1 Introduction

This study explores the differences in the volumetric properties between proteated and deuterated water as a function of temperature, especially as these differences relate to the difference in the extent of hydrogen bonding between them. In a sense, it serves as a prequel to this laboratory’s efforts to investigate the solvation properties of water [1,2,3,4,5] by considering the thermodynamic properties and modeling of pure proteated and deuterated water as a function of temperature. Each of the previous studies compares the degree of hydrogen bonding in the bulk water to that in the vicinity of the solute molecule, which is referred to as either the “solvated” or “coordinated” water. The conclusion drawn from these studies is that water in the solute’s hydrophobic region exhibits a greater degree of hydrogen bonding than that of the bulk water. Such regions are referred to as “structure makers.”. The opposite obtains for the hydrophilic regions which are referred to as “structure breakers.” Simple electrolytes are expected to be entirely hydrophilic leading to the expectation that the “coordinated water” will exhibit a lesser degree of hydrogen bonding than the bulk water. Previously, the volumetric properties of aqueous solutions of hexaamminecobalt(III) nitrate were determined in both proteated and deuterated water as functions of temperature. The results were supportive of the notion that hydrophilic regions are “structure breaking” and the assumption that deuterated water exhibits a greater degree of hydrogen bonding than does proteated water. In that paper, the following relationship was developed (ref. [4], Eq. 5):

$$q_{0} = \frac{{\left( {V_{ \ 2D}^{0} - V_{2H}^{0} } \right)}}{{\left( {V_{1H}^{ \cdot } - V_{1D}^{ \cdot } } \right)}},$$

where \(q_{0}\) is the coordination number for water; \(V_{2D}^{0}\) and \(V_{2H}^{0}\) are the solute’s partial molar volumes at infinite dilution in deuterated and proteated water, respectively; and \(V_{1H}^{ \cdot }\) and \(V_{1D}^{ \cdot }\) are the molar volumes of pure proteated and deuterated water, respectively. Thus \(q_{0}\) at a given temperature depends upon the difference between the molar volumes of pure proteated and deuterated water. The aim of this study, in part, is to precisely determine this difference.

The effect of a solute’s hydrophobic region is perhaps most clearly demonstrated in the volumetric study of aqueous solutions of a homologous series of primary alcohols which shows that the extent of hydrogen bonding of water in the vicinity of the aliphatic group increases systematically with size of the aliphatic group compared to bulk water [5]. Information about the extent of hydrogen bonding in pure and, by extension, bulk water, would contribute to a more complete understanding of aquation in both proteated and deuterated water.

Since the intention of this study is to closely compare the volumetric properties of proteated and deuterated water, their densities as functions of temperature were measured in this laboratory under nearly identical experimental conditions. This comparison allows for the development of a heuristic molecular model that rationalizes the difference between the extents of hydrogen bonding expected for proteated and deuterated water as a function of temperature.

The unusual volumetric feature of water at ca. 4 °C has been known and remarked upon for a long time [1, 3,4,5,6,7,8,9,10,11,12]. Less noticed and discussed are thermodynamic properties related to this volumetric feature. Any molecular description of water’s volumetric properties must also take into account these related properties.

At ambient pressure, the molar volumes of proteated and deuterated water exhibit minima at ca. 4 °C and ca. at 11 °C, respectively (see Fig. 1). Their isothermal compressibilities remain positive over the entire liquid temperature range [13], thus \(\left( {\frac{\partial V}{{\partial P}}} \right)_{T}\) < 0 at all temperatures which is consistent with physical intuition and is standard behavior for liquids.

Fig. 1
figure 1

The molar volumes of proteated (diamonds) and deuterated (squares) water at the measured temperatures

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The expectations for single-component liquids are as follows:

$$\left( {\frac{\partial V}{{\partial T}}} \right)_{P} > 0,$$
$$\left( {\frac{\partial P}{{\partial T}}} \right)_{V} > 0,$$
$$\left( {C_{P} - C_{V} } \right) > 0,$$
$$\left( {\frac{\partial S}{{\partial V}}} \right)_{T} > 0,$$
$$\left( {\frac{\partial S}{{\partial P}}} \right)_{T} < 0.$$

These expectations are met by most single-component liquids and are conveniently rationalized using statistical thermodynamic arguments as well as physical intuition.

It can be shown that the difference between the constant-pressure and constant-volume heat capacities may be written in several ways [c.f. 14], to wit

$$\left( {C_{P} - C_{V} } \right) = - T\frac{{\left( {\frac{\partial V}{{\partial T}}} \right)_{P}^{2} }}{{\left( {\frac{\partial V}{{\partial P}}} \right)_{T} }},$$
(1)
$$\left( {C_{P} - C_{V} } \right) = T\left( {\frac{\partial V}{{\partial T}}} \right)_{P} \left( {\frac{\partial P}{{\partial T}}} \right)_{V} ,{\text{ and}}$$
(2)
$$\left( {C_{P} - C_{V} } \right) = - T\left( {\frac{\partial P}{{\partial T}}} \right)_{V}^{2} \left( {\frac{\partial V}{{\partial P}}} \right)_{T} .$$
(3)

These equations are related through Euler’s cyclic chain rule. Since \(\left( {\frac{\partial V}{{\partial P}}} \right)_{T}\) is always negative (see above), the difference between the heat capacities is always positive except when \(\left( {\frac{\partial V}{{\partial T}}} \right)_{P}\) is zero (see Eq. 1). Thus \(\left( {C_{P} - C_{V} } \right) > 0\) at all temperatures except at ca. 4 °C where the difference is identically zero. Equations 2 and 3 prove that \(\left( {\frac{\partial V}{{\partial T}}} \right)_{P}\) and \(\left( {\frac{\partial P}{{\partial T}}} \right)_{V}\) are both negative below ca. 4 °C, equal to zero at the minimum, and positive at temperatures above the minimum. Thus, the unusual volumetric feature of water is accompanied by unusual behavior for the heat capacity difference as well as for \(\left( {\frac{\partial P}{{\partial T}}} \right)_{V}\).

The appropriate Maxwell relations reveal how the entropy varies with pressure and volume under isothermal conditions. Thus,

$$\left( {\frac{\partial P}{{\partial T}}} \right)_{V} = \left( {\frac{\partial S}{{\partial V}}} \right)_{T} {\text{and}}$$
$$\left( {\frac{\partial V}{{\partial T}}} \right)_{P} = - \left( {\frac{\partial S}{{\partial P}}} \right)_{T} ,$$

indicating that the entropy changes under isothermal conditions exhibit sign changes at ca. 4 °C and are equal to zero at the volumetric minimum. Below ca. 4 °C they feature the following unusual properties:

$$\left( {\frac{\partial S}{{\partial V}}} \right)_{T} < 0\,{\text{and}}$$
$$\left( {\frac{\partial S}{{\partial P}}} \right)_{T} > 0.$$

All conclusions drawn from proteated water hold for deuterated water except for the latter the molar volume minimum is at ca. 11 °C. Table 1 summarizes the conclusions.

Table 1 Summary of the characteristics of the thermodynamic properties
Full size table

2 Materials

Deuterated water was obtained from Sigma-Aldrich (99.96 atom % D) and used without further purification. Deionized proteated water was used. Our departmental ion exchange system from Evoqua Water Technologies produces de-ionized water with a measured conductance of 0.0550 ± 0.0001 microSiemans/cm.

3 Density Measurements

Density was measured over a range of 1.00 (or 5.00 in the case of deuterated water) to 30.00 (± 0.02)  °C with an Anton Paar DMA 4500 density meter with a separate Peltier temperature controller. The techniques, standardization, accuracy, and precision are given elsewhere [1]. The range for the density is 0.99992 to 1.13048 \({\text{g}}\cdot{\text{cm}}^{ - 3}\) which falls well within the manufacturer’s recommendations for the standardization methods used here. The measured densities of pure water at 20.00 °C were identical for all sets (0.99821 \({\text{g}}\cdot{\text{cm}}^{ - 3}\)) and agree with the accepted value for water at 20 °C of 0.998206 \({\text{g}}\cdot{\text{cm}}^{ - 3}\) (one additional significant figure). The vendor reports a density repeatability of \(1 \times 10^{ - 5}\) \({\text{g}}\cdot{\text{cm}}^{ - 3}\) which results in an uncertainty in molar volumes of \(< 1 \times 10^{ - 4}\) \({\text{mL}}\cdot{\text{mol}}^{ - 1}\) for the molar masses of proteated and deuterated water as well as for the so-called intrinsic and expansive volumes (see below).

The standardization procedure allows the user to either enter height above sea level or the atmospheric pressure measured using a barometer. The latter option was used for these measurements. In addition, the pressure was measured after the density measurements and found to be identical to the value used for the standardization procedure.

Since the object of this study is a detailed comparison of the volumetric properties of proteated and deuterated water, both sets of data were gathered on the same day ensuring nearly identical experimental conditions. A previous study of the densities of proteated and deuterated water measured the densities for the deuterated water at intervals of 5 °C which was not deemed sufficient for the aim of this study [13]. The differences between the densities of proteated water measured in this study and those in Kell’s study [13] exhibit differences below 0.00003 \({\text{g}}\cdot{\text{cm}}^{ - 3}\) in absolute value over the entire range of temperature with no discernible trend with temperature.

4 Results

Tables 2 and 3 report the densities and molar volumes of the proteated and deuterated water, respectively, at various temperatures. Each set of molar volumes and densities were fit to a quartic polynomial in temperature (°C) using the Excel ANOVA routine. Thus,

$$V = a + bt + ct^{2} + dt^{3} + et^{4} \,{\text{and}}$$
$$\rho = a^{\prime} + b^{\prime}t + c^{\prime}t^{2} + d^{\prime}t^{3} + e^{\prime}t^{4} .$$
Table 2 The densities and molar volumes of \({\text{H}}_{{2}} {\text{O}}\) at the measured temperatures
Full size table
Table 3 The densities and molar volumes of \({\text{D}}_{{2}} {\text{O}}\) at the measured temperatures
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Tables 4 and 5 report the values of the polynomial coefficients. The molar volumes over the entire temperature ranges for proteated and deuterated water calculated using either the coefficients for molar volume or for density vary at most by 0.00010 \({\text{mL}}\cdot{\text{mol}}^{ - 1}\) from the experimental values, and this in only one case. Most deviations are below 0.00005 \({\text{mL}}\cdot{\text{mol}}^{ - 1}\). Moreover, there is no evidence that the deviations are more prevalent at either high or low temperature.

Table 4 The coefficients for the quartic regressions of the molar volumes for proteated and deuterated water
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Table 5 The coefficients for the quartic regressions of the densities for proteated and deuterated water
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The temperature derivatives of the molar volumes were calculated employing the coefficients for the molar volume. Using the coefficients of the density yields nearly identical values with those of the molar volume. The temperatures for the volumetric minima of proteated and deuterated water were estimated to be 3.9 and 11.2 °C, respectively. These values compare favorably with the literature values of 4.0 and 11.2 °C [9].

5 Discussion

Over the course of many years, a picture for liquid water has emerged suggesting that water be considered as a “mixture” of species, one set of which is comprised of water molecules exhibiting extensive hydrogen bonding networks sometimes called “icebergs” or ice-like structures while another set of which is comprised of populations exhibiting little or no hydrogen bonding herein referred to as “collapsed” structures [1, 3,4,5,6,7,8,9,10,11,12, 15, 16]. In this spirit, the following simple two-species heuristic model is presented in anticipation of rationalizing the volumetric features described above. The terms used in this model are described here:

  1. 1.

    The average molar volumes for the ice-like regions (\(V_{i}\)) and for the collapsed regions (\(V_{c}\)) and

  2. 2.

    The fractions of molecules contained in each region (\(f_{i}\) and \(f_{c}\), respectively).

The molar volumes for proteated (\(V_{H}\)) and deuterated (\(V_{D}\)) water may be reported as

$$V_{H} = f_{c,H} V_{c,H} + f_{i.H} V_{i,H} \,{\text{and}}$$
(4)
$$V_{D} = f_{c,D} V_{c,D} + f_{i.D} V_{i,D}$$
(5)

where the additional subscripts, “H” and “D,” indicate proteated and deuterated species, respectively. It is plausible to assume that the only difference between proteated and deuterated water is in the fractions of the collapsed and ice-like populations such that \(f_{iD} > f_{iH}\) and \(f_{cD} < f_{cH}\) since the D–O hydrogen bond is stronger than that for the H–O hydrogen bond [7, 9]. Given that the sum of fractions is one, the eqs. 4 and 5 become

$$V_{H} = V_{c} + f_{i,H} \left( {V_{i} - V_{c} } \right) = V_{c} + f_{i,H} \Delta V_{ic} \,{\text{and}}$$
(6)
$$V_{D} = V_{c} + f_{i,D} \left( {V_{i} - V_{c} } \right) = V_{c} + f_{i,D} \Delta V_{ic}$$
(7)

where it is assumed that \(V_{i,H} = V_{i,D} = V_{i}\) and \(V_{c,H} = V_{c,D} = V_{c}\).

This assumption is, in part, based upon crystallographic studies of solid proteated and deuterated water which show that the lengths of the H–O and D–O hydrogen bonds differed by a mere 0.0001 nm [9]. Further, there is evidence that \(V_{i} > V_{c}\) [1, 3,4,5, 15, 16] which leads to the conclusion that \(\Delta V_{ic} > 0\) and the prediction that \(V_{D} > V_{H}\) over the entire temperature range which is the experimental finding (see Fig. 1). Vedamuthu et al. employed the same set of assumptions in their modeling of the volumetric properties of proteated and deuterated water [17].

The temperature derivative of \(V_{H}\) gives

$$\left( {\frac{{\partial V_{H} }}{\partial T}} \right)_{P} = \left( {\frac{{\partial V_{c} }}{\partial T}} \right)_{P} + \Delta V_{ic} \left( {\frac{{\partial f_{i,H} }}{\partial T}} \right)_{P} + f_{i,H} \left( {\frac{{\partial \Delta V_{ic} }}{\partial T}} \right)_{P} .$$
(8)

The first term on the right is expected to be positive owing to normal thermal expansion for this population. The second term is expected to be negative because the fraction of hydrogen bonding decreases with increasing temperature since hydrogen bonding is exothermic. The sign of the third term depends upon the sign of \(\left( {\frac{{\partial \Delta V_{ic} }}{\partial T}} \right)_{P}\) which may be written as \(\left( {\frac{{\partial V_{i} }}{\partial T}} \right)_{P} - \left( {\frac{{\partial V_{c} }}{\partial T}} \right)_{P}\). It has been argued previously that hydrogen bonding of the ice-like population reduces the degree of its thermal expansion compared to the thermal expansion of the collapsed population which leads to the conclusion that the sign of the third term is negative [3,4,5]. Thus, this situation affords the possibility of both positive and negative values for the temperature derivative of molar volume and depends upon temperature. At the very least one expects that the positive first term in Eq. 8 to increase with temperature since the second temperature derivative of the molar volume of a species is normally positive for reasons discussed in previous work [5, 6]. It is not clear how the second and third terms change with temperature. In any event, this model provides a plausible rationalization for the finding that the slope of the temperature derivative of molar volume is negative at low temperature and positive at high temperature exhibiting a minimum. The same set of arguments obtains for deuterated water.

The so-called second thermodynamic equation of state [6] may be written as

$$V = \left( {\frac{\partial H}{{\partial P}}} \right)_{T} + T\left( {\frac{\partial V}{{\partial T}}} \right)_{P} = V_{{\text{int}}} + V_{\exp }$$
(9)

where it is useful to designate the first term after the equal sign as the intrinsic volume (\(V_{{\text{int}}}\)) and the second term as the expansive volume (\(V_{\exp }\)). The Appendix offers a rationale for these subscripts. Figure 2 shows \(V_{\exp ,H}\) (\(= T\left( {\frac{{\partial V_{H} }}{\partial T}} \right)_{P}\)) and \(V_{\exp ,D}\) as functions of temperature highlighting the properties described above. It is worth noting that, for the region where the overall temperature derivative of molar volume and therefore \(V_{\exp }\) (\(= T\left( {\frac{\partial V}{{\partial T}}} \right)_{P}\)) are negative, each population—the ice-like and collapsed—experience positive values, i.e., both \(\left( {\frac{{\partial V_{i} }}{\partial T}} \right)_{P}\) and \(\left( {\frac{{\partial V_{c} }}{\partial T}} \right)_{P}\) are positive but \(\left( {\frac{{\partial \Delta V_{ic} }}{\partial T}} \right)_{P}\) is negative (see Eq. 8).

Fig. 2
figure 2

The expansive volumes (\(V_{\exp } = T\left( {\frac{\partial V}{{\partial T}}} \right)_{P}\)) for proteated (diamonds) and deuterated (squares) water at the measured temperatures

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As noted earlier, in the region where \(\left( {\frac{\partial V}{{\partial T}}} \right)_{P} < 0\), then \(\left( {\frac{{\partial S_{{}} }}{\partial P}} \right)_{T}\) > 0. This may be rationalized by assuming the following:

  1. 1.

    The molar entropy for the collapsed region is greater than that for the ice-like region. Intuitively, the ice-like population is more geometrically structured than the collapsed population.

  2. 2.

    Increasing the pressure isothermally promotes the conversion of ice-like water to collapsed water, i.e., \(\left( {\frac{{\partial f_{i} }}{\partial P}} \right)_{T} < 0\). Above ca. 4 °C, the conversion is less pronounced because of the reduced ice-like population. Intuitively, increasing pressure promotes the “collapse” of the ice-like structures.

  3. 3.

    Applying the chain rule, \(\left( {\frac{\partial S}{{\partial V}}} \right)_{T} = \frac{{\left( {\frac{\partial S}{{\partial P}}} \right)_{T} }}{{\left( {\frac{\partial V}{{\partial P}}} \right)_{T} }}\), shows that the entropy changes under isothermal conditions take different signs at all temperatures.

A mathematical consequence for the region where the liquid exhibits a negative slope for molar volume is that the intrinsic volumes exhibit values greater than molar volumes (see Fig. 3). This again is a consequence of the increase in the fraction of molecules in the collapsed population with increasing temperature.

Fig. 3
figure 3

The intrinsic volumes (\(V_{{\text{int}}} = V - T\left( {\frac{\partial V}{{\partial T}}} \right)_{P}\)) for proteated (diamonds) and deuterated (squares) water at the measured temperatures

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Subtracting Eq. 5 from Eq. 4 gives.

$$V_{diff} = V_{D} - V_{H} = \left( {f_{i,D} - f_{i,H} } \right)\Delta V_{ic} = \left( {\Delta f_{i} } \right)\Delta V_{ic} .$$
(10)

Figure 4 reports the subtraction of their quartic polynomial regressions as a function of temperature. This approach was necessary since the measured densities for the two species were not always done at the same temperature. For the cases where they were, the values from the regressions and those from the measured data are basically indistinguishable.

Fig. 4
figure 4

The difference between the molar volumes of deuterated and proteated water (\(V_{diff} = V_{D} - V_{H}\)) at various temperatures

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\(V_{diff}\) is a positive though decreasing function of temperature exhibiting no extrema even though the molar volumes of both the proteated and deuterated species exhibit minima at different temperatures. The positive values for \(V_{diff}\) are predicted by Eq. 10 since both \(\Delta f_{i}\) and \(\Delta V_{ic}\) are positive. Furthermore, as the temperature increases, \(\Delta f_{i}\) decreases as the population of the ice-like molecules diminishes which, in turn, results in a decrease in \(V_{diff}\). This suggests that as the temperature increases the properties of the proteated and deuterated water converge. This convergence is observed in all related terms (see below).

Taking the temperature derivative of \(V_{diff}\) gives

$$\left( {\frac{{\partial V_{diff} }}{\partial T}} \right)_{P} = \Delta f_{i} \left( {\frac{{\partial \Delta V_{ic} }}{\partial T}} \right)_{P} + \Delta V_{ic} \left( {\frac{{\partial \Delta f_{i} }}{\partial T}} \right)_{P} ,$$
(11)

where the first term after the equal sign is expected to be negative as discussed above. The sign of the second term depends upon the sign of \(\left( {\frac{{\partial \Delta f_{i} }}{\partial T}} \right)_{P}\). Since \(f_{i,D} > f_{i,H}\), it is plausible to conclude that \(\left( {\frac{{\partial f_{i,D} }}{\partial T}} \right)_{P} < \left( {\frac{{\partial f_{i,H} }}{\partial T}} \right)_{P}\) reflecting the notion that the decrease in the fraction of molecules engaged in ice-like structures in deuterated water is greater than in proteated water because of the larger strength of the D–O hydrogen bond. This leads to the expectation that \(V_{diff}\) monotonically decreases with temperature as discussed above. Figure 4 confirms this expectation.

As temperature increases, the fraction of molecules of the collapsed population increases and, as discussed above, does so more rapidly for deuterated than for proteated water even though the fraction is lower for the deuterated species at all temperatures. This is consistent with the finding that the molar, expansive, and intrinsic volumes converge as temperature increases. Figures 5 and 6 reveal this behavior for the differences between deuterated and proteated water of their expansive and intrinsic volumes. To summarize, the inequalities

$$\begin{gathered} V_{D} > V_{H} , \hfill \\ V_{{D,{\text{int}} }} > V_{{H,{\text{int}} }} , \hfill \\ V_{D,\exp } < V_{H,\exp } \hfill \\ \end{gathered}$$
Fig. 5
figure 5

The difference between the expansive volumes of deuterated and proteated water (\(V_{\exp ,D} - V_{\exp ,H}\)) at various temperatures

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Fig. 6
figure 6

The difference between the intrinsic volumes of deuterated and proteated water (\(V_{{{\text{int}} ,D}} - V_{{{\text{int}} ,H}}\)) at various temperatures

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hold throughout the temperature region but the differences diminish with increasing temperature.

This simple approach where water is considered to be a “two-species mixture,” the composition of which is temperature-dependent, might prove useful for other systems. Another area where this analysis may prove useful is the temperature-dependent isobaric heat capacity for water which also exhibits a minimum [18].

6 Appendix

There are two well-established thermodynamic equations, developed from first principles, describing pressure and molar volume [19]. They are

$$P = - \left( {\frac{\partial U}{{\partial V}}} \right)_{T} + T\left( {\frac{\partial P}{{\partial T}}} \right)_{V} \,{\text{and}}$$
(12)
$$V = \left( {\frac{\partial H}{{\partial P}}} \right)_{T} + T\left( {\frac{\partial V}{{\partial T}}} \right)_{P} .$$
(13)

\(\left( {\frac{\partial U}{{\partial V}}} \right)_{T}\) is often called the internal pressure, designated here as \(P_{{\text{int}}}\), because it is associated with molecular attractions in liquids at or near ambient conditions. The values for this term are positive and considerably larger than P under these conditions and their values normally decrease with increasing pressure [20]. This behavior suggests that the internal pressure is not merely a function of molecular attractions but of repulsions also where the effect of attractions dominate for liquids near ambient conditions. The second term on the right of Eq. 12, where \(T\left( {\frac{\partial P}{{\partial T}}} \right)_{V} = T\left( {\frac{\partial S}{{\partial V}}} \right)_{T}\), employing the pertinent Maxwell relation, may then be considered as a thermal expansive pressure representing the natural tendency for molecules to separate due to thermal motion. Note that this term is related to entropic change consistent with chemical intuition. Clearly, for most liquids, it is also positive and at least as large as the internal pressure under near ambient conditions given that the exerted pressure is negligible compared to the internal pressure. For a perfect gas the first term on the right side of Eq. 12 may be shown to be identically zero consistent with the assumption that the molecules of a perfect gas exhibit neither attractive nor repulsive forces.

Equations 12 and 13 may be combined to give

$$\left[ {P + \left( {\frac{\partial U}{{\partial V}}} \right)_{T} } \right]\left[ {V - \left( {\frac{\partial H}{{\partial P}}} \right)_{T} } \right] = T^{2} \left( {\frac{\partial V}{{\partial T}}} \right)_{P} \left( {\frac{\partial P}{{\partial T}}} \right)_{V} .$$
(14)

Both \(\left( {\frac{\partial U}{{\partial V}}} \right)_{T}\) and \(\left( {\frac{\partial H}{{\partial P}}} \right)_{T}\) are identically zero for a perfect gas. Comparing this equation to the van der Waals (vdW) [21] empirical equation of state for gasses and, in some rare instances, liquids may provide a basis for gaining insight into some of the terms in Eq. 12. The empirical vdw equation may be written as

$$\left( {P + \frac{a}{{V^{2} }}} \right)\left( {V - b} \right) = RT$$
(15)

where a is generally regarded as a measure of molecular attraction and b, the excluded volume, as a measure of repulsion. It has been noted that the \(\frac{1}{{V^{2} }}\)-term has units of \({\text{m}}^{ - 6}\) which is the functionality of the Lennard-Jones attractive potential [22]. The association of the \(\frac{a}{{V^{2} }}\)-term with \(\left( {\frac{\partial U}{{\partial V}}} \right)_{T}\), the internal pressure, is strongly suggestive.

The excluded volume is equal to four times the molecular volume of the gas molecule assumed to be a sphere, and so determined as the molecule moves creating a cylinder of exclusion for other gas molecules. Clearly, an alternate interpretation of the repulsion is required for liquids since intermolecular distances are less than molecular diameters. Still, it is tempting to associate the vdw b-term with \(\left( {\frac{\partial H}{{\partial P}}} \right)_{T}\) and to interpret it as an apparent intrinsic volume occupied by the molecule and, perhaps, an additional contribution (see below). On the other hand, the second term after the equal sign of Eq. 12, \(T\left( {\frac{\partial V}{{\partial T}}} \right)_{P}\), is considered to be a measure of normal thermal expansion which is consistent with the Maxwell relation, \(\left( {\frac{\partial V}{{\partial T}}} \right)_{P} = - \left( {\frac{\partial S}{{\partial P}}} \right)_{T}\).

Considering these associations, one may write

$$V = \left( {\frac{\partial H}{{\partial P}}} \right)_{T} + T\left( {\frac{\partial V}{{\partial T}}} \right)_{P} = V_{{\text{int}}} + V_{\exp } ,$$

where the subscripts after the equal sign refer to the intrinsic and expansive volumes.

A feature of Eq. 12 is illustrated by rearranging the equation to give

$$P_{{\text{int}}} = \left( {\frac{\partial U}{{\partial V}}} \right)_{T} = T\left( {\frac{\partial P}{{\partial T}}} \right)_{V} - P.$$

Below ca. 4 °C for proteated and ca. 11 °C for deuterated water, \(P_{{\text{int}}}\) is negative [20]. This unexpected result is clearly related to the negative expansive volume exhibited for these two species below their volumetric minima (see Fig. 2).