II

Volume - Relation to Temperature

This is the second of a series of publications, which will present a complete new theoretical treatment of the liquid state. As brought out in the first paper, the results reported herein have been derived entirely by extension and elaboration of the consequences of two new postulates as to the nature of space and time which were formulated and explained by the author in a previously published work.¹ The first paper gave a brief outline of the general theory of liquids thus derived. We now begin a detailed discussion of the application of this general theory to specific liquid properties. It will be convenient to start with the property of volume inasmuch as this is a relatively simple item which plays an important part in most of the more complex physical properties that will be discussed later. The volume presentation will be divided into three sections.' Since the available experimental values which will be used for comparison with the results calculated from theory include a temperature effect which varies widely from substance to substance the first section will establish the relation between volume and temperature so that the basic volumetric factors characteristic of each substance can be identified. The next section will show how these volumetric factors can be derived from the chemical composition and molecular structure, and the final section will develop the relation between volume and pressure.

Theoretically the initial point of the liquid state is at zero temperature; that is, when the thermal energy of a solid molecule reaches the limiting value the molecule undergoes a transition to the liquid state at zero temperature. Inasmuch as the surrounding molecules are at a higher temperature this zero temperature condition cannot persist and the molecule immediately absorbs enough heat from its environment to bring it into thermal equilibrium with the neighboring molecules. The theoretical initial point of the liquid is therefore a level that cannot be reached in practice but it does constitute a convenient reference point-for our calculations. From the basic theory of the liquid state as previously outlined it follows that the thermal motion beyond the initial point of the liquid is the one-dimensional equivalent of the thermal motion of a gas. It therefore conforms to the gas laws; in particular, the volume generated by this motion is directly proportional to the temperature. At the unit temperature level this volume should equal the initial liquid volume, V₀, the volume at zero temperature. The factors affecting the magnitude of the temperature unit will be analyzed in a subsequent publication and for present purposes we will merely note that the unit applicable to most organic liquids and a large number of common inorganic liquids has been evaluated as 510.2º K. The volume of a liquid molecule between absolute zero and the critical temperature can then be expressed as

In most cases the effective value of the initial volume applicable to the motion in the second dimension differs somewhat from that applicable to the initial dimension because of geometric factors which will be discussed later, and if we represent the two values of V₀ by V₁ and V₂ respectively, equation 1 becomes

The volume of a liquid aggregate deviates from the linear relation of equation 2 in two respects. At the lower end of the liquid temperature range the aggregate contains a certain proportion of solid molecules and the average volume per molecule is therefore either slightly above or slightly below the true liquid volume, depending on whether the volume of the solid is greater or less than that of the liquid. At the upper end of the liquid temperature range the aggregate contains a similar proportion of what we may call critical molecules; that is, molecules which have individually reached the critical temperature and have acquired freedom of movement of the liquid type in the third dimension but have not yet made the transition to the unidirectional translational motion characteristic of the gaseous state. On assuming the critical status each molecule acquires a volume component in the third dimension similar to the components in the other two dimensions and these additional volumes increase the average molecular volume of the liquid aggregate above the value given by equation 2.

In order to calculate the volume of the liquid-aggregate over the entire liquid temperature range it will thus be necessary to determine the proportion of solid molecules and the proportion of critical molecules existing in the aggregate at each temperature and then to apply these figures to the volume increments accompanying the change of state in the individual molecule. Since the existence of other-than-liquid molecules in the liquid aggregate is the result of the distribution of molecular velocities, the number of such molecules is a probability function of the temperature and its numerical evaluation is simply a question of using the appropriate probability expression.

Thus far in all of the applications of probability mathematics that have been encountered in the course of the investigation of which this liquid study forms a part, it has been found that sufficient accuracy for present purposes can be obtained by the use of one variation or another of the so-called "normal" probability function. Whether this mathematical expression is an exact representation of the true relationship or merely a very close approximation is a question that can be left for later treatment. Because of the extremely broad scope of this investigation it has been physically impossible to study the "fine structure" at every point and any question of this kind which is beyond the limits of accuracy of the work as a whole has been passed up for the time being. It should be noted, however, that eliminating consideration of these fine-structure factors has very little effect on the accuracy of the liquid volume calculations.

Ordinarily the only uncertain element entering into the application of the normal probability function is the size of the probability unit. Ultimately it will no doubt be possible to develop methods of determining this unit from, purely theoretical considerations but in the meantime it can be identified quite readily on an empirical basis since this present study has disclosed that the unit is a simple fraction of the appropriate reference temperature. For example, the reference temperature for the solid-liquid transition is the melting point and the unit applicable to this transition in the paraffin hydrocarbons is one-fourth of the melting temperature For the critical transition the reference temperature is not the critical temperature as might be expected but the critical temperature plus half of the 510º temperature unit. Furthermore, the change in the dimensions of motion at the critical point results in a corresponding change in-the probability unit and we find that the unit applicable to half of the molecules is only one-third as large as that applicable to the other half. If ire designate the larger unit, which we find is (T_c + 255)/9, as A, the smaller unit as B, and the corresponding probability functions as f_A and f_B, we may express the proportion of critical molecules in the saturated or orthobaric liquid aggregate as ½(f_A + f_B). The transition of the individual molecule from the liquid to the critical condition is necessarily instantaneous since it is simply the result of breaking the inter-molecular bond in the third dimension. The third-dimensional volume increase therefore takes place isothermally so far as the individual molecule is concerned and the added volume per critical molecule is V₀. Where the proportion of critical molecules ½(f_A + f_B) the average volume increase for the liquid aggregate as a whole is ½(f_A + f_B) V₀. Here again the value of V₀ applicable to this particular dimension may differ somewhat from the values that apply to the other dimensions and we will therefore identify this effective initial volume in the third dimension as V₃. The complete volume equation for all three liquid components is then

As previously indicated, a small additional adjustment is required in the range just above the melting point to compensate for the effect of the solid molecules which are present in the aggregate at these temperatures. In computing this adjustment by means of the probability relations, one of the points which must be taken into consideration is the location of the equal division between solid and liquid molecules, On a temperature basis the end point of the solid and the initial point of the liquid are coincident. From an energy standpoint, however, there is a substantial difference between the two: a difference, which is represented by the heat of fusion. If we continue adding heat to liquid aggregate, which has just reached the melting point, we find that the first additions of this kind do not result in any increase in temperature but are absorbed in the change of state. According to the theoretical principles developed in this study the change of state or the individual molecule is completed instantaneously and an isothermal absorption of heat in an aggregate of this kind can only result from these complete changes of state on the part of the individual molecules. It is apparent; therefore, that the aggregate reaches the melting temperature when the proportion of liquid molecules contained therein arrives at some limiting value A, which is less than 50 percent. Further additions of heat then enable more solid molecules to make the transition into the liquid state until the proportion of liquid molecules reaches another limiting value B, above 50 percent, beyond which part of the added thermal energy goes into an increase of the temperature of the aggregate. It thus follows that the location of equal division between solid and liquid molecules is not at the end point of the solid nor at the initial point of the liquid but midway between the two; that is, it is offset from each of these points by half of the temperature equivalent of the heat of fusion. In order to calculate the volume deviation due to the presence of solid molecules in the liquid aggregate it will therefore be necessary to know the amount of this temperature offset as well as the difference between the pure solid and pure liquid volumes. For present purposes we may simplify the calculations by using average values applicable to entire classes of substances rather than computing these factors on an individual basis, as the volume deviations due to this cause are small in any event and the basic factors for substances of similar structure are almost identical. The theoretical aspects of this situation will be discussed-in detail in a subsequent publication, which will-examine the process of freezing liquids by the application of pressure.

In the Immediate vicinity of the critical temperature still another factor enters into the picture, as some of the gas molecules remain in solution in the liquid aggregate. It will be convenient, however, to terminate the present study at the lower limit of this zone, about 20 degrees below the critical temperature, and to defer the discussion of the gas adjustment to a later paper in which the results of a study of vapor volume will be published.

As an example of the method of calculation of the solid-state volume increments shown in the columns headed D_s in the tabulations Included here with, let us look at the figures for hexane at -50º C. First we divide the melting temperature, 178º K, by 4 to obtain the probability unit 44.5º. Next we divide the 45 degrees difference between -50º C and the melting point by the unit value 44.5º, obtaining 1.01 as the number of probability units above the melting point. For present purposes the offset of the melting point from the location of equal division between solid and liquid molecules will be taken as .40 units, which is an average value that can be applied in all of the calculations of this kind that will be made in this paper. Adding the .40 units to 1.01 units we arrive at a total of 1.41 units. The corresponding value of the integral of the normal probability function, which we will designate f, is .158. This probability function is 1.00 at the point of equal division between the two states and the value .158 therefore indicates that 7.9 percent of the total number of molecules in the liquid hexane aggregate at -50º C are in the solid state. We then need only to multiply the difference in volume between solid and liquid molecules by .079 to obtain the average Increment for the aggregate as a whole. Again we will use average values to simplify the calculations, and for the lower paraffin hydrocarbons (C₁₄ and below) we will take the molecular increment as +.080. The slightly higher value +.084 will be applied to the paraffins above C₁₄, including hexadecane, one of the compounds covered by the tabulations. The product .079 x .80 gives us .006 as the amount to be added to the true liquid volume calculated from equation 3 to obtain the volume of the actual liquid aggregate.

Calculation of the critical volume increment, V₃, is carried out in a similar manner. Again the first step is to determine the probability unit. As indicated in the preceding discussion, this unit is 1/9 of (T_c + 255), and for hexane amounts to 84.8º. In the computation for +50º C, for examples, we next subtract 50º from the critical temperature, 235º C, obtaining a difference of 185º. Dividing 185 by 84.8, we find that the number of probability units below the critical temperature is 2.18. The corresponding value of ½(f_A + f_B) is .015. Here the 1.00 probability factor indicates the situation in which 100 percent of the molecules have reached the critical temperature and the result of our calculation therefore means that 1.5 percent of the total number of molecules at +50º C are in the critical condition. We then multiply .015 by .9778, the critical volume increment per molecule, which gives us .0147 as the critical increment (V₃) for the aggregate.

The quantity used in the foregoing multiplication, the critical volume increment per molecule or third dimensional value of the initial volume, V₀, and the corresponding initial volumes for the first and second dimensions can be derived from the molecular composition and structure by methods which will be discussed in the next paper in this series. For the present it will merely be noted that in most cases the basic value of the initial volume remains constant in all dimensions and the differences between the initial values of V₁, V₂, and V₃ are due to the modification of the basic value V₀ by a geometric factor which varies from .8909 to 1.00. In the base of hexane, or example, V₀ is .9778 and the geometric factors for the three dimensions of motion are .9864, .9727, and 1.000 respectively.

Volumetric data for a number of representative liquids are given in Table 11-1. In this table the D_s and V₃ volumes calculated in the manner described are added to the constant V₁ volume and the value Of V₂ obtained from the linear relation of equation 2 to arrive at the total volume of the liquid aggregate for comparison with the experimental volumes.¹² In those cases where the solid-state volume incremental, D_s negligible except for a few of the lowest temperatures of observation, calculation of this volume component has been omitted. All volumes are expressed in cm³/g.

The extent of agreement between the calculated and experimental values in these tables is typical of the results obtained in the study of several hundred substances. In the most accurate experimental temperature range, in the neighborhood of room temperature, the deviations for the compounds which have been studied most thoroughly are within the general range of accuracy of the mathematical treatment, about 0.1 percent. At higher or lower temperatures and with less reliable experimental values the deviations are greater, as would be expected, but in most cases remain below one percent. The next paper in this series will present additional comparisons of the same kind for a wide variety of liquids at a few selected temperatures.

In this initial presentation of the liquid volume relations the discussion has been confined to liquids of the simplest type. It may be mentioned, however, that the modifications required in equation 3 to make it applicable to the more complex liquids are quite simple and usually amount to nothing more than replacing the temperature unit 510.2 degrees by 510.2 n degrees. In such liquids as water, the glycols and many condensed aromatic compounds the value of n is 2.

Supplement

This supplement to the original paper II in the liquid series has been prepared as a means of answering some questions that have been raised concerning the application of equation (3), the volume-temperature relationship, to liquids other than those of the simple organic type.

The particular advantage of a mathematical relation of this kind derived entirely from sound theoretical premises by logical and mathematical processes is that such a relation has no limitations. In its most general form this volume relationship is universally applicable throughout the entire range of the liquid state. The original paper showed that it is valid at all liquid temperatures and stated that it is applicable to all types of liquids, although the tabulated examples were limited to simple organic Compounds. The present supplement amplifies this statement by adding examples of other liquid types., including inorganic liquids., liquid metals and other elements, and fused salts. In the next paper in the series it will be shown that the same mathematical expressions can be applied to the calculation of liquid volumes under pressure, thus completing the coverage of the entire area in which the liquid state exists. The opening statement of this paragraph can then be applied in reverse; that is,, the demonstration that there are no limitations on the applicability of the mathematical relationship is strong evidence of the validity of the theoretical premises and of the processes by which the relationship was derived from those premises.

In equation (3) the term T in its general significance refers to the effective temperature rather than to the measured temperature. As long as the application of the equation is limited to simple organic compounds of the type covered in Tables II - 1 and III - 2 this distinction can be ignored as the effective temperature for these compounds is equal to the measured temperature. In general., however., the effective temperature is T/n, where n is an integral value ranging from 1 to 16. For general application the expression T/510.2 in equation (3) must therefore be modified to T/510.2 n as indicated in the last paragraph of paper II. The volume calculations for any liquid can then be carried out in the manner previously described.

In order to distinguish between this temperature factor n and the number of volumetric groups in the liquid molecule the symbols set and n_v will be used in the following discussion. Most of the cam, inorganic compounds which are liquid at room temperature have the same unit value of n_t as the organic compounds of the previous tabulations. Table II - 2 gives the volumetric data for CCl₄, which can quality either as organic or inorganic., depending on the definition that-is used, and for SO₂ and HCl, which are definitely inorganic. Also included in this table are similar data for hydrogen and fluorine, two elements with n_t = 1.

One of the influences which may increase the temperature factor n_t is a greater degree of molecular complexity such as that which characterizes the-condensed aromatic compounds, for example. Most of the complex aromatic liquids have n_t = 2. Table II - 3 gives the volumetric data for water (n_t = 2), an inorganic liquid with a similarly complex molecular structure. Because of the relatively large solid state increments the quantity V_S -V_L has been determined individually for each temperature in this table using V_S = 1.085. Otherwise the calculations involved in the determination of these volumes are identical with those previously described.

The liquids thus far discussed are composed entirely of electronegative elements (for this purpose carbon and hydrogen which are on the borderline between electropositive and electronegative, are included in the electronegative class), and principally of those elements in this class which either (1) have atomic weights below 11 or (2) have unit valence. If both the mass and the valence of the principal constituent or constituents exceed these limits the temperature factor n_t is greater than unity. Thus sulfur and phosphorus have n_t values of 4 and 3 respectively. We may sum up the foregoing by saying that the extreme electronegative liquids ordinarily take the minimum n_t value, unity, and n_t increases as the liquid components move toward the electropositive side., either by increase of valence or by increase in the atomic mass. Conversely, the extreme electropositive liquids, the heavy liquid metals, ordinarily take the maximum n_t value, 16.

Table II - 4 shows the volumes of several liquids with temperature factors above 2. In calculating these volumes it has been assumed that the first and second dimension values of V₀ are equal. This appears to be the general rule in this class of compounds and in any event it would not be possible to verify the existence of any small difference as the experimental volumes of these liquids are subject to considerable uncertainty because of the unfavorable temperature conditions under which the measurements must be made. There is no appreciable third dimension component in the temperature range of Table II - 4 and only one V₀ value is therefore shown.

The n_t values for compounds of electropositive and electronegative elements are intermediate between the two extremes, as would be expected. Table II - 5 shows the pattern of values for the simplest compounds of this type, the alkali halides. Here we find some half-integral values: evidently averages of integral values for each of the positive and negative components. In Table II - 6 which follows, the number of volumetric units per formula molecule, n_v, is indicated for each of these same compounds. Table II - 7 then gives the calculated and experimental volumes at two different temperatures within the liquid range. The previous comments with respect to Table II - 4 also apply to Table II - 7.