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 paper^² (and Larson’s original work⁸),

(4)

 where

            V_L = overall specific volume of liquid (cm^³/g) (total volume/total mass)

            V₁ = specific volume increment at 0^ºK 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 pressure^³, the volume is multiplied by , where P_o 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 n_v 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, n_p, and divided by the ratio of the base volume to 1, raised to the 2/3 power.  (The solid expression just uses volume, or s_o³.)  Therefore,

              atm                                                         (12)

Substituting eq. 10 in eq. 12, we get

              atm                                             (13)

n_p is the number of atoms effectively acting against the external pressure.  It is sometimes, but not usually, equal to the number of volumetric units, n_v.  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

            n_v, the number of volumetric units

            _s1, geometric factor

            _s2, geometric factor

            n_t, the number of temperature units

            n_p, the number of pressure units

Example Calculations and Comparisons with Experiment^5,6

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

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

The n_p 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 n_v, n_t, and n_p 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 P_lnu is unique.