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 pointfor 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 onedimensional 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, V_{0}, 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
V_{L} = (1 + T/510.2) V_{0 }

(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 V_{0} by V_{1} and
V_{2} respectively, equation 1 becomes
V_{L} = V_{1} + (T/510.2) V_{2}

(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 liquidaggregate 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 otherthanliquid 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 socalled
"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 finestructure 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 solidliquid transition is
the melting point and the unit applicable to this transition in the paraffin
hydrocarbons is onefourth 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 inthe probability unit and we
find that the unit applicable to half of the molecules is only onethird
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 intermolecular bond in the third dimension. The thirddimensional
volume increase therefore takes place isothermally so far as the individual
molecule is concerned and the added volume per critical molecule is V_{0}.
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}) V_{0}. Here again the value of V_{0}
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 V_{3}. The
complete volume equation for all three liquid components is then
V_{L} = V_{1} + (T/510.2) V_{2}
+ ½(f_{A} + f_{B})
V_{3}

(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 discussedin detail in a subsequent
publication, which willexamine 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 solidstate 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 (C_{14} and below) we
will take the molecular increment as +.080. The slightly higher value
+.084 will be applied to the paraffins above C_{14}, 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, V_{3}, 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 (V_{3}) for the aggregate.
The quantity
used in the foregoing multiplication, the critical volume increment per
molecule or third dimensional value of the initial volume, V_{0},
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 V_{1}, V_{2}, and V_{3}
are due to the modification of the basic value V_{0} by a geometric
factor which varies from .8909 to 1.00. In the base of hexane, or example,
V_{0} 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 111. In this
table the D_{s} and V_{3} volumes calculated in the manner
described are added to the constant V_{1} volume and the value
Of V_{2} 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 solidstate 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 cm^{3}/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

T_{c}
= 508
 f
unit = 84.8

V_{0}=
.9645  .9512  .9778 cm³/g
 V_{S}
 V_{L} = .080

T
 V2
 V3
 D_{s}

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

T_{c}
= 728
 f
unit = 109.2

V_{0} = .8195  .7764  .8373 cm³/g
 V_{S}
 V_{L} = .084

T
 V2
 V3
 D_{s}
 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

T_{c}
= 564
 f
unit = 91.0

V_{0} =.7208  .7208  .8091 cm³/g
 V_{S}
 V_{L} = 0

T
 V2
 V3
 D_{s}
 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

T_{c}
= 596
 f
unit a 94.55

V_{0}
=.6346  .5469  .7016 cm³/g

T
 V2
 V3
 D_{s}
 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

T_{c}
= 522
 f
unit = 86.35

V_{0}
= .7043  .7043  .7771 cm^{3}/g

T
 V2
 V3
 D_{s}
 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

T_{c}
= 461
 f
unit = 79.55

V_{0}
= .7346  .6545  .7346 cm^{3}/g

T
 V2
 V3
 D_{s}
 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

T_{c}=
498
 f
unit = 83.65

V_{0}
= 7552  .7552  .7629 cm^{3}/g

T
 V2
 V3
 D_{s}
 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 volumetemperature 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 CCl_{4}, which can quality either as organic
or inorganic., depending on the definition thatis used, and for SO_{2}
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 thecondensed
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 V_{0} 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 V_{0} 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 halfintegral 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.
TABLE
II  2

LIQUID
VOLUME (n_{t} = 1)

Hydrogen
V_{0} = 9.318  9.318  10.459 cm^{3}/g

T
 V_{2}
 V_{3}
 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 (n_{t} = 1)

Fluorine
V_{0} = .5241  .4939  .5543 cm^{3}/g

T
 V_{2}
 V_{3}
 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 (n_{t} = 1)

Hydrochloric
Acid V_{0} = .6104  .6025  .6498 cm^{3}/g

T
 V_{2}
 V_{3}
 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 (n_{t} = 1)

Sulfur
Dioxode V_{0} = .4394  .4663  .4932 cm^{3}/g

T
 V_{2}
 V_{3}
 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 (n_{t} = 1)

Carbon
Tetrachloride V_{0} = .4108  .3772  .4108 cm^{3}/g

T
 V_{2}
 V_{3}
 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 (n_{t})

Water
n_{t} = 4
 V_{0}
= .7640  .7640  .8769 cm^{3}/g

T
 V_{2}
 V_{3}
 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

n_{t}
= 4
 V_{0}
= .4578

T
 V_{2}
 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

n_{t}
= 9
 V_{0}
= 1.7729

T
 V_{2}
 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

n_{t}
= 16
 V_{0}
= .0923

T
 V_{2}
 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

n_{t}
= 16
 V_{0}
= .1331

T
 V_{2}
 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
 V_{0}
 V_{2}
 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

