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This paper provides a step-by-step procedure for the calculation of liquid specific volume as a function of composition and temperature, based on the Reciprocal System of D. B. Larson1. In this theory, each individual molecule may be in the solid, liquid, or gaseous (or vapor) state, regardless of the state of the majority of molecules of the substance. First let's define some terms: Then, The initial values of these
three components are designated Just as the volume
of a gas is determined by the number of molecules, so the volume of a
liquid is determined by the number of volumetric groups which it contains.
In an organic compound, for instance, each of the common interior groups,
such as CH2, CH, or CO, constitutes one volumetric group.
The CH3 groups in the end positions of the aliphatic chains
occupy two units each. So hexane, represented as CH3CH2CH2CH2CH2CH3,
has 8 volumetric groups. Let Let
The natural unit of mass is
1 atomic mass unit, so For hexane, For the critical (gaseous
or vapor) specific volume increment, the geometric factor In a multi-group molecule,
the value of the geometric factors For hexane, for instance, From eq. 10 it's clear that
ordinarily Now that we have the initial
values as a function of composition, we can determine the values of the
three components as a function of temperature. The solid specific volume
increment not only includes the initial volume at 0 oK but
also a factor proportional to the number of solid molecules in the substance
at any temperature, To use the normal probability
function or table we need to know the value of the normal random variable,
zs, applicable. It should be proportional to the difference
between the liquid temperature We want the right tail of
the distribution, so we subtract the value of the normal function, denoted
by erf(zs), from 1 and then multiply by the average
difference in specific volume between solid and liquid molecules, denoted
by Larson uses an average value
of The thermal motion
beyond the initial point of the liquid (considered as starting at 0 oK)
is the one-dimensional equivalent of the thermal motion of a gas, and
thus the volume generated is directly proportional to the temperature,
In Ref. 2, Larson derived
the value of The gaseous or
vapor increment of specific volume depends on the proportion of critical
molecules existing in the aggregate at each temperature, which can be
computed from probability considerations. Larson uses two random variables
for this computation, both a function of the critical temperature, Then the specific volume increment due to critical molecules in the substance is For hexane, The .5747 factor means that 57.47% of the molecules at this temperature are in the critical state. Having determined
To automate the task of comparing the theoretical values with those observed, I've prepared a computer program and run it on most of the same liquids Larson used in the original series of papers: hexane, hexadecane, benzene, acetic acid, ethyl acetate, ethyl chloride, ethanethiol, fluorine, hydrochloric acid, sulfur dioxide, carbon tetrachloride, and water. Printouts from the program for all of these liquids follow. The observed values come from the same sources Larson used: Timmermans' Physico-chemical Constants of Pure Organic Compounds, the American Petroleum Institute, and the International Critical Tables. Most of the computer
results are in harmony with Larson's manual calculations. The two seeming
exceptions are for acetic acid and water. For acetic acid, Larson used
a value of initial liquid specific volume of .5469, which is .7795 that
of his base initial volume, .7016; but .891 is supposedly the smallest
allowed fraction. For water, Larson used a value of .7640 for both the
initial solid and liquid specific volumes, but this is only .8713 that
of his base initial volume, .8769, not .891. Actually. these differences
are due to "hydrogen bonding", which can allow closer packing
than normal. In a second calculation for water, I input 1.78 for To compute the specific volume for any liquid of your choice, follow these steps: 1. Determine the formula of the compound and its molecular weight. 2. From the formula, determine the number of volumetric units and number of temperature units. 3. Use equation 4 to obtain the base initial volume. 4. Use equations 9 and 10 to compute the geometric factors; some iteration here may be required to the get the right values. 5. Compute the initial volumes with equations 6, 7, and 8. 6. Using equations 12 and 13, compute the solid specific volume increment, equation 11. 7. Use equation 14 to compute the liquid specific volume increment. 8. Using equations 15 and 16, compute the critical specific volume increment, equation 17. 9. Sum the results to get the final value, from equation 1. References: 1. D. Larson, The Liquid State , privately circulated series of papers on the liquid state, circa. 1960-1964. Note: I made use of the papers numbered I, II, II-supplement, and III. I've reorganized all of the equations and changed some of the symbols for the sake of clarity. I've also used the latest values of the conversion constants. The computer program is entirely original. 2. D. Larson, Basic Properties of Matter (Salt Lake City, UT: International Society of Unified Science, 1959-1988), pp. 59-60. Appendix: The Computer Program The following pages show the input screens of the program. The data base language is filePro Plus and the computation language is TrueBasic. This is the first of what will be a comprehensive series of programs for the calculation of all properties of matter based on the Reciprocal System of theory. Eventually the programs will be made available for purchase. |
. These differ
only by a geometric factor (designated 
) applied to
a base initial value, 
, determined
as follows.
be the number
of volumetric groups and recall that the factor .707 expresses the geometric
reduction obtained by the close-packed arrangement of the liquid groups
because of their flexibility of movement. Then, in natural units, the
base initial volume is directly proportional to the number of volumetric
units, reduced by close-packing:
be the molecular
weight (non-dimensional) of the molecule of the substance, 
be the value
of the natural unit of atomic mass in g, and 
be the value
of the natural unit of liquid volume expressed in cm3. Then,
in conventional units, the basic initial value is 
is not
the cube of the natural unit of space in the time-space region, which
is applicable only to the gaseous state. Rather, 
is the cube
of the natural unit of space in the time region, which is 1/156.45 (the
inter-regional ratio) of that in time-space region, or 2.9139 x 10-8
cm. Cubing this we get
is 1.65979x10-24
g. Putting these values in eq. 3, we get
is 8 and the
molecular weight is 86.18. Therefore,
is always 1.00.
For the solid specific volume increment, the geometric factor 
is .891 (the
cube root of .707) where close-packing in the solid state can be achieved.
Where such packing cannot be achieved, the geometric factor 
is 1.000.
The same applies to the geometric factor for the liquid specific volume
increment, 
. Therefore,
the initial values of the three volume components may be expressed as
and 
represent averages,
since some groups may be at .891 while others at 1.000. Let 
= the number
of close-packed groups per molecule in the solid state, and let 
= the number
of close-packed groups per molecule in the liquid state. Then
is .9864 (with
1 group at .891 and 7 groups at 1.0000, the average is 7.891/8 or .9864)
and 
is .9728 (with
2 groups at .891 and 6 groups at 1.0000). Therefore, for hexane, the
initial values of the specific volume increments are
, which can
be determined by probability considerations.
and the melting
point 
, in degrees
K, divided by the melting point. The coefficient and the intercept have
unfortunately not been worked out theoretically, but are given empirically
by Larson (Ref. 1) as follows:
:
= 178 K (-95
oC). At 
= -50 oC,
zs=1.41 and from the normal probability table,
erf(zs) 
.9207. Subtracting
this from 1.0000, we get .0793, which means that 7.93 % of the molecules
in the liquid hexane aggregate at -50 oC are in the solid state.
Multiplying this figure by the approximate difference in specific volume
between solid and liquid molecules, .080, we get .0063 cm3/g
for the value of 
. Let 
be the natural
unit of temperature in the time region (for the condensed states of matter)
and 
be the temperature
factor. Then
to be 510.8
K. For simple substances, 
is 1. More
complex or more electropositive substances have values of 
of 2 up to
16. Hexane has a value of 1; water, 2; silver, 16. Compounds of electropositive
and electronegative elements have intermediate values (some with half-integral
values, which are averages), as would be expected.
:
= 508 K. At
= 210 oC,
= .2947 and
=.8106. The
corresponding values of the normal probability function are .6144 and
.8109. Then, from eq. 17, 
we can now
calculate 
from eq. 1.
and 
so as to get
the initial volumes to be .7640. The theoretical results computed came
out to be much closer to the experimental ones than the previous run.