The Molecular Basis for the Heat Capacity and Thermal Expansion of Natural Waters

The high heat capacity of seawater has been cited as why 93% of the heat trapped by anthropogenic greenhouse gases is absorbed by the ocean. Specific heats (CP) are closely tied to molecular weight. The mean molecular weight of pure water over the range 0–40 °C is 86.1–80.7 and 89.4–84.5 for seawater. Warming of water increases the kinetic energy of the molecules and induces breaking of hydrogen bonds (8.364 kJ/mol); both effects increase the volume of the fluid. Warming pure water from 0–10 °C increases the single H2O molecular form by 1.64%, accounting for 36.3% of the energy consumed. The specific heat of pure water is thus attributable (63.7%) to increasing the kinetic energy of the water, and (36.3%) to the energy required to break hydrogen bonds. For seawater, 34.7% of the energy goes to breaking hydrogen bonds while the rest (65.3%) is attributable to increasing the kinetic energy of the molecules.


Introduction
The molecular basis for trapping of heat within the atmosphere by the CO 2 molecule and related greenhouse gases has long been known, and it provides the foundation underlying all studies of modern climate change and greenhouse gas driven global warming. But the molecular basis and consequences for the large scale opposing process-the remarkable absorption by the ocean of some 93% of the heat trapped by anthropogenic greenhouse gases (Cheng et al., 2017;Rhein et al., 2013)-is poorly understood. This seems unwise given the importance of the process in the planetary heat balance. The effect is usually attributed to the large size of the ocean, the omnipresent stirring by wind, and the "high heat capacity" of water and seawater. What is meant by high heat capacity is typically stated simply within the Earth science literature as a bulk property without further inquiry or explanation. Heat capacity is an extensive property; it depends upon the total amount of the substance present. Specific heat (C P ) is the heat required at constant pressure per unit amount (typically mass or mole) to raise the temperature of a substance one degree; here we use the units of J·g −1 ·C −1 .
In marked contrast the molecular basis for the absorption by the ocean of some 25% of industrial CO 2 emissions via the reaction with carbonate ion and the resulting reduction of pH is now well known (Brewer, 1997;Gattuso & Hansson, 2011). Yet there is a substantial, albeit confusing, literature on the molecular basis for the absorption of heat as a property of water. The absorption of heat also increases the specific volume of the fluid, and this phenomenon is closely tied to molecular properties governing C p . The purpose of this paper is to introduce the topic as applicable to natural waters and to begin to quantify where possible the processes underlying this remarkable phenomenon without which we would not have a habitable planet.
Various web sites do provide some introductory material noting that the high specific heat is related to the fact that water is a hydrogen bonded fluid but typically do not develop the argument much beyond that. Water is essentially a bi-molecular fluid with free non-hydrogen bonded (nHB) H 2 O molecules in a temperature controlled dynamic equilibrium with a complex of many hydrogen bonded (HB) forms. Seawater is 96.5% water by weight and recent work (Brewer et al., 2018) has shown that it is possible to quantify the water species present so that the amounts of the nHB form, and the ensemble of HB forms, can be precisely determined. The ensemble of HB forms has no specific identity-it is a labile population of rapidly exchanging forms with local structural heterogeneities where "hydrogen bonding states associated with the formation of four hydrogen bonds by a water molecule exhibit the largest survival probability and corresponding lifetime" (Guardia et al., 2015). The specific heat of seawater is dominated by the water terms, and the specific heat of pure water is higher than that of seawater.
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10.1029/2019GL085117
Key Points: • The specific heat of water and sea water is a function of the molecular structure governed by hydrogen bonding. • Warming water and sea water breaks hydrogen bonds and releases single H 2 O molecules accounting for 36% of the heat absorbed. • The molecular weight of water ranges from 86 to 81 from 0 to 40 degrees C accounting for the remaining 64% of the specific heat.
There have been efforts to describe the specific heat of water in molecular terms (Benson & Siebert, 1992) based on a "simple two structure model" of water octamers and tetramers, but this does not seem to correlate with modern Raman spectroscopic measurements (Carey and Korenowski, 1998;Smith et al., 2005;Brewer et al., 2018). There are also efforts to solve the reverse problem-to deduce the molecular structure of liquid water based on the evidence provided by the observed specific heat (Dougherty & Howard, 1998); the topics are tightly coupled.
Here we use recent Raman spectroscopic data to provide a first-order description of the molecular processes involved when greenhouse gas generated heat is transferred to ocean waters. It will not be possible to solve the problem exactly but the lack of papers discussing this subject matter in the oceanographic literature suggests that a simple description here will be useful.

Analysis
The first question we might ask is why the specific heat of water is so large, second only to liquid ammonia. A general theory of the specific heat of liquids has not yet been achieved. On a per atom basis all substances have a specific heat close to the Dulong-Petit limit (Petit & Dulong, 1819) of 3 R per mole of atoms where R is the universal gas constant. On a per mole basis specific heat is strongly correlated with molecular weight-it takes more work to increase the kinetic energy of a large (heavy) molecular structure than a smaller (lighter) one and for water the molecular size distribution has no single or unique value. Within the water population family of HB and nHB species, molecular size and weight are strongly correlated.
The effect of heating a fluid is first to increase the kinetic energy of the molecules-immediately raising the question of which molecules are present. And in the case of water we must also consider the energy taken to break hydrogen bonds. Observations show that on warming of water and seawater the population of nHB molecules increases, and the ensemble of HB forms decreases (Walrafen, 1964(Walrafen, , 1967Brewer et al., 2017Brewer et al., , 2018. The breaking of hydrogen bonds is the inescapable cause of this process. The strength of the hydrogen bond has been reliably estimated (Silverstein et al., 2000;Smith et al., 2005) as 2 kcal/mol (8.364 kJ/mol). Thus, in order to describe ocean warming in molecular terms, we need to determine the abundance of the various molecules present at any temperature, and how many bonds are broken for a specific temperature change. This is no easy task.

The Ensemble Molecular Weight of Water
Water has no fixed or easily determinable molecular weight. The formula weight for water is 18.0153. But the single nHB H 2 O molecules represent only a small fraction of the liquid population, and that fraction is temperature dependent. The dominant HB species is the Walfrafen tetrahedral pentamer, with a molecular weight of 90.0765, but this exists within a sea of other molecules of both lower (trimer and tetramer) and higher (hexamer, heptamer, etc.) molecular weights.
There is no agreement on the ensemble or mean molecular weight for pure water although, taking the stable isotope populations into account, the various n-mers must be discretized in multiples of 18.0153. For simplicity we have assumed the most abundant n-mer to be the Walfrafen tetrahedral pentamer bounded by a Gaussian-like distribution of the other species. We note also the simultaneous presence of a planar pentamer form, also of molecular weight 90.0765. The actual distribution of the various n-mers is unknown, as Raman spectroscopy can only distinguish between nHB molecules and those within the HB species. This distribution is also the subject of much discussion (Ludwig, 2001). However, as long as the molar distribution is symmetrical and centered on the pentamer, the mean molecular weight of the clusters remains the same. One then only needs to know the ratio of nHB H 2 O species to the HB species to calculate the ensemble molecular weight (Table 1).
The ubiquitous presence of pentamers and hexamers in ocean water is readily observed in the rapid formation of gas hydrates. These HB ice-like solids hosting gas molecules are found in huge quantities in gas rich sediments and are readily synthesized in the ocean water column, and in sediment pore waters, by simple gas injection Hester et al., 2007;Kleinberg et al., 2003). The structures are based on pentamer and hexamer water species recruited from the liquid state, and a common form of the hydrate water cage is the 14 sided tetradecahedron with 12 pentagonal and 2 hexagonal faces (Sloan & Koh, 2008). Note. N-mer weight % is 100% for n = 1 and a Gaussian distribution for n = 2-8 (where a = 0.443012, b = 90.0765, and c = 18.0153) times the n-mer molecular weight. The ensemble molecular weight is calculated as the sum of n-mer mass × molecular weight products divided by the sum of the n-mer masses (plus the hydrated cations in the case of seawater). In Figure 1a we show a schematic of one possible distribution of water species in pure water at temperatures of 0 and 40°C. The ensemble molecular weight decreases slightly with rising temperature as the population of large HB forms decreases. From the example shown in Figure 1a, we estimate the ensemble molecular weight of water at 0°C to be 86.1 and at 40°C to be 80.7 (see Table 1).
In the case of seawater the presence of salts adds further complexity. The effect of dissolved salts, dominantly the cations, is to recruit a hydration shell of typically six individual water molecules arranged around the ion. These molecules are recruited from the bulk fluid and diminish the population of all HB forms and the free nHB form proportionally (Brewer et al., 2018). The molecules in the hydration shell are not hydrogen bonded, nor are they free. They are held in place by a dative covalent bond-some 20× stronger than the hydrogen bond-and the quantity appears to be independent of temperature and pressure. The matter of possible anion effects on the HB structures is far less certain, but the presence of anions does tend to affect the ordering of neighbor water molecules (Hribar et al., 2002).
The molecular weight of the salts contributes to the overall mass; but only the sulfate ion (SO 4 2− ), with a molecular weight of 96.0626, exceeds that of the water pentamer. The water component in the hydration shell must be added to the weight of the cation since the two are inseparable in aqueous solution. Thus, a sodium ion of mass 22.9898 with its six water molecule hydration shell will have a formula mass of 131.0816. From the example shown in Figure 1b, we estimate the ensemble molecular weight of seawater (S P = 35) at 0°C to be 89.4 and at 40°C to be 84.5 (see Table 1).

Specific Heats
The specific heats of water and seawater of various salinities at constant pressure (C p ) are well known. For a convenient example we use here the compact and accurate data of Cox and Smith (1959). In Figure 2 we replot their data for pure water, and for seawater (S P = 35), as a function of temperature.
The data shown in Figure 2 show two clear effects that require explanation: (1) the specific heat of pure water is higher than that of seawater, and (2) the trends with temperature are reversed; the specific heat of pure water decreases with rising temperature as the population of the single H 2 O form increases thus reducing the average molecular weight, yet the specific heat of seawater increases with rising temperature.
The overall reduction in C p for seawater versus pure water can be attributed to the recruitment of water molecules primarily from the high molecular weight HB pool into the hydration shells of cations (Brewer et al., 2018) where it is present in single molecular form. The anion contributions to changing water structures are less well known, but they plainly represent a "local structural heterogeneity" by which the tendency of the water molecules to form large temporary structures is disrupted. Overall there are simply fewer HB pentamers in a kg of seawater than in a kg of pure water and this accounts for a large fraction of the reduced specific heat of salt waters. Beyond this the seawater trend towards increasing C p with rising temperature requires further explanation.
These data reveal the competing effects of two simultaneous processes: the ensemble molecular weight of water in seawater must decrease (and the specific volume increase) with rising temperature just as for pure water for in both cases the population of nHB forms increases. But this effect is more than compensated for by the increasing energy required to thermally excite the water molecules in the cation solvation shell. In place these molecules possess vibrational energy but not translational energy independent of their associated Figure 2. Specific heats (C P , Cox & Smith, 1959) for pure water (blue diamonds) and seawater (S P = 35, red dots) as a function of temperature (a) and the molar ratio of HB to nHB water molecules (b). Pure water has a higher C P which decreases as temperature rises while seawater has a smaller C P that increases with increasing temperature and decreasing salinity.
cation-they must move together as a heavy unit. The dative covalent bond makes it increasingly difficult for the hydrated cations to achieve the mean kinetic energy of the water molecules in the bulk fluid.

Calculation
From the background information given above it is possible to make some estimates of the relative importance of the two major components that account for the unusually high specific heat of water and seawater. We need make no judgement as to which HB forms are disrupted by warming-the entire population is shifted. Walrafen (1964) noted that "two hydrogen bonds are broken when one H 2 O molecule is liberated in the complete break-down of a tetrahedral network" but such an assumption is not needed here. We can simply take the observed quantity of single H 2 O molecules formed and the enthalpy of the hydrogen bond. Spectroscopic observations show that warming of water produces this effect and the molar quantities can now be calculated (Brewer et al., 2018).

Pure Water
Pure water presents the simplest case and here we provide a first-order estimate of the changes forced by the uptake of heat. The formal concentration of water in water is 55.5084 M and at the freezing point we have 90% of the molecular species in HB forms, and 10% as single H 2 O molecules (Walrafen, 1964).
It takes 42.042 kJ to heat 1 kg of water from 0 to 10°C. Taking data from Brewer et al. (2018)  This was accomplished by the input of 15.27 kJ. Thus, the fraction of the specific heat that can be assigned to hydrogen bond breaking is some 36.3% of the total-the remainder is due to the energy required to increase the kinetic energy of molecules with an average molecular weight of~85.5.

Seawater
For seawater of S P = 35, we find very similar conclusions. It takes 39.86 kJ to heat 1 kg of seawater from 0 to 10°C. Taking data from Table 1, at S P = 35, we find that this liberates 0.827 moles of single H 2 O molecules which can be accomplished by the input of some 13.83 kJ-less than is required for the same mass of pure water. There is simply a smaller quantity of HB forms in seawater than in pure water. Nonetheless, the phenomenon of hydrogen bond breaking on ocean warming accounts for some 34.7% of the total heat absorbed. The bulk of the specific heat is due to the large molecular weights of the combined population of HB forms and hydrated cations. The difference is that the HB population is responsive to changing temperature whereas the hydrated cation population is independent of temperature. Thus, this component plays an increasing role as the ocean warms, and its greater molecular weight requires more kinetic energy.
From the data presented in Figure 2a and where 35.179 is the mass of dissolved salts, 0.54198 is the number of moles of dissolved cations, 6 is the solvation number, and 18.0153 is the molecular weight of the solvation water molecules. The variation of the C P of the dissolved salts as a function of temperature is shown in Figure 3.

Conclusions
The evidence presented here is further testimony to the huge and under-recognized role of the hydrogen bond in Earth Science, for without this, water would not be in the liquid state. For specific heat the bonding plays two roles: first in creating an average molecular weight of water of approximately 83-87 at 20°C (depending upon the salinity),~4.7 times that of the single molecule; and second in the energy required to break apart the dominant HB forms. Although heating of water by 10°C results in an increase of only 1.64% in the population of the nHB form, this accounts for a remarkable 35-36% of the energy input. The increase in specific volume with warming is a direct result of these processes, and while we cannot provide a precise analysis it seems likely that something similar to this ratio holds true for the volumetric changes as well.
In the case of seawater, the recruitment of single water molecules from the ensemble population into the hydration sphere around a cation reduces the population of the large molecular weight HB forms more than the nHB form. This results in the lower specific heat of seawater compared to pure water within the oceanic temperature range. The retention of water molecules in the hydration shell by the dative covalent bond makes it increasingly difficult for the cation molecular population to achieve the mean kinetic energy defined by rising temperature when the scaling factor is provided by 1/T where T is in Kelvins.
The ocean absorbs some 25% of fossil fuel CO 2 emissions; in doing so carbonate ion is consumed and this observably reduces the buffer capacity as ocean pH decreases. In contrast the ocean absorbs some 93% of all anthropogenic greenhouse gas trapped heat-and for salt water the specific heat actually increases by a small amount as the ocean warms.