V. The Liquid-Solid Transition

As pointed out in paper I of this series, development of the consequences of the postulates on which this work is based indicates that physical state is a property of the individual molecules and not, as generally assumed, a "state of aggregation". Because of the distribution of molecular velocities due to the operation of the probability principles it follows that a liquid aggregate in the vicinity of the melting point is not a homogeneous structure but a composite which includes both solid and liquid molecules. The effect of the presence of solid molecules on the volume of the liquid aggregate was discussed briefly in paper II. Further consideration of this situation will now be required in order to permit extending the liquid compressibility calculations into the extreme high-pressure range.

The procedure is essentially simple. The proportion of solid molecules in the liquid aggregate at any specified pressure is determined by the same technique utilized in paper IV for the calculation of the critical volume component; that is, by the use of a composite probability index obtained by adding the equivalent of the applied pressure to the index corresponding to the prevailing temperature. The percentages thus determined are then multiplied by the difference between the pure solid and pure liquid volumes at each individual pressure to arrive at the increase (or decrease) in the volume of the aggregate due to the presence of the solid molecules.

Table V-1 shows how this procedure is applied to the calculation of the volumes of liquid methyl alcohol at 50° C and various pressures.

The first step is to determine the pure liquid volume by the methods of paper IV. The values thus obtained for each of the pressures of column 1, relative to the volume at 0° C and atmospheric pressure, are given in column 2. Column 3 lists the corresponding volumes of the pure solid. Since the theoretical study of the volume of solid organic compounds is still incomplete, the solid volumes used in this paper have been derived from whatever measured volumes are available adjusted to the various pressures by the methods explained in a previous paper.⁵ Column 4 is the difference between the pure liquid and pure solid volumes.

The next operation is the determination of the percentage of solid molecules in the aggregate at each pressure. Paper II has already indicated how the number of probability units corresponding to the difference between any specified temperature and the location of equal division between solid and liquid can be obtained. This number, the probability index, for methyl alcohol at 50° C is 3.04. The pressure probability unit applicable to the liquid side of the solid-liquid transition is the initial pressure P₀, and we now subtract the pressure equivalent P/P₀ for each of the pressures of column 1 from 3.04, entering the results in column 5.

A detailed study of the liquid-solid transition process has revealed that the probability unit of pressure, which is P₀ on the liquid side of the neutral point, is 2P₀ on the solid side. The probability index increment due to a change in pressure therefore undergoes a decrease from P/P₀ to P/2P₀ in the middle of the transition zone.

This shift from one probability unit to the other is itself governed by probability and it has been found that the transition curve can be defined by taking 0.5 on the probability index scale as the probability unit. Since the value of ½ f is very small beyond 2 units, this means that the effective probability index stays on the P₀ basis down to the vicinity of a 1.00 index, then follows a transition curve of the probability type to a point midway between P₀ and 2P₀ at zero index, and continues on a reverse curve of the same kind to the vicinity of index -1.00, beyond which the 2P₀ basis prevails. Column 6 gives the adjusted values of the probability index where adjustment for the change in the probability unit is required.

From the probability tables we now obtain the values of ½f corresponding to the effective probability index for each pressure, entering them in column 7. These figure which represent percentage of solid molecules, are then multiplied by the values in column 4, giving the volume decrease due to the presence of the solid, which we show in column 8. Subtracting this amount from the volume of the pure liquid, column 2, we arrive at the volume of the aggregate, column 9. Bridgman's results are shown in column 10 for comparison.

Table V-2 gives similar data for a number of other liquids in the same pressure range. To conserve space some of the columns of Table V-1 have been omitted from these additional tabulations, but the calculations are identical.

In Table V-3 the calculations are extended to the maximum pressure of Bridgman's liquid experiments, 50,000 kg/cm², taking ethyl acetate at 75° C and at 125° C as the example.

The procedure is exactly the same as that described in the explanation of Table V-1 except that the reference temperature is 20° C. When the volume of the liquid aggregate (column 9) is obtained, this value is subtracted from the volume at 5000 kg/cm² to obtain the volume decrease for comparison with the experimental results.

When we determine the probability index at the normal freezing point by the method explained in paper II we usually find it to lie within the range of .40 to .60, which means that the liquid normally freezes when the proportion of solid molecules reaches a level somewhere in the neighborhood of 30 percent. Under pressure, however, we find the ethyl acetate aggregate at 75° C still liquid when over 75 percent of the molecules are in the solid state. At first glance this may seem to be an impossibly high figure, but fortunately Bridgman has given us another set of observations, which enables us to put the situation into the proper perspective. He finds that under more favorable conditions ethyl acetate at 75° C will 1 freeze at 23,800 kg/cm² and at 125° it will freeze at 31,860 kg/cm². These pressures correspond to solid percentages of 41.4 and 46.7 respectively. At 25° freezing was experienced at a pressure of 12,100 kg/cm² (28 percent solid molecules) in this series of experiments, whereas in the other set of observations the liquid state persisted up to the pressure limit of 50,000 kg/cm².

From these figures we may obtain a consistent pattern. Ethyl acetate at the lower temperatures and pressures freezes when the proportion of solid molecules reaches the normal limit at about 30 percent. As pointed out in paper I, however, the requisite proportion of solid molecules is not in itself sufficient to insure freezing. These solid molecules must not only be present in adequate numbers but they must be able to make contact and to maintain that contact against the disruptive forces long enough to establish the nucleus of a crystal lattice. Although pressure and temperature are opposed from many standpoints they are both in the category of disruptive forces and when higher pressures are applied at higher temperatures the formation of the crystal lattice-becomes progressively more difficult. Consequently we find (1) that even under the most favorable conditions an increasing margin above the normal 30 percent solid molecules is required for freezing and (2) where conditions are unfavorable (mechanical agitation, asymmetric molecules, etc.) the super-saturated liquid may persist to very high solid percentages.

Table V-4 gives some additional comparisons of calculated and experimental volumes in this extreme high-pressure range. As in Table V-2 some of the less significant columns of figures have been omitted for economy of space but the calculations have been carried out just as described previously.

The last three tables presented in this paper constitute an appropriate climax to the entire discussion of liquid volume as they confirm the validity of the volume calculation process developed herein by producing theoretical volumes for water which agree almost exactly with the volume pattern determined by experiment: a pattern so complicated that Bridgman was led to doubt that it could ever be reproduced by any mathematical expression.³⁴

Actually all of this complexity originates from a simple causes the existence of distinct high temperature and low temperature forms of the water molecules both in the solid state and in the liquid state. Aside from the necessity of determining the proportionality between these two forms, the entire volume calculation for water is carried out in exactly the same manner as heretofore described and without introducing any additional numerical constants. The polymorphism of the solid is well established by the work of Bridgman and other investigators. Several low temperature forms have actually been identified but the densities are nearly the same and for present purposes it will be sufficiently accurate to make the low-pressure computations on the basis of the familiar ice of our everyday experience. The two forms of the liquid are difficult to distinguish and in this study it has been determined that the essential difference between the two is in the nature of the atomic association under pressure. These findings indicate that the high temperature water molecule is (H₂O)₄ and all atoms act as independent liquid units. The value of n_v for use in equation 7, the initial pressure equation, is therefore 12, and the resulting initial pressure is 5965 kg/cm². In the low temperature form the number of independent liquid units in the molecule drops to eight, which evidently means that OH associations have been formed and the molecule has become (H.OH)₄. Substitution of 8 for 12 in equation 7 gives us 3976 kg/cm² as the initial pressure.

The transition from the low temperature liquid form to the high temperature form begins at the normal melting point, 0° C, and continues linearly to completion at 59° C. This liquid, in whatever stage it may be at the temperature under consideration, with the appropriate proportion of ice molecules of the low temperature form, constitutes the low temperature water aggregate, which we will designate L_I. The high temperature water aggregate, L_II, consists of high temperature ice molecules dispersed in the high temperature form of the liquid.

Table V-5 shows the calculation of the volumes of the super-saturated liquid at -10° C, the highest temperature at which the pure low-pressure liquid (L_I) exists throughout the entire observed liquid range. Values of L_I at other temperatures are obtained in the same manner. It should be noted that the pressure increment applied to the probability index is positive in these calculations because the solid volume exceeds that of the liquid.

Table V-6 is a similar presentation of the volumes at 250° C where only the high temperature form of the liquid (L_II) is present. Here there are no solid molecules in the aggregate but the V₃ volume component has an appreciable magnitude and the critical volume calculation is necessary as in Table IV-3.

Table V-7 summarizes the complete calculation for water at 30° C, an intermediate temperature at which the normal liquid aggregate is a mixture of L_I and L_II. The first section of the table shows the computation of L_I, following the pattern of Table V-5.

A study of the transition from L_I to L_II indicates that this change begins at the upper transition point of the pure liquid, 59° C, and follows a probability curve with the same probability units as the liquid-solid transition curve; that is, 40 degrees and 5965 kg/cm². The probability increment is positive for both temperature and pressure as an increase in either of these quantities favors L_II. The third section of Table V-7 is set up in the same manner as the two preceding sections but deals with the relative proportions of the two types of liquid aggregate rather than with the proportions of liquid and solid molecules. The transition from L_I to L_II is asymmetrical; that is, the liquid is 100 percent L_I at all negative probability indexes and the probability value corresponding to each index is therefore f rather than the ½ f value which prevails in the symmetrical transitions where the condition at the probability base is an equal division between the two alternates. The quantity f represents the proportion of L_I in the aggregate and is applied in the usual manner to compute the volume increment due to the presence of the less dense component.

Table V-8 lists the basic volume and pressure factors used in the calculations in this paper. These are; of course, the same factors that are utilized in computing the volume components at atmospheric pressure, which constitute the starting point for the compression calculations.

References

5. Larson. D. B., Compressibility of Solids. privately circulated paper available from the author on request.

14. All experimental values are from Bridgman unless otherwise specified. For a bibliography of Bridgmants reports see his book “The Physics of High Pressure.” G. Bell Sons, Ltd., London. 1958.

34. Bridgman. P. W., Ibid., Page 153.

35. Kennedy. George C., American Journal of Science. 248-540.