# 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

 VL = (1 + T/510.2) V0 (1)

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

 VL = V1 + (T/510.2) V2 (2)

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 (Tc + 255)/9, as A, the smaller unit as B, and the corresponding probability functions as fA and fB, we may express the proportion of critical molecules in the saturated or orthobaric liquid aggregate as ½(fA + fB). 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(fA + fB) the average volume increase for the liquid aggregate as a whole is ½(fA + fB) 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

 VL = V1 + (T/510.2) V2 + ½(fA + fB) V3 (3)

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 Ds 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 (Tc + 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(fA + fB) 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 Ds 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, Ds 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.

 TABLE II - 1 LIQUID VOLUME Hexane Tc = 508 f unit = 84.8 V0= .9645 - .9512 - .9778 cm³/g VS - VL = .080 T V2 V3 Ds V(calc.) V(obs) -100 .3229 .031 1.318 1.313 -90 .3415 .024 1.330 1.327 -80 .3601 .018 1.343 1.342 -70 .3788 .014 1.357 1.357 -60 .3974 .010 1.372 1.373 -50 .4361 .006 1.387 1.389 -40 .4347 .004 1.403 1.405 -30 .4534 .0010 .003 1.422 1.422 -20 .4720 .0010 .002 1.440 1.440 -10 .4907 .0020 .001 1.457 1.458 0 .5093 .0029 1.477 1.477 10 .5280 .0039 1.496 1.496 20 .5466 .0059 1.517 1.516 30 .5652 .0078 1.538 1.537 40 .5839 .0108 1.559 1.560 50 .6025 .0147 1.582 1.583 60 .6212 .0196 1.605 1.607 70 .6398 .0254 1.630 1.633 80 .6585 .0332 1.656 1.661 90 .6771 .0430 1.685 1.690 100 .6958 .0548 1.715 1.720 110 .7144 .0694 1.748 1.753 120 .7330 .0851 1.783 1.790 130 .7517 .1056 1,822 1.829 140 .7703 .1293 1.864 1.872 150 .7890 .1564 1.910 1.920 160 .8076 .1877 1.960 1.975 170 .8263 .2278 2.019 2.035 180 .8449 .2767 2.086 2.105 190 .8636 .3461 2.174 2.188 200 .8822 .4351 2.282 2.291 210 .9009 .5593 2.425 2.425

 TABLE II - 1 LIQUID VOLUME Hexadecane Tc = 728 f unit = 109.2 V0 = .8195 - .7764 - .8373 cm³/g VS - VL = .084 T V2 V3 Ds V(calc.) V(obs) 20 .4461 .028 1.294 1.293 30 .4613 .024 1.305 1.305 40 .4766 .020 1.316 1.316 50 .4918 .017 1.328 1.328 60 .5070 .014 1.341 1.341 70 .5222 .011 1.353 1.353 80 .5374 .009 1.366 1.366 90 .5527 .007 1.379 1.379 100 .5679 .005 1.392 1.392 110 .5831 .0008 .004 1.407 1.407 120 .5983 .0008 .003 1.422 1.421 130 .6135 .0008 .002 1.436 1.436 140 .6287 .0017 .002 1.452 1.451 150 .6440 .0025 .001 1.467 1.466 160 .6592 .0033 .001 1.483 1.483 170 .6744 .0042 1.498 1.499 180 .6896 .0050 1.514 1.516 190 .7048 .0067 1.531 1.533 200 .7201 .0084 1.548 1.551 210 .7353 .0109 1.566 1.570 220 .7505 .0134 1.583 1.590 230 .7657 .0167 1.602 1.611 240 .7809 .0201 1.621 1.633 250 .7961 .0251 1.641 1.655 260 .8114 .0310 1.662 1.679 270 .8266 .0385 1.685 1.704 280 .8418 .0461 1.707 1.730 290 .8570 .0553 1.732 1.759

 TABLE II - 1 LIQUID VOLUME Benzene Tc = 564 f unit = 91.0 V0 =.7208 - .7208 - .8091 cm³/g VS - VL = 0 T V2 V3 Ds V(calc.) V(obs) 0 .3859 .0000 1.107 1.111 10 .4000 .0008 1.121 1.124 20 .4142 .0008 1.136 1.138 30 .4283 .0016 1.151 1.151 40 .4424 .0024 1.166 1.166 50 .4566 .0032 1.181 1.181 60 .4707 .0049 1.196 1.197 70 .4848 .0065 1.212 1.212 80 .4989 .0081 1.228 1.228 90 .5131 .0113 1.245 1.244 100 .5272 .0146 1.263 1.262 110 .5413 .0186 1.281 1.281 120 .5555 .0243 1.301 1.300 130 .5696 .0307 1.321 1.321 140 .5837 .0388 1.343 1.344 150 .5978 .0494 1.368 1.368 160 .6120 .0607 1.394 1.392 170 .6261 .0744 1.421 1.420 180 .6402 .0898 1.451 1.448 190 .6544 .1084 1.484 1.480 200 .6685 .1295 1.519 1.514 210 .6826 .1545 1.558 1.555 220 .6967 .1837 1.601 1.599 230 .7109 .2217 1.653 1.649 240 .7250 .2702 1.716 1.709 250 .7391 .3358 1.796 1.783 260 .7532 .4215 1.896 1.877 270 .7674 .5292 2.017 2.006

 TABLE II - 1 LIQUID VOLUME Acetic Acid Tc = 596 f unit a 94.55 V0 =.6346 - .5469 - .7016 cm³/g T V2 V3 Ds V(calc.) V(obs) 20 .3143 .0000 .949 .953 30 .3250 .0007 .960 .962 40 .3357 .0007 .971 .972 50 .3464 .0014 .982 .983 60 .3571 .0021 .994 .994 70 .3679 .0028 1.005 1.005 80 .3786 .0035 1.017 1.017 90 .3893 .0049 1,029 1.029 100 .4000 .0063 1.041 1.042 110 .4107 .0084 1.054 1.055 120 .4215 .0112 1.067 1.068 130 .4322 .0147 1.082 1.083 140 .4429 .0182 1.096 1.100 150 .4536 .0239 1.112 1.116 160 .4643 .0302 1.129 1.133 170 .4751 .0372 1.147 1.150 180 .4858 .0463 1.167 1.169 190 .4965 .0554 1.187 1.189 200 .5072 .0681 1.210 1.210 210 .5179 .0807 1.233 1.233 220 .5287 .0968 1.260 1.259 230 .5394 .1151 1.289 1.288 240 .5501 .1361 1.321 1.321 250 .5608 .1621 1.358 1.358 260 .5715 .1936 1.400 1.401 270 .5823 .2343 1.451 1.449 280 .5930 .2884 1.516 1.509 290 .6037 .3578 1.596 1.579 300 .6144 .4448 1.694 1.681

 TABLE II - 1 LIQUID VOLUME Ethyl Acetate Tc = 522 f unit = 86.35 V0 = .7043 - .7043 - .7771 cm3/g T V2 V3 Ds V(calc.) V(obs) 0 .3771 .0016 1.083 1.082 10 .3909 .0023 1.098 1.096 20 .4047 .0031 1.112 1.110 30 .4185 .0047 1.128 1.135 40 .4323 .0062 1.143 1.141 50 .4461 .0078 1.158 1.158 60 .4599 .0109 1.175 1.175 70 .4737 .0148 1.193 1.194 80 .4875 .0194 1.211 1.213 90 .5013 .0256 1.231 1.233 100 .5151 .0326 1.252 1.254 110 .5290 .0420 1.275 1.277 120 .5428 .0528 1.300 1.302 130 .5566 .0653 1.326 1.327 140 .5704 .0808 1.356 1.355 150 .5842 .0971 1.386 1.387 160 .5980 .1181 1.420 1.422 170 .6118 .1422 1.458 1.460 180 .6256 .1710 1.501 1.503 190 .6394 .2067 1.550 1.553 200 .6532 .2549 1.612 1.610 210 .6670 .3225 1.694 1.682 220 .6808 .4088 1.794 1.771

 TABLE II - 1 LIQUID VOLUME Ethyl Choride Tc = 461 f unit = 79.55 V0 = .7346 - .6545 - .7346 cm3/g T V2 V3 Ds V(calc.) V(obs) 20 .3761 .0125 1.123 1.119 30 .3889 .0169 1.140 1.138 40 .4018 .0228 1.159 1.159 50 .4146 .0309 1.180 1.181 60 .4274 .0397 1.202 1.204 70 .4402 .0507 1.226 1.229 80 .4531 .0639 1.252 1.256 90 .4659 .0801 1.281 1.287 100 .4787 .0984 1.312 1.320 110 .4916 .1212 1.347 1.357 120 .5044 .1477 1.387 1.399 130 .5172 .1814 1.433 1.447 140 .5300 .2255 1.490 1.504 150 .5429 .2872 1.565 1.572 160 .5557 .3746 1.665 1.661 170 .5685 .4856 1.789 1.789

 TABLE II - 1 LIQUID VOLUME Ethanethiol Tc= 498 f unit = 83.65 V0 = 7552 - .7552 - .7629 cm3/g T V2 V3 Ds V(calc.) V(obs) 0 .4044 .0031 1.163 1.160 10 .4192 .0038 1.178 1.183 20 .4340 .0053 1.195 1.196 30 .4488 .0076 1.212 40 .4636 .0107 1.230 1.227 50 .4784 .0137 1.247 1.245 60 .4932 .0183 1.267 1.266 70 .5080 .0244 1.288 1.287 80 .5228 .0320 1.310 1.312 90 .5376 .0412 1.334 1.335 100 .5524 .0519 1.360 1.361 110 .5672 .0648 1.387 1.387 120 .5820 .0809 1.108 1.416 130 .5968 .0969 1.449 1.451 140 .6116 .1182 1.485 1.488 150 .6264 .1442 1.526 1.531 160 .6412 .1747 1.571 1.577 170 .6560 .2144 1.626 1.631 180 .6708 .2647 1.691 1.695 190 .6856 .3364 1.777 1.773 200 .6904 .4318 1.887 1.873

## 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 nv 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 nt 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 nt = 1.

One of the influences which may increase the temperature factor nt 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 nt = 2. Table II - 3 gives the volumetric data for water (nt = 2), an inorganic liquid with a similarly complex molecular structure. Because of the relatively large solid state increments the quantity VS -VL has been determined individually for each temperature in this table using VS = 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 nt is greater than unity. Thus sulfur and phosphorus have nt values of 4 and 3 respectively. We may sum up the foregoing by saying that the extreme electronegative liquids ordinarily take the minimum nt value, unity, and nt 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 nt 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 nt 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, nv, 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.

 TABLE II - 2 LIQUID VOLUME (nt = 1) Hydrogen V0 = 9.318 - 9.318 - 10.459 cm3/g T V2 V3 V calc. V obs. -257 .298 3.598 13.21 13.35 -253 .363 4.466 14.15 14.03 -250 .419 5.449 15.19 14.87 -246 .503 7.269 17.09 16.53 -243 .550 8.409 18.28 18.52

 TABLE II - 2 LIQUID VOLUME (nt = 1) Fluorine V0 = .5241 - .4939 - .5543 cm3/g T V2 V3 V calc. V obs. -208 .0632 .0233 .610 .610 -203 .0677 .0268 .621 .620 -198 .0731 .0366 .634 .634 -193 .0775 .0438 .645 .646 -190 .0810 .05l0 .656 .657 -188 .0825 .0543 .661 .662

 TABLE II - 2 LIQUID VOLUME (nt = 1) Hydrochloric Acid V0 = .6104 - .6025 - .6498 cm3/g T V2 V3 V calc. V obs. -80 .2283 .0l36 .852 .849 -70 .2398 .0l95 .870 .869 -60 .2518 .0273 .890 .891 -50 .2639 .0377 .912 .915 -40 .2753 .0513 .937 .940 -30 .2874 .0676 .965 .970 -20 .2988 .0884 .998 1.003 -10 .3109 .1124 1.034 1.040 0 .3229 .1443 1.078 1.082 10 .3344 .1891 1.134 1.135 20 .3464 .2541 1.211 1.203

 TABLE II - 2 LIQUID VOLUME (nt = 1) Sulfur Dioxode V0 = .4394 - .4663 - .4932 cm3/g T V2 V3 V calc. V obs. -50 .2042 .0015 .645 .642 -40 .2131 .0025 .655 .652 -30 .2224 .0035 .665 .663 -20 .2313 .0049 .676 .674 -10 .2406 .0069 .687 .686 0 .2499 .0094 .699 .697 10 .2588 .0128 .711 .710 20 .2681 .0173 .725 .723 30 .2770 .0232 .740 .738 40 .2863 .0301 .756 .754 50 .2956 .0385 .774 .772 60 .3045 .0483 .792 .792 70 .3138 .0617 .815 .814 80 .3227 .0764 .839 .838 90 .3320 .0942 .866 .866 100 .3413 .1159 .897 .898 110 .3502 .1450 .935 .936 120 .3595 .1850 .984 .982 130 .3684 .2436 1.051 1.045 140 .3777 .3191 1.136 1.136

 TABLE II - 2 LIQUID VOLUME (nt = 1) Carbon Tetrachloride V0 = .4108 - .3772 - .4108 cm3/g T V2 V3 V calc. V obs. 0 .2019 .0004 .613 .612 10 .2093 .0004 .691 .620 20 .2167 .0008 .628 .627 30 .2241 .0012 .636 .635 40 .2315 .0016 .644 .643 50 .2388 .0021 .652 .651 60 .2462 .0029 .660 .660 70 .2536 .0037 .668 .668 80 .2610 .0053 .677 .677 90 .2684 .0070 .686 .687 100 .2758 .0090 .696 .697 110 .2832 .0115 .706 .708 120 .2906 .0148 .716 .719 130 .2980 .0189 .728 .731 140 .3054 .0234 .740 .744 150 .3128 .0292 .753 .757 160 .3201 .0357 .767 .770 170 .3275 .4444 .783 .785 180 .3349 .0534 .799 .802 190 .3423 .0637 .817 .820 200 .3497 .0760 .837 .841 210 .3571 .0g04 .858 .864 220 .3645 .1126 .888 .891 230 .3719 .1323 .915 .921 240 .3793 .1651 .955 .958 250 .3867 .2070 1.005 1.002 260 .3941 .2604 1.065 1.063

 TABLE II - 3 LIQUID VOLUME (nt) Water nt = 4 V0 = .7640 - .7640 - .8769 cm3/g T V2 V3 Ds V(calc.) V(obs.) 0 .2048 .0312 1.0000 1.0002 10 .2124 .0242 1.0006 1.0004 20 .2193 .0185 1.0018 1.0018 30 .2269 .0137 1.0046 1.0044 40 .2345 .0099 1.0084 1.0079 50 .2422 .0009 .0068 1.0139 1.0121 60 .2493 .0009 .0048 1.0188 1.0171 70 .2567 .0009 .0032 1.0248 1.0228 80 .2643 .0018 .0020 1.0321 1.0290 90 .2720 .0038 .0012 1.0390 1.0359 100 .2796 .0026 .0007 1.0469 1.0435 110 .2865 .0035 .0004 1.0544 1.0515 120 .2941 .0053 .0002 1.0636 1.0603 130 .3018 .0061 1.0719 1.0697 140 .3094 .0079 1.0813 1.0798 150 .3171 .0114 1.0925 1.0906 160 .3239 .0140 1.1019 1.1021 170 .3316 .0184 1.1140 1.1144 180 .3392 .0228 1.1260 1.1275 190 .3469 .0289 1.1398 1.1275 200 .3545 .0360 1.1545 1.1565 210 .3614 .0447 1.1703 1.1726 220 .3690 .0544 1.1874 1.1900 230 .3767 .0658 1.2065 1.2087 240 .3842 .0789 1.2272 1,2291 250 .3919 .0947 1.2506 1.2512 260 .3988 .1114 1.2742 1.2755 270 .4064 .1315 1.3019 1.3023 280 .4141 .1543 1.3324 1.3321 290 .4217 .1806 1.3663 1.3655 300 .4294 .2131 1.4065 1.4036

 TABLE II - 4 LIQUID VOLUME Sulfur nt = 4 V0 = .4578 T V2 V(calc.) V(obs.) 115 .0869 .545 .552 134 .0911 .549 .557 158 .0966 .555 .563 178 .1012 .559 .565 210 .1085 .567 .570 239 .1149 .573 .576 278 .1236 .582 .584 357 .1415 .600 .602

 TABLE II - 4 LIQUID VOLUME Lithium nt = 9 V0 = 1.7729 T V2 V(calc.) V(obs.) 200 .1826 1.96 1.97 400 .2606 2.03 2.04 600 .3369 2.31 2.11 800 .4149 2.19 2.19 1000 .4911 2.26 2.27

 TABLE II - 4 LIQUID VOLUME Silver nt = 16 V0 = .0923 T V2 V(calc.) V(obs.) 960 .0139 .106 .105 1092 .0154 .108 .109 1195 .0166 .109 .110 1300 .0178 .110 .111

 TABLE II - 4 LIQUID VOLUME Tin nt = 16 V0 = .1331 T V2 V(calc.) V(obs.) 300 .0093 .142 .145 450 .0118 .145 .147 600 .0142 .147 .149 700 .0158 .149 .150 800 .0174 .150 .151 900 .0192 .152 .153 1000 .0208 .154 .154 1100 .0224 .155 .156 1200 .0241 .157 .156

 TABLE II - 5 TEMPERTURE FACTORS Li Na K Rb Cs F 4 4 3½ 3½ 3 Cl 4 3½ 3½ 3 3 Br 3½ 3½ 3 3 3 I 3½ 3 3 3 2½

 TABLE II - 6 VOLUMETRIC UNITS Li(½) Na(1) )K(1½) Rb(2) Cs(2½) F(½) 1 1½ 2 2½ 3½ Cl(1½) 2 2½ 3 3½ 4 Br(2) 2½ 3 3½ 4 4½ I(2½) 3 3½ 4 4½ 5

 TABLE II - 7 LIQUID VOLUME T V0 V2 V calc. V obs. LiF 887 .3617 .2058 .568 .558 1058 .2358 .598 .587 LiCl 626 .4697 .2071 .677 .668 900 .2701 .740 .727 LiBr 547 .2701 .1240 .394 .392 700 .1472 .417 .410 NaF 1017 .3351 .2118 .547 .517 1214 .2443 .579 .549 NaCi 809 .4013 .2432 .645 .650 1010 .2885 .690 .697 NaBr 785 .2735 .1625 .436 .433 954 .1882 .462 .460 NaI 675 .2267 .1406 .367 .367 724 .1478 .375 .374 KF 913 .3230 .2145 .538 .534 1054 .2400 .563 .563 KCl 785 .4161 .2467 .663 .658 958 .2871 .703 .706 KBr 751 .2856 .1911 .477 .473 945 .2273 .513 .512 KI 700 .2538 .1614 .415 .411 751 .1698 .424 .420 RbF 820 .2245 .1374 .362 .347 1006 .1610 .386 .372 RbCl 734 .2858 .1881 .474 .476 822 .2046 .490 .493 RbBr 697 .2269 .1439 .371 .372 780 .1561 .383 .384 RbI 700 .2177 .1385 .356 .357 800 .1526 .370 .372 CsF 720 .2238 .1452 .369 .368 824 .1605 .384 .386 CsCl 661 .2229 .1360 .359 .360 741 .1478 .371 .372 CsBr 662 .1984 .1212 .320 .321 743 .1317 .330 .333 CsI 639 .1806 .1291 .310 .315 701 .1380 .319 .323