HEAT OF DISSOLUTION OF GASES IN WATER

N. P. Kurin and S. A. Sigov

Modern chemical industry is very rich in processes involving the interaction of gaseous and liquid phases. In particular, the treatment of gases with water is now carried out on a large industrial scale, and therefore a comprehensive study of this case of interaction between the gaseous and liquid phases is highly desirable not only from a theoretical but also from a practical point of view.

On questions of the solubility of gases in water there is a considerable amount of experimental material in the literature; as for the study of the thermal effects of dissolution, however, this aspect of the question has not yet received sufficiently complete treatment. Nevertheless, knowledge of the thermal effects of the dissolution of gases in water is of interest both from the standpoint of filling the gap that exists in this field and from the standpoint of using these quantities to elucidate certain energy questions connected with the mechanism of the process of absorption of gases by water.

In the present work an attempt is made to obtain, by calculation, those thermal effects which accompany the process of dissolution of gases in water, on the basis of the temperature dependence of the solubility of gases at atmospheric pressure.

Most of the gases discussed in the present work have low solubility and therefore, at such a low partial pressure as 1 atm. and in the temperature interval \(0—100^\circ\mathrm{C}\), they give only very dilute aqueous solutions, which may be regarded as ideal solutions.

As is known\(^1\), for ideal solutions thermodynamics gives the following dependence of the mole fraction of the dissolved substance on temperature:

\[ \left(\frac{\partial \ln N_2}{\partial T}\right)_P=\frac{Q}{RT^2}, \tag{1} \]

where \(N_2\) is the mole fraction of the dissolved gas,
\(Q\) is the differential heat of dissolution of one mole of gas at constant pressure, in cal/mole,
\(T\) is the absolute temperature,
\(R\) is the gas constant, equal to \(1.987\) cal/degree.

In turn,

\[ N_2=\frac{n_2}{n_1+n_2}, \]

where \(n_2\) is the number of moles of gas dissolved under the given conditions in \(n_1\) moles of water.

Passing from natural logarithms to common logarithms, and also modifying equation (1) somewhat, we obtain equation (2)

\[ \left[\frac{\partial \log N_2}{\partial \left(\frac{1}{T}\right)}\right]_P = -\frac{Q}{2.303R}. \tag{2} \]

Equation (2) indicates that the differential heat of dissolution of a gas in a liquid at constant pressure and at a given temperature will be equal to

the first derivative of \(\log N_2\) with respect to \(\left(\frac{1}{T}\right)\), multiplied by \(2.303R\).

If \(\left(\frac{1}{T}\right)\) is plotted on the abscissa axis and \(\log N_2\) on the ordinate axis, then, if the heat of dissolution of a gas in water is independent of temperature, a straight line should be obtained; the tangent of the angle of inclination of this line (determined, for example, graphically), multiplied by \(2.303R\), must be equal to the heat of dissolution.

However, such constructions carried out by us (Figs. 1, 2, 3, 4, and 5) for 23 gases over a wide range of temperatures did not yield straight lines. This circumstance indicated the presence of variation of the heat of dissolution with temperature and, thus, the possibility of using the above-described method for finding the desired quantities was eliminated.

The data on the solubility of gases in water necessary for constructing the curves \(\log N_2 = f\left(\frac{1}{T}\right)\) were taken by us

for CO from the works of Winkler \(^{2}\)

Determination of the tangent of the angle of inclination of the tangent, or of the first derivative of \(\log N_2\) with respect to \(\left(\frac{1}{T}\right)\), for a given gas can be carried out either by differentiating the empirically found equation \(\log N_2 = f\left(\frac{1}{T}\right)\) or by using graphical methods for this purpose. Establishing empirical equations is, first, associated with a great expenditure of time and, second, those equations based on the law of the parabola, while giving good results on integration, can lead to completely incorrect results on differentiation, differing not only in absolute magnitude but even in sign.

Fig. 1. Dependence of the mole fraction of dissolved gas in an aqueous solution on temperature for CO; H₂; Ar; Kr; NO. Gas pressure = 1 atm.

Fig. 2. Dependence of the mole fraction of dissolved gas in an aqueous solution on temperature for SO₂; Br₂; H₂S; Xe; Gas pressure = 1 atm.

Fig. 3. Dependence of the mole fraction of dissolved gas in an aqueous solution on temperature for \(C_2H_2\); \(CO_2\); \(N_2O\); \(COS\); Rn; \(C_2H_4\). \(P_{\text{gas}} = 1\) atm.

Fig. 4. Dependence of the mole fraction of dissolved gas in an aqueous solution on temperature for \(O_3\) and \(CH_4\). \(P_{\text{gas}} = 1\) atm.

Fig. 5. Dependence of the mole fraction of dissolved gas in an aqueous solution on temperature for \(O_2\), air, Ne, and He. \(P_{\text{gas}} = 1\) atm.

Fig. 6.

In view of the foregoing, for solving the problem posed we used a very elegant, comparatively rapid, and quite accurate graphical “mirror” method. This method consists in the following: a diagram of sufficiently large dimensions is constructed (Fig. 6), expressing the dependence of \(\log N_2\) on \(\left(\frac{1}{T}\right)\). To find the first derivative at the point (0) corresponding to the temperature at which it is desired to find the heat of dissolution, a tangent is drawn and then, from the inclination of the latter to the abscissa axis \(\left(\frac{1}{T}\right)\), the tangent of the angle of inclination is determined. The construction of the tangent in this method is carried out with the aid of a mirror ruler. For this purpose the ruler is placed on edge so that the planes of the drawing and of the ruler (ABCD) are perpendicular to one another, and then it is moved about a vertical axis passing through the point (0) until the images of the curve in front of the mirror, in the mirror, and behind the mirror merge into one smooth curve. In this case the line of sight must be perpendicular to the line AD. With the ruler thus positioned, a straight line RS is drawn on the drawing; this is nothing other than the perpendicular to the desired tangent. Along this perpendicular, using a triangle, the tangent itself (ST) is drawn. Incorrect positions of the mirror ruler for points O₁ and O₂ are illustrated by constructions I and III, and in both cases a discontinuity is observed in the image of the curve in the mirror and behind the mirror.

In determining the tangent of the angle of inclination of the tangent, it is not recommended to use a protractor, since the angle \((\varphi)\) read on this instrument, and consequently the tangent as well, will depend on the scales on which the argument and the function are plotted; therefore, to find the actual angle, additional corrections for the difference between the ordinate and abscissa scales would have to be introduced. It is simpler, faster, and much more accurate to determine not the angle, but the tangent of the angle \((\varphi)\) by dividing the opposite (US) leg by the adjacent (UT) leg of the angle \((\varphi)\) formed by the tangent and the abscissa axis. The magnitudes of the legs should be read not in millimeters, but in the units in which the argument and the function are laid off on the coordinates. In our constructions we used a metallic mirror ruler, 15 mm wide. Compared with a glass one, a metallic ruler has the advantage that there is no gap between the curve on the drawing and its image in the ruler. For a glass mirror ruler this gap, owing to the presence of the layer of glass, is completely unavoidable, and this considerably hampers the work. Before the work was begun, the ruler and the triangle were carefully calibrated.

By multiplying the values of the tangents of the angles of inclination of the tangent found in this way by \(2.303 \cdot R = 2.303 \cdot 1.987\), the differential heats of dissolution of gases in water at different temperatures and at constant pressure equal to 1 atm were obtained. The calculated differential heats of dissolution for 23 gases are collected in Table (1) and presented in Figs. 7, 8, 9, 10, 11, 12.

From the table and figures given, it is seen that the heat of dissolution of gases in water at atmospheric pressure depends on the nature of the absorbed gas. Thus, at a temperature of \(0^\circ\text{C}\) the positive heat of dissolution has the minimum value for hydrogen, 1530, and the maximum value for \(\mathrm{Br}_2 = 8080\) cal/mol; in addition, such gases as helium and neon at this temperature have a negative heat of dissolution, i.e., upon their dissolution heat is not evolved, but absorbed.

Table 1. Differential molar heats of dissolution of gases in water at various temperatures and at a partial pressure equal to 1 atm

Differential heats of dissolution in cal per mole for gases

Continuation of Table 1.

Fig. 7. Dependence of the differential heat of dissolution of gases in water on temperature for (O₂; N₂O; COS; Xe; Ar; and air), in cal/mol.

Fig. 8. Dependence of the differential heat of dissolution of gases in water on temperature for (SO₂); Rn; Kr; CO.

Fig. 9. Dependence of the differential heat of dissolution of gases in water on temperature for (CO₂); (H₂S); NO and (N₂).

Fig. 10. Dependence of the differential heat of dissolution of gases in water on temperature for \(\mathrm{C_2H_6}\), \(\mathrm{C_2H_4}\), \(\mathrm{C_2H_2}\), and \(\mathrm{CH_4}\).

Fig. 11. Dependence of the differential heat of dissolution of gases in water on temperature for \(\mathrm{O_3}\), \(\mathrm{Ne}\), and \(\mathrm{Br_2}\).

\(\mathrm{Ne}\)—left scale; \(\mathrm{O_3}\), \(\mathrm{Br_2}\)—right scale.

Fig. 12. Dependence of the differential heat of dissolution of gases in water on temperature for H₂, He, and Ne.

Fig. 13.

Besides the nature of the gas, the heat of dissolution also depends on temperature. The character of this change is formally determined by the temperature course of solubility. For the gases whose heat of dissolution is discussed in the present work, the temperature course of solubility, and consequently the temperature dependence of the heat of dissolution, is covered by the following four cases: the 1st case—the solubility of the gas decreases with increasing temperature and the functional dependence of the logarithm of the mole fraction of the dissolved gas on the reciprocal temperature,

\[ \left[\text{i.e. } \log N_2 = f\left(\frac{1}{T}\right)\right], \]

is determined by curve I, shown in Fig. 13.

This case includes the dissolution of the majority of the gases considered by us, namely: CO; N₂; NO; O₂; Ar, air, Kr; Xe; H₂S; CH₄; C₂H₂; C₂H₄; C₂H₆; CO₂; SO₂; Rn; COS; Br₂. For this group of gases the first derivative of \(\log N_2\) with respect to \(\left(\frac{1}{T}\right)\), and accordingly the heat of dissolution, have a positive sign and decrease with increasing temperature.

Thus, for example, the positive heat of dissolution at 0°C for this group of gases, depending on their nature, varies from 3750 cal/mole for nitrogen to 8080 cal/mole for bromine. With increasing temperature, the heats of dissolution of all gases of this group fall, and at temperatures of the order of 100° they are measured in several hundred calories. The exceptions are radon and bromine, since the heat of dissolution at 100°C for the former is 1110 calories and for the latter, apparently, approximately the same. The 2nd case is determined by the character of curve II in the same figure. The first derivative of \(\log N_2\) with respect to \(\left(\frac{1}{T}\right)\) and the heat of dissolution likewise have a positive sign, but their absolute values will increase with increasing temperature. This case includes two gases: nitrous oxide (Fig. 3) and ozone (Fig. 4), whose positive heat of dissolution increases with increasing temperature (Figs. 7 and 11). Thus, for example, the differential heats of dissolution of N₂O at temperatures of 18 and 36°C are respectively 4170 and 5990 cal/mole. It should be noted that the heat of dissolution of ozone increases especially markedly with increasing temperature; at 0°C this quantity proved to be 3460, and at 55°C—24400 cal/mole. The 3rd case is represented by curve III in Fig. 13. With these relationships between \(\log N_2\) and \(\left(\frac{1}{T}\right)\), when the solubility of the gas in the liquid increases with temperature, the first derivative

\[ \left| \frac{\partial \log N_2}{\partial \left(\frac{1}{T}\right)} \right|_P \]

and the heat of dissolution must have negative values, and their absolute magnitudes will increase with increasing temperature. An illustration of exactly what has been set forth is the process of dissolution of neon. For this gas the heat of dissolution is negative, and its absolute value increases with increasing temperature (at 0°C—1760 and 50°C—9030 cal/mole) (Fig. 11). The 4th case—the solubility of the gas increases with increasing temperature, but the relationships between \(\log N_2\) and \(\left(\frac{1}{T}\right)\) are determined by a curve of type IV (Fig. 13). In this case

\[ \left| \frac{\partial \log N_2}{\partial \left(\frac{1}{T}\right)} \right|_P , \]

and consequently the heats of dissolution, have negative-

value, but their absolute magnitudes will decrease with increasing temperature.

Other, more complex relationships between the logarithm of the molar fraction of the dissolved substance and the reciprocal temperature are also possible, but all of them can be obtained by combining these four cases. This, in particular, is the case for the dissolution of hydrogen and helium in water (Figs. 1 and 5). \(\left|\dfrac{\partial \log N_2}{\partial \left(\dfrac{1}{T}\right)}\right|_P\) and the heat of dissolution for hydrogen (Fig. 12) at temperatures from 0 to 50°C have a positive sign and their absolute magnitudes decrease. This is expressed by curve V (Fig. 13), which can be obtained by combining curves I and III. For helium, the relationship between \(\log N_2\) and \(\left(\dfrac{1}{T}\right)\) is expressed by a curve of type IV, which can be obtained by combining curves III and IV. In this case the first derivative of \(\log N_2\) with respect to \(\left(\dfrac{1}{T}\right)\), and the heat of dissolution, throughout the temperature interval studied have negative values; however, the absolute magnitude of the heat of dissolution first decreases as the temperature is raised from 0 to 20°C, and then, with a further increase in temperature to 50°C, increases (Fig. 1).

However, in discussing the solubility of neon and helium in water, one should point to the later studies of Lonning\(^{21}\). The latter, using a more accurate experimental method, found that Antropov’s\(^{21}\) data on the solubility of Ne and He in water were erroneous. Using Lonning’s data, we plotted the curves \(\log N_2 = f\left(\dfrac{1}{T}\right)\) for Ne and He. In this case, for Ne the curve expressing the dependence between \(\log N_2\) and \(\left(\dfrac{1}{T}\right)\) can be obtained by combining curves I and II in Fig. 13. Accordingly, the differential heat of dissolution for Ne (Fig. 12) has a positive sign and, as the temperature rises, first decreases and then, having reached a minimum value, begins to increase. As for He, for it, in the temperature interval studied (15–37°), \(\lg N_2\) is connected with \(\left(\dfrac{1}{T}\right)\) by a straight-line dependence, and the calculated heat of dissolution proved to be a positive quantity, independent of temperature and equal to 382 cal/mole.

It should be noted that the heat of dissolution is determined not by the absolute magnitude of the gas solubility, but by the magnitude of its change with temperature.

For many gases, within certain temperature limits, the change in the heat of dissolution is related to temperature by the linear dependence \(Q_t = Q_0 + at\), where \(Q_t\) and \(Q_0\) are the molar differential heats of dissolution of the gas in water at temperatures 0° and \(t^\circ\)C, respectively.

\(a\) — temperature coefficient.
\(t\) — temperature in °C.

The equations found by us are given in Table 2.

Table 2

As a rule, a rectilinear dependence of the heat of dissolution on temperature occurs, first, not for all gases and, second, it is replaced by a more complex dependence of the type: \(Q_t = Q_o + \alpha t + \beta t^2 + \alpha t^3 +\) when the temperature limits are extended from 0 to 100°C.

It is of interest to note that the heats of dissolution (Table 3) of air, calculated by the rule of additivity from the heats of dissolution of nitrogen and oxygen, are in good agreement with the heats found directly from data on the solubility of air.

Table 3

Calculation of the heat of dissolution of air by the rule of additivity from the heats of dissolution of nitrogen and oxygen

The data necessary for these calculations on the ratio between oxygen and nitrogen dissolved in water, at an air pressure equal to 1 atm, were borrowed from the work of Winkler \(^{24}\).

It may be expected that also in other cases of the joint dissolution of several gases the molar heat of dissolution of the mixture will be additive.

be composed of the heats of dissolution of its components; however, the absence of data on joint solubility does not permit more definite conclusions to be drawn on this question.

The appearance of a heat effect in the dissolution of gases in water, its absolute magnitude, and its change with temperature are determined by those forces that manifest themselves in the process of interaction of the gas with water. These forces may be of a chemical and a physical nature.

In favor of the possibility of chemical bonding are the facts that many gases, dissolving in water, form hydrates which in some cases, at a sufficiently high gas pressure and low temperature, precipitate from the solution in the solid state.

Thus, according to the investigations of Vrublevskii^25), \(CO_2\), dissolving in water at \(0^\circ C\) and \(P_{CO_2} = 12.4\) atm, forms the compound \(CO_2 \cdot 8H_2O\), the dissociation pressure of which increases strongly with increasing temperature, reaching 26.1 atm at \(6.8^\circ C\).

Hempel and Seidel^26) confirmed the correctness of Vrublevskii’s results and also found in addition that \(CO_2\), dissolving in water, can give a nonahydrate, i.e. \(CO_2 \cdot 9H_2O\). Zel’venskii^27), who studied the solubility of \(CO_2\) in water, also came to this conclusion. According to Villard^28), in the interaction of \(CO_2\) with \(H_2O\) a hexahydrate \((CO_2 \cdot 6H_2O)\) can also be formed. The formation is known of a monohydrate, called carbonic acid, \(CO_2 \cdot H_2O\) or \(H_2CO_3\).

All this shows that, when a gas dissolves in water, in principle chemical interaction may arise between the dissolved gas and the liquid solvent, as a result of which, depending on the particular conditions of dissolution (temperature, pressure), definite chemical compounds, very different in their composition and only slightly stable under ordinary conditions (since their dissociation pressure is high), may be obtained in solution. Besides carbon dioxide, hydrate formation occurs in the dissolution in water of such unstable gases as \(H_2S\) \((H_2S \cdot 12H_2O)\), \(Cl_2\) \((Cl_2 \cdot 5\) or \(8H_2O)\); \(N_2O\) \((N_2O \cdot 6H_2O)\); \(C_2H_2\) \((C_2H_2 \cdot 6H_2O)\); \(C_2H_4\) \((C_2H_4 \cdot 6H_2O)\)^29) and others.

Further, Forcrand^30) showed that even the inert gases Ar, Kr, Xe, dissolving in water at high pressures and low temperatures, form crystalline hydrates; true, their dissociation pressures proved to be very large, with the exception of xenon, which gives the most stable hydrate, with a dissociation pressure of 1.4 atm at \(1.4^\circ C\). The amounts of water combining with one atom of inert gas for Ar, Kr, and Xe are respectively equal to 5, 6, and 6–7 molecules.

Everything set forth here is in known agreement with the hydrate theory proposed by Mendeleev^31), according to which solutions may be regarded as liquid, unstable, definite chemical compounds in a state of dissociation.

In this connection, the heat effect of the dissolution of a gas in water will to a considerable extent be determined by the heat effect of hydrate formation, while the changes of \(Q_t\) with temperature will be conditioned by mutual transitions of some forms of hydrates into others.

The transitions of some hydrate forms into others can apparently explain the abrupt changes in the temperature course of \(Q_t = f(t)\) for a number of gases. Thus for Ar (Fig. 7) this break occurs in the temperature interval \(15\text{–}20^\circ C\), for \(SO_2\) (Fig. 8)—\(30^\circ C\), \(O_2\) (Fig. 7)—\(50^\circ C\), \(H_2S\) (Fig. 9)—\(50\text{–}60^\circ C\), He—\(20^\circ C\). The curves \(Q = f(t)\) for \(H_2\) and \(Br_2\), Figs. 12 and 11, have two breaks each, respectively at 20 and \(30^\circ C\), and at 11 and \(47^\circ C\). It is quite clear that the compositions of the hydrates formed under particular conditions of gas dissolution in water and their mutual proportions can be determined only by special investigations.

Besides the hydrate theory, the appearance of a thermal effect of dissolution may, in our opinion, also be elucidated from another point of view. Recently, in connection with X-ray studies of liquids, the latter have been ascribed a quasi-crystalline structure, and thus the structure of a liquid is likened more to that of a solid than to that of a gas.

According to Stewart\(^{32}\), a liquid consists of a large number of regions within which the relative orientation of the molecules approaches the uniform distribution found in the solid state. These ordered regions have been called “cybotactic groups.” According to Stewart, one cybotactic group contains from 100 to 1000 molecules. Sharp boundaries between regions of greater and lesser order are absent.

In 1936 a paper by Stutzer\(^{33}\) appeared, in which the author ascribes to liquids a glissant structure. According to this conception, upon melting a solid breaks up along the least strong cleavage planes into separate thin plates, “glissants,” capable of sliding over one another, which determines the mobility of liquids. Without deciding the question of which model of liquid structure (cybotactic or glissant) is more plausible, we note that, according to Bernal\(^{34}\) and Fowler\(^{34}\), liquid water has a quasi-crystalline structure of quartz-like form. Recognition of the quasi-crystalline structure of liquids permits one to suppose that, when a gas dissolves in a liquid—in particular, in water—the gas is adsorbed on the surface of quasi-crystallites, and in this case the thermal effect of dissolution may be reduced to the thermal effect of adsorption; moreover, the magnitude and sign of the thermal effect of adsorption, and consequently also of dissolution, will depend on whether, in addition to van der Waals attractive forces understood in the broad sense of the word (as forces arising from the action of permanent dipoles, quadrupoles, induced dipoles, etc.), forces of chemical nature also participate in this process. In this connection, it is highly expedient to compare the thermal effects of dissolution of gases in water obtained by us with the thermal effects of adsorption of these gases on the surfaces of solid bodies. However, this comparison, as well as the determination of the degree of participation of physical van der Waals and chemical forces in the appearance of the thermal effect of dissolution, lies outside the scope of the present communication.

CONCLUSIONS.

1. From solubility data, differential heats of dissolution were calculated for 23 gases in water (CO, N\(_2\), NO, O\(_2\), air, Ar, Kr, Xe, H\(_2\)S, CH\(_4\), C\(_2\)H\(_2\), C\(_2\)H\(_4\), C\(_2\)H\(_6\), CO\(_2\), SO\(_2\), Rn, COS, Br\(_2\), H\(_2\), He, Ne, N\(_2\)O, O\(_3\)) at a pressure of 1 atm and temperatures from 0 to 100°C. A method is described for calculating differential heats of dissolution, based on the laws of ideal solutions and on the graphical “mirror” method.

2. At a pressure of 1 atm, the differential heats of dissolution of gases in water depend both on the nature of the dissolved gas and on temperature.

3. A formal classification is given of possible cases of the change of solubilities and heats of dissolution with temperature.

4. For a group of gases, within certain temperature limits, a linear dependence is observed between the differential heat of dissolution and temperature. The presence of sharp breaks on the curves \(Q_t = f(t)\) is attributed to the formation of definite chemical compounds (hydrates) when the gas dissolves in water.

1. The heats of dissolution of air, calculated by the rule of additivity from the heats of dissolution of nitrogen and oxygen, are in good agreement with the heats of dissolution found directly from the data on the solubility of air.

2. The presence of thermal effects during the dissolution of gases in liquids is due to the action of the physical van der Waals and chemical forces of the medium. In connection with the quasi-crystalline structure of liquids, the possibility is indicated of reducing the heat of dissolution to the thermal effect of adsorption of the dissolved gas on the surface of quasi-crystals.

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