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THE LIQUID STATE IN THE RECIPROCAL SYSTEM:
THE VOLUME/PRESSURE RELATION,
A CONTEMPORARY MATHEMATICAL TREATMENT, PART II


From thermodynamics,¹ the general equation of state of a pure substance is

 
(1)

where

 
volume expansivity
(2)

and

 
isothermal compressibility
(3)

(Of course, V = volume, P = pressure, T = temperature.)

From my previous paper² (and Larson’s original work8),

 
(4)

 where

            VL = overall specific volume of liquid (cm³/g) (total volume/total mass)

            V1 = specific volume increment at 0ºK and that due to the solid molecules in solution of the liquid (solid volume/total mass)

            V2 = specific volume increment due to the liquid molecules of the substance, temperature above 0ºK (liquid volume/total mass)

            V3 = specific volume increment due to the critical (gaseous or vapor) molecules in solution of the liquid (gaseous volume/total mass)

In this paper we will consider the effect of pressure on a liquid at temperatures below the liquid natural temperature unit, 510.8ºK.  At low temperature, .  Pressure has a different effect on V3 than it has on V2.  Also, pressure has a different effect on a liquid at a temperature above, rather than below, 510.8ºK.  These differences will be handled in another paper.

For a solid under pressure³, the volume is multiplied by , where Po is the internal pressure and P is the external pressure.  For a liquid under pressure, the volume is multipled by the square of the solid factor, or simply .  So,

                                                                           (5)

It follows that isothermal compressibility is

                   (6)

It's often easier to work with the bulk modulus, B, which is the inverse of

                                                                       (7)

From my previous paper,

              cm3/g                                                             (8)

since  is negligible for most liquids above the melting point.

              cm3/g                                                                       (9)

              cm3/g                                                                        (10)

where nv is the number of volumetric units.

The internal pressure of a liquid is obviously different from that of a solid.  The natural unit of pressure in the Reciprocal System is4 15,538,642 atm.  To calculate the internal pressure of a solid we divide this number by the interregional ratio, 156.45.  For a liquid, we divide by the square of the interregional ratio.  Because liquid cohesion is two-dimensional rather than three-dimensional we must also multiply the expression by 2/3.  Therefore,

              atm                                  (11)

This expression is then multiplied by the number of pressure units, np, and divided by the ratio of the base volume to 1, raised to the 2/3 power.  (The solid expression just uses volume, or so3.)  Therefore,

              atm                                                         (12)

Substituting eq. 10 in eq. 12, we get

              atm                                             (13)

np is the number of atoms effectively acting against the external pressure.  It is sometimes, but not usually, equal to the number of volumetric units, nv.  Using eq. 8, 9, and 10, B can be expressed as

                         atm                             (14)

Now let's turn to calculating the volume expansivity.

              K-1                     (15)

where  is the value of the expansivity at the end point of the solid.

One could plug (or 1/B) and into eq. 1 and integrate, but the resulting equation is more complex than eq. 5 and thus not useful.

            In summary, to calculate bulk modulus and volume expansivity of a liquid, it is necessary to determine

            m, the molecular weight

            nv, the number of volumetric units

            s1, geometric factor

            s2, geometric factor

            nt, the number of temperature units

            np, the number of pressure units

Example Calculations and Comparisons with Experiment5,6

I selected four important liquids:  acetic acid, carbon tetrachloride, ethyl acetate, and water.  Here are the results, in table format.

Chemical

Formula

M

Nv

Nt

Np

P atm

T ok

Balc atm

Bobs atm

Acetic Acit

CH3CO2H

60.05

.9046

.7820

4

1.0

7

1

288.16

11441.503

11279.014

Carbon Tetrachloride

CCl4

153.81

1.0

.9183

6

1.0

5

1

250.26

12334.317

11878.218

Ethyl Acetate

CH3CO2C2H5

88.10

.9818

.9818

6

1.0

6

1

293.16

8687.0274

8733.6283

Water

H2O

18.0153

.8707

.8707

1.5

2.0

9

1

273.16

19697.992

19698.877

Chemical

calc k1

obsK-1

acetit acid

1.1377x10-3

1.269x10-3

Carbon
Tetrachloride

1.240x10-3

1.2987x10-3

Ethyl Acetate

1.24398x10-3

1.304x10-3

Water

7.72383x10-4

7.992x10-4

(The values of  have not yet been determined, which explains the descrepancy between calc and obs.)

The np values are easy to understand.  In acetic acid, the CH3 contributes 3 units and the CO2H contributes 4.  In carbon tetrachloride, each atom contributes 1 unit.  In ethyl acetate, each volumetric group contributes a unit.  In water, 3 molecules of 3 atoms each act against the external pressure, for a total of 9.  All values of nv, nt, and np are integral or half-integral, as required by the nature of the Reciprocal System.  This is very different from the empirical correlations used by other investigators.7

In the coming years I hope some member of ISUS will calculate the results for thousands of liquids following the equations given here.

References:

1.  M. Abbott, H. Van Ness, Thermodynamics (New York:  McGraw-Hill Book Company, 1972), p. 105.

2.  R. Satz, "The Liquid State in the Reciprocal System:  The Volume/Temperature Relation, a Contemporary Mathematical Treatement," Reciprocity, Vol. XXIII, No. 2, Autumn 1994.  Incidentally, the normal function should have been denoted by , not erf

           

           

(The numerical results of the paper do not change, because  was actually used.)

3.  R. Satz, "The Equation of State of Solid Matter," Reciprocity, Vol. X, No. 2, Spring-Summer 1980.

4.  D. Larson, Nothing But Motion (Portland, Oregon:  North Pacific Publishers, 1979), p. 160.

5.  Handbook of Chemistry and Physics, 72nd Edition (Cleveland:  The Chemical Rubber Company, 1991-1992), pp. 6-108 to 6-110.

6.  American Institute of Physics Handbook (New York:  McGraw-Hill Book Company, 1972).  values are difficult to find.  If you know the volume at temperature i and temperature f (and the pressure is constant), then from equation 1, .

7.  R. Reid, J. Prausnitz, B. Poling, The Properties of Gases and Liquids, 4th Edition (New York:  McGraw-Hill, Inc., 1987).

8.  D. Larson, The Liquid State, privately circulated series of papers on the liquid state, circa. 1960-1964.  Note:  for this work, I made use of his paper IV.  Larson used the semi-empirical value 415.84 atm for the liquid natural pressure unit.  My derivation of Plnu is unique.


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