IV

Volume - Relation to Pressure

The preceding papers in this series have developed the general characteristics of the liquid state from new fundamental theory and have shown that on this new theoretical basis the volume of a liquid molecule consists of three separate components which respond to changes in temperature in the foil owing manner: the initial component remains constant, the second component varies in direct proportion to the effective temperature, and the third component is generated isothermally at the critical temperature. Because of the distribution of molecular velocities in the liquid aggregate the number of molecules which are individually at or above the critical temperature is a matter of probability and the third volume component of a liquid aggregate therefore followers a probability function which represents the proportion of critical molecules in the total.

This paper will extend the volume relationships to liquids under pressure and will show that in its general aspects the response to variations in pressure is identical with the response to variations in temperature; that is, the initial component remains constant, the second component varies in direct proportion to the reciprocal of the effective pressure, and the third volume component of the aggregate follows a probability function for the same reasons as in the case of temperature variations. Equation (3), the volume-temperature relation previously developed, can therefore be extended to apply to liquids under pressure.

In calculating the volume of a liquid at temperature T and pressure P, we first determine the three volume components at temperature T and saturation pressure in the manner described in paper II. We will call these components V_I, V_II, and V_III. The initial component, V_I, is not affected by either temperature or pressure. The second component, V_II, responds to an increase in effective pressure in the same manner as to a decrease in effective temperature. It should be noted, however, that this effective pressure includes the pressure equivalent of the cohesive force between the liquid molecules and an evaluation of this initial pressure, as we will call it, is the first step toward a determination of the second volume component at pressure P.

The unit of pressure corresponding to the 510.2 degree temperature unit is 415.84 atm. or 429.8 kg/cm², where the initial specific volume, V₀, is 1.00. In order to avoid an extended theoretical discussion at this point we will consider this as an empirically determined value for the present, as was done with the temperature unit. For any value of V₀ other than unity the pressure unit becomes 415.84./V₀^2/3 atm. This is the pressure exerted against each independent liquid unit within the liquid molecule. The external pressure is exerted against the molecule as a whole rather than against the individual units and where there arc n_v units in the liquid molecule, the pressure exerted against each unit is P/n_v. For purposes of calculation, however, it will be more convenient to use the external pressure as the reference value and on this basis the external pressure is P and the initial pressure is

Since the application of pressure is not exactly equivalent to a decrease in thermal energy it is quite possible that the nature of the atomic association that participates in the pressure process may differ from that which participates in the temperature process. The values of n_v applicable to equation (7) are therefore not necessarily identical with those, which were arrived at in paper III in connection with the evaluation of V₀. Such equality is quite common but there is a tendency to split up into a larger number of units in the pressure process, particularly in the case of the smaller molecules. In the limiting condition each atom is acting independently.

It should also be remembered that the previous determination of n_v was concerned only with a ratio: the number of volumetric units corresponding to the mass represented by the formula molecule. The initial pressure calculation, on the other hand, requires a knowledge of the absolute number of individual liquid units in the actual molecule and where the liquid molecule comprises two or more formula molecules the value of n_v applicable to equation (7) is the corresponding multiple of the value previously found. The value of n_v used in calculating the Cs₂ volumes in Table II-3, for instance, is 3, where we now find that the value that must be used in equation (7) is 9. This does not conflict with the previous determination; it merely means that the true liquid molecule is (CS₂)₃.

Another factor, which enters into the calculation of VII, is that above 510.2° K part of the V_II component is subject to only one-sixteenth of the total initial pressure. A complete theoretical explanation of this situation which exists beyond the unit temperature level is not available as yet, but it has been found that the proportion of high temperature volume at any temperature of observation can be computed from the normal probability function using 510.2° K as the base and one-fourth of this value as the probability unit. Up to 2/3 of 510.2° the lower initial pressure is applicable to the full amount thus calculated, beyond 8/9 of 510.2° it is applicable to half of the calculated value, and in between these points the effective proportion decreases linearly.

Turning now to the third component, V_III, we first obtain from our previous calculations the figure representing the number of probability units between temperature T and the critical temperature. Since this quantity will play an important part in the volume determinations it will be desirable to give it a name for convenient reference and we will therefore call it the probability index. To this probability index at saturation pressure we now add the increment corresponding, to the applied pressure, taking the previously established value 415.84 atm. as the probability unit. If the index is above 1.15 at saturation pressure we can proceed directly to a determination of V_III, first obtaining from the probability tables the probability value corresponding to the probability index at each individual pressure and then multiplying each of these probabilities by V₃, the third dimensioned value of V₀, to obtain V_III.

If the probability index is below 1.15 at saturation pressure the B component of the probability expression ½(f_A + f_B) has an appreciable magnitude and this introduces an additional operation into the calculations. The nature of this B component was not indicated very clearly by the way in which it enters into the computation of the saturation volume but its behavior under pressure is more enlightening. We have previously found that the A probability represents the proportion of the total number of molecules which have individually reached the critical temperature and consequently have acquired a volume component in the third dimension. These molecules are still subject to the cohesive forces of the liquid; that is, to the liquid initial pressure. Now we find that as the average temperature of the aggregate approaches closer to the critical temperature and more thermal energy is available some of the molecules escape from the cohesive forces, doubling their volume in the process. The B component of the probability represents the proportion of molecules in this condition and the expression ½f_B V₃ is the volume added by this process at saturation pressure. The total volume of these B molecules at saturation is then twice this amount, or 0_B V₃, and the A portion of the V_III volume, the part still subject to the initial pressure, is ½(f_A + f_B) V₃. Dividing ½(f_A + f_B) by ½f_A gives us the percentage reduction in the A volume due to molecules shifting to the B status.

We now calculate the total A volume at each pressure by means of the expression ½ f V₃ and apply the foregoing reduction factor to arrive at the portion of the volume still remaining in the A condition. The B volume is subject only to the externally applied pressure and it varies in in_verse proportion to that pressure. The effective volume at each pressure P is therefore obtained by application of the factor P_S/P to f_B V₃, the B volume at saturation pressure P_S.

As can be seen from this description, the whole operation of calculating the liquid volumes under pressure is carried out entirely on the basis of values previously determined in the course of computing the volumes at saturation pressure, with the exception of those cases where n_v must be redetermined, either because of an actual difference in the internal behavior of the molecule or because the liquid molecule is composed of more than one formula molecule. There are no "adjustable constants" which can be manipulated to fit the observed values; the volumes under pressure must conform to a fixed pattern in each case, or if there is any element of uncertainty present, must conform to some one of two or three possible alternate patterns. These are very stringent requirements and the degree of correlation between the calculated and observed volumes as shown by the tabulations, which follow, is therefore highly significant as an indication of the validity of the new theoretical principles on which the work is based.

To illustrate the method of calculation let us consider heptane at 30° C. By the methods of paper III we determine that n_v for heptane is 9 and the three values of the geometric factor are .9878, .9636, and 1.000. From these figures we obtain V₁ = .9346, V₂ = .9117, and V₃ = .9461. Entering equation (3) with these three values we then calculate the volume components at 30° C and saturation pressure, obtaining V_I = .9346, V_II = .5417, and V_III = .0038. From our probability tables we find that at 30° C the volume originating above 510.2° K is 5.3 percent of the total V_II component, and on this basis we separate V_II into two parts: V_II(L) = .5130 and V_II(H) = .0287. Applying the previously determined values n_v = 9 and V₀ = .9461 to equation (7) we find that the initial pressure, P₀, effective against V_II (L) is 3884 atm. The initial pressure effective against V_II(H) is then 1/16 x 3884 = 243 atm. To find the V_II components at each pressure we now reduce the saturation values of V_II(L) and V_II(H) by the effective pressure ratios. P_n/(P + P₀) and P₀/(16P + P₀) respectively. The results are shown in columns 2 and 3 of Table IV-1.

Next we evaluate the probability index at 30° C and saturation pressure by the methods of paper II, obtaining the value 2.68. To this we add the increment corresponding to each pressure, which we obtain by dividing the increase in pressure above the saturation level by 415.84 atm. The composite probability indexes thus derived are shown in column 4 of the table. Column 5 gives the values of ½f corresponding to each index. Multiplying each of these values of ½f by .9461 we arrive at the V_III component for each pressure as shown in column 6. Column 7 then indicates the total theoretical volume of the liquid aggregate, the sum of V_I (constant at .9346), V_II(L) from column 2, V_II(H) from column 3, and V_III from column 6. Column 8 shows the corresponding measured volumes for comparison.

In order to carry the comparisons into the pressure range above 351 atm., the highest pressure reached in the set of measurements listed in Table IV-1, we now turn to the work of Bridgman who gives us a set of values at 50° C, with the first observation at 1000 kg/cm² (approximately 1000 atm.) and increasing by steps of 1000 kg/cm² to a maximum of 10,000 kg/cm². Bridgman's results are reported as relative volumes based on the volume at 0° C and atmospheric pressure as the reference level. Our first requirement, therefore, is to compute from equation (3) the volume under these reference conditions, which we find to be 1.424 cm³/g. m is value can then be used as a conversion factor to reduce the calculated volume components at 50° C and saturation pressure to Bridgman's relative basis. By this means we arrive at the following volumes: V_I = .656, V_II(L) = .377, and V_II(H) = .029. V_III is negligible in the pressure range of this work and can be disregarded. The volumes under pressure are then calculated in the manner described in the preceding paragraphs. Table IV-2 compares the results with Bridgman's values.

Table IV-3 summarizes the results of a number of similar calculations in the relatively low-pressure field. Since all of these calculations follow the regular pattern without exception, intermediate data such as the probability indexes have been omitted and the table shows only the separate volume components and the tot al calculated and measured volumes. The objective of the comparisons in this table is to show that there is a wide range of temperatures and substances in which the calculated and measured volumes agree within 0.5 percent at all experimental pressures. In some of the other sets of measurements, which have been examined during this investigation, the agreement is less satisfactory in certain portions of the pressure range but the general trend of the values follow the theoretical pattern in all cases.

The preceding papers have stressed the fact that the temperature term in equation (3) refers to the effective temperature: a quantity which is commonly identical with the measured temperature, but not necessarily so. The same is true of the pressure factors with which we are dealing in this paper. We have already seen that the pressure effective against the V_II volume component is substantially reduced beyond the unit temperature level (510.2° K). In some substances, chiefly outside the organic division, the pressure applicable to the V_III component is also subject to a reduction from P to P/n_p and two examples of this kind are included in Table IV-3: H₂S (n_p = 2) and NH₃ (n_p = 3).

Table IV-4 presents some further comparisons with Bridgman's measurements in the range up to 12,000 kg/cm². Some of his more recent work has extended to considerably higher pressures' reaching a level of 50,000 kg/cm² in a few instances. At these extreme pressures the transition to the solid state is well under way and the volumes of the liquid aggregates are modified quite substantially by the presence of solid molecules. Consideration of the volume situation in this pressure range will therefore be deferred to the next paper in this series, which will examine the characteristics of the liquid-solid transition. Some of the results at 12,000 kg/ cm² and below are also subject to this solid state effect and in these cases the tabular comparisons have not been carried beyond the point where the volume decrease due to solid molecules amounts to more than about .002. Double asterisks in the column of observed volumes indicate omissions due to this cause.

As mentioned in a previous paper, the scope of this investigation has been so broad that it has been physically impossible to study the "fine structure" of all of the relationships that have been covered, and it is quite possible that there may be factors of this kind which would alter the results slightly. Some additional uncertainty has been introduced by the use of the measured values of the vapor pressure at saturation. Since these uncertainties probably amount to something in the neighborhood of 0.1 percent there is no particular advantage in carrying the calculations to any higher degree of accuracy and it does not appear that such refinements as additional decimal places, fractional values of the probability indexes, etc., are justified at this stage of the project.

In the second section of this table, which follows, the values of the individual volume components are given in the following units: cm³/g x 10⁴, cu.ft./lb. x 10⁶, cm³/g mole x 10², cu.ft./lb. mole x 10⁴. Total volumes are exnpessed in the units listed above.

Specific Volumes