characteristics is a result of a cer- f 4 ¼ Cexp Æ ; f 5 ¼ Cexp Æ 2 ½3; 4: tain type of molecular dynamics including reversible reactions of Н- T Т Fig. 2. fa-approximation for TD of self-diffusion coefﬁcient of Н17О (D). Values of D(t) for (а) and (b) were imported from [13,14], respectively. 2
A. Kholmanskiy, N. Zaytseva / Journal of Molecular Liquids 275 (2019) 741–748 743 Fig. 3. fa-approximations for TDs of water dynamic viscosity (η) and shear viscosity (ηs). Values of η(T) and ηs(T) were imported from [15,16], respectively. Lines of trends are colorless and dotted. Here, R is speciﬁc gas constant (8.3 J·mol−1·K−1), Сi, are constant of points tE to form Т-intervals for which linear approximations had dif- values, Еа and Еi are effective activation energies, α is slope ratio for cor- ferent values of Еа. responding anamorphosises [3,4]. In this work, in order to study the speciﬁcity of molecular dynamics Applicability of functions fa is conditioned by structural uniformity of responsible for abnormal properties of water, physically adequate func- water, in macro-scale: (2) is Fraenkel's formula for TD of dynamic vis- tions have been applied for approximation of temperature dependences cosity (η), f1 is derived with the account of two ice-like clusters, while of water basic characteristics, in the range of −30 °C to 100 °C, and effec- f2 takes account of difference in dynamics of free water molecules in tive activation energies have been compared with earlier reported clusters occurring at low (Е1 = 39.4 kJ/mol) and high (Е2 = values for energies of atom-molecular motions and processes in liquid 13.4 kJ/mol) values of Т. Function f3 is not adequate in relation to prop- water. erties of liquid water since temperature value То ~ 150 K is far beyond the range of real conditions. In , function f4 was applied to approxi- 2. Materials and methods mate TDs for η, D and dielectric relaxation time (τD), while physically inadequate function f5 was used for approximation of TDs for ρ, V, CР Empirical data on TDs of water characteristics were taken from ref- and γ. In works [3,4] the entire range of Т was divided with the help erence books or original works, and graphs were digitized, when Fig. 4. fa-approximation for TD of water relaxation time of the proton spins (T2). Values T2 were imported from .
744 A. Kholmanskiy, N. Zaytseva / Journal of Molecular Liquids 275 (2019) 741–748 marked with arrows. ‘MS Excel’ application was used to plot TD approx- imations. The extent of proximity of value R2 to 1 was chosen as reliabil- ity criterion for approximations, in various Т-intervals. The accuracy of values Еа derived from approximations was deﬁned by inherited error that might depend on water pureness degree, particularly in the range of t b 25 °C. Physical sufﬁciency of form fa was determined by ﬁtting the sign of exponent and the value of power coefﬁcient Т±β in pre- exponential factor with reported relationships between water charac- teristics. In this case, optimal values of β were chosen from numerical series 0, ±1/2, ±3/2, ±2. The form of functions fa, for D and η, was brought into match with function fη and Stokes-Einstein formula: Fig. 5. fa-approximation for TD of water dielectric relaxation time (τD), the initial data from kT . D¼ 6πηr in which k is Boltzmann constant and r is radius of particle. The form of functions fa for spin-lattice relaxation time (Т1) was chosen with the ac- count of proportional relationship between Т1 and self-diffusion coefﬁ- cient D . It has been also taken into consideration that spin-spin relaxation time (Т2) and τD are proportional to rotational relaxation time (τr) that can be described with the use of Debye formula : 4πηr 3 τr ¼ : 3kT The form of selected functions fa for γ, V and ρ complies with the fol- lowing well-known relationship: V ¼ ðγ Á ρÞ−1=2 : 3. Results Fig. 6. Approximation for experimental TD of spin-lattice relaxation time (Т1): а) is approximation by sum of two exponential functions (С1 and С2 are constant values) Graphs for approximations and those for reference data are shown  and b) is fa-approximation. in Figs. 1 to 13, while forms of functions fa and values of Еа are presented in the Table 1. necessary, with the use of ‘Paint’ computer application. The borders of 4. Discussion the entire temperature range for TDs were deﬁned based on those of available data varying in the limits of −30 °C to 100 °C. Intervals of Т se- The fact that values of Еа substantially differ for all characteristics lected for approximations of water characteristics TDs having extremal of water provides a reason to divide them onto dynamic parameters points included, in all cases, one demarcation point corresponding to (δ, Р, D, η, τD, Т1, Т2) and conditionally static ones (Ср, ρ, V, γ, σ). Re- the extremum value that, in some circumstances, was complemented actions responsible for abnormalities of TDs corresponding to char- with tE values for other TDs. To become aware of participation of hexag- acteristics D, Т1, V, Р and δ, for separate molecules, are endothermic onal ice-like clusters in molecular dynamics, point 25 °C was chosen as while those corresponding to η, τD, Т2, Ср, γ, σ and δ for bound mol- demarcation point, for all dynamic characteristics of water. On the ap- ecules, are exothermic. In the former case, bonds between molecules proximation graphs, junction points of adjacent Т-intervals were get weaker as a result of structure transformation while, in the latter Fig. 7. Temperature dependence of water speciﬁc heat capacity (Ср) on Т (а) and its fa-approximations (b), for Т N 273 from . Data for Т b 273 were obtained by digitizing Ср(Т) graph imported from . Lines of trends are colorless.
A. Kholmanskiy, N. Zaytseva / Journal of Molecular Liquids 275 (2019) 741–748 745 Fig. 8. Temperature dependence of water density (ρ) on Т (а) and its fa-approximations (b), for Т N 273 from . Data for Т b 273 were obtained by digitizing ρ(T) graph imported from . Lines of trends are colorless. case, they get stronger. Values of Еа for dynamic characteristics cor- Values of Еа for static characteristic of water are close to those of en- relate with those of Н-bonds energies (8 kJ/mol to 25 kJ/mol ) ergies of translational and torsional molecular motions that belong to and with each other, within their related Т-intervals. It stands to rea- the ranges 0.2 kJ/mol to 1 kJ/mol and 2.4 kJ/mol to 7.2 kJ/mol . The son that, in water structure transitions responsible for abnormalities following relationship can be derived from formula (2) by substitution of TDs for dynamic characteristics, those reactions dominate that of corresponding functions fa: lead to breakage of H-bonds that were deformed to various extents. That is their kinetics that has no explicit extremum within any of 1À γ Á ЕV ¼ а Е −Еρ ; T-intervals. Therefore, derivatives of functions fa with respect to T 2 а а do not become zero, for these characteristics. Values of Еа change step-wise for all dynamic characteristics, in point that is fairly applicable in the range of 0 °C to 75 °C (see Table 1). For 25 °C, which correlates with sharp growth of the role of ice-like hexag- static characteristics with extremal TDs, derivatives for their fа- onal clusters in molecular dynamics responsible for TDs of correspond- functions become zero in point ТE. In this case, the following formulas ing characteristics, at t b 25 °C. Arithmetic mean value of Еа difference, can be deduced for ρ, Ср, γ and V, in point TE: for Т-intervals in the ranges of 0 °C to 25 °C and 25 °C to 100 °C, equals ρ;C to 5.4 ± 10% kJ/mol, for all endo- and exothermic reactions of water ρ;C p Ea p γ Eγ 2EV TE ¼ ; TE ¼ a ; TV ¼ a : E structure transitions. This value correlates well with that of water spe- R 2R R ciﬁc heat of crystallization and speciﬁc heat of melting for ice (6 kJ/mol), as well as with the difference between values of Еа for evap- In Table 1, values 2.3 kJ/mol and 2.6 kJ/mol of Еа for ρ and Ср, respec- oration reactions from supercooled water and ice, i.e. 5.8 kJ/mol (see tively, correspond to values 277 K (4 °C) and 313К (40 °C) of ТE (tE), re- Table 1). Therefore, restructuring of hexagonal ice-like clusters within spectively. By substitution, in equations for γ and V, the value of Еа equal tetrahedral lattice of Н-bonds, in interval of 0 °C to 25 °C, can be, to a cer- to their average values (6 + 4.5)/2 kJ/mol and (1.8 + 1.1)/2 kJ/mol, for tain extent, regarded as continuation of processes of melting and crys- temperature intervals adjacent to 46°С and 75°С we obtain ТE (tE) equal tallization of Ih-ice. to 319 K (46 °C) and 349 K (76 °C), respectively. These results prove Fig. 9. Dependence of sound velocity in water (V) on Т (a) and its approximation (b). Data on V(T) were imported from . Arrows show the borders of intervals (in °C). Lines of trends are colorless.
746 A. Kholmanskiy, N. Zaytseva / Journal of Molecular Liquids 275 (2019) 741–748 Fig. 10. Temperature dependence compressibility (γ) on Т (а) and its fa-approximation (b). Data on γ(T) were imported from . Arrows show the borders of intervals (in °C). Lines of trends are colorless. physical viability of selected functions fа for static characteristics of spin states of water molecule, as well as by the change of clusterization water having extremal points on the corresponding TDs. ratio of its structure. The latter factor depends on the value and sign of For static characteristics, sign of reaction heat will be deﬁned by level change of reaction entropy (ΔS). For the case of exothermic reactions of population and energy of transition between rotation-vibrational and of supramolecular water structure transitions, the share of entropic con- tribution to the thermal effect (ТΔS) will increase with decreasing Т (see Еа for Ср, V and γ). In exothermic reactions of structure transforma- tion, the maximum value of parameter |ТΔS| has not to exceed reaction heat value for water crystallization, equals to 0.33 kJ/mol, at 0 °C. In the process of structure transitions at temperatures close to tE, and in the range of supercooled state, water retains its ﬂuid phase which is indica- tive of low energy of molecular links in clusters. It means that the value Fig. 11. fa-approximations for TDs of water surface tension coefﬁcient (σ). Data on σ(t) were imported from  (1) and from  (2). Fig. 13. fa-approximations for TDs of the share (δ) of free water molecules (а) and of those Fig. 12. fa-approximations for TDs of saturated vapor pressure over supercooled water (y1) bound by hydrogen bonds (b). Reference data for (а) and (b) were imported from [5,23], and ice (y2). Data on P(T) were imported from . respectively.
A. Kholmanskiy, N. Zaytseva / Journal of Molecular Liquids 275 (2019) 741–748 747 of |TΔQ| is lower than 0.33 kJ/mol and, therefore, substantially lower with those related to lifetime of Н-bond in  into account, the follow- than Еа even for static characteristics of water, at t N 0 °C. ing inequalities were considered: Taking into account conditionally additive property of structure transition mechanisms in ﬂuid water, these mechanisms have been rep- k1 ≫k−1 ; k2 ≫k−2 ; k3 ≫k−3 and k3 ≫k−1 : resented in form of kinetic scheme : With such assumptions, the following relationships can be derived from the condition of thermostable equilibrium: k1 ¼ k2 and Nk−3 ¼ k3 B: Combinations of speciﬁc reaction rates, on kinetic scheme, deﬁne values of К1 and К−1, in Eq. (1), and consequently the form of fа-func- tions, for reactions prevailing in structure transitions. In this case, value and sign of Еа (ΔQ) depend on the number and energy of Н- bonds broken and formed, in direct and reverse reactions. For dynamic On scheme, A is lattice of hydrogen bonds comprising n ice-like characteristics, the average difference between these numbers, for di- hexagonal clusters (Н2О)6, B is supramolecular structure comprising rect and reverse reactions, will vary in the range from ~1 to ~3 depend- m identical elements built of clusters with g = 2, 3, 4 and 5. N desig- ing on particular water characteristic and Т-interval. In case of static nates molecules incorporated into clusters, as well as those partici- characteristics, variation of translational-torsional states of molecules pating in self-diffusion process. Such molecules can be categorized will occur either without changing the number of Н-bonds in clusters as ‘virtually’ free of Н-bonds in moments of ‘capturing’ by a cluster or in form of their constant-energy transformations, in direct and re- or ‘jump’ . verse reactions. In a closed system, for each Т value, thermodynamic equilibrium es- Dependence of pre-exponential factor of fа-functions on Т±β may be tablishes with its particular complex of clusters and values of number N. conditioned by the following factors. Molecular mechanisms of water It is known from experience that with water temperature tending to structure transformation include translational component and, there- 100 °C n (in A) and m (in B) tend to 0 and 1, respectively, g does not ex- fore, they depend on Maxwell velocity distribution function whose ceed 2 (dimers) and share N amounts to ~20%. Taking these data along pre-exponential factor contains variable Т−3/2. Torsional oscillations Table 1 Extreme points, the form of the functions fa and the activation energy of the temperature dependences of the water characteristics. N Water characteristics fa tE (°С) Interval Еа Δt (°С) (kJ/mol) 1 Coefﬁcient self-diffusion, the data from [13,14] (D, cm2 s−1) Техр(−Еа/RT) 0; 25 −23–0 27.8 0–25 17.9 (17.1) 26–100 14.5 (13.5) 2 The spin-lattice relaxation time (T1, s) Техр(−Еа/RT) 0; 25 −20–0 28.9 5–25 19.2 30–100 13.1 3 Dynamic viscosity (η, сP) ехр(Еа/RT) 0; 25 −10–0 22.4 0–25 18.6 25–100 14.0 4 Shear viscosity (ηs, сP) ехр(Еа/RT) 0; 25 0–28 19.8 28–100 13.5 5 Time of the dielectric relaxation (τD, s) Т−1ехр(Еа/RT) 0; 25 −20–0 22.7 0–25 17.8 25–100 14.0 6 The spin-spin relaxation time of protons (T2, s) Т−1ехр(Еа/RT) 0; 25 −20–0 23.0 ~0–25 17.6 30–100 13.7 7 Isobaric heat capacity (Cp, J g−1 K−1) Техр(Еа/RT) 35 −27–0 4.5 0–35 2.6 35–100 2.7 8 Density (ρ, kg m−3) Т−1ехр(−Еа/RT) 4 −30–4 2.4 4–100 2.3 9 Sound speed (V, m s−1) Т−1/2eхр(−Еа/RT) 75 0–25 2.9 26–46 2.3 47–75 1.8 76–100 1.1 10 Isothermal compressibility (γ, bar−1) Т2ехр(Еа/RT) 46 −10–0 10 0–25 7.9 25–46 6 47–75 4.5 76–100 2.8 11 Surface tension (σ, mN/m) ехр(Еа/RT) 25 −22–25 1.3 0–35 1.5 12 Saturated vapor pressure above water (ice) (P, Pa) eхр(−Еа/RT) – −30–0 45.1 (50.9) 13 Fraction of molecules (δ, %) Free eхр(−Еа/RT) – −9–100 7.6 Bound eхр(Еа/RT) 25 0–25 17.2 30–90 11.4
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