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The formulation of the so-called law of rectilinear diameter for the determination of the critical volume of substances in the concluding decades of the nineteenth century became in a very useful and acceptably exact alternative tool for researchers in the field of critical phenomena. Its corresponding original expression, and even those of its early few modifications, were so mathematically simple that their use did not limit to exclusively contribute to remove the by then experimental obstacle for the estimating of this critical parameter, but also extended along several decades in the increasing applications of the principle of corresponding states.
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Quim. Nova, Vol. 33, No. 9, 2003-2010, 2010
Assuntos Gerais
Simón Reif-Acherman
Escuela de Ingeniería Química, Universidad del Valle, A. A. 25360 Unicentro, Cali, Colombia
Recebido em 5/1/10; aceito em 29/4/10; publicado na web em 1/9/10
The formulation of the so-called law of rectilinear diameter for the determination of the critical volume of substances in the
concluding decades of the nineteenth century became in a very useful and acceptably exact alternative tool for researchers in the field
of critical phenomena. Its corresponding original expression, and even those of its early few modifications, were so mathematically
simple that their use did not limit to exclusively contribute to remove the by then experimental obstacle for the estimating of this
critical parameter, but also extended along several decades in the increasing applications of the principle of corresponding states.
Keywords; rectilinear diameter; critical density; history of science.
The discovering of the critical phenomena by the French Baron
Charles Cagniard de la Tour (1777-1859) in 1822, and its later de-
velopment by the Scottish scientist Thomas Andrews (1813-1885)
in the second half of the nineteenth century, became a decisive step
in the efforts for studying the matter behaviour.1,2 The definitive
establishment of the real role and meaning of the concept of critical
point would come with the Johann Diderik van der Waals’s (1837-
1923) formulation of the principle of corresponding states several
years later, which did not only allowed the definition of a new set of
reduced variables, but also the initial possibility of needing only one
universal equation of state as a basis for the estimation of different
properties of several classes of compounds.3
A basic requirement for the use of this principle was the avai-
lability of the experimental values of the three critical constants,
pressure, temperature and volume. The usual, and practically the
only by then known, methods for the measurement of these critical
parameters were essentially either Caignard de la Tour’s, based on
the vanishing of the meniscus on heating a liquid in a sealed tube, or
Andrews’s, based on the determination of isotherms at various tem-
peratures and the selection of the one for which the horizontal part
just disappeared. While the measurement of the critical temperature
by using vapours of pure liquids boiling under known pressures was
the easiest experimental procedure, and that of critical pressure only
required the guarantee of absence of impurity in the compound of
interest for a relatively simple procedure, the determination of the
critical volume or density of a substance was a much more difficult
matter. It was clear from that time that the reasons of this extreme
difficulty were due to the natural peculiarities of the critical point,
and, specifically, to the small curvature of the coexistence curve
close to it, which leaded to great volume sensitivity with variations
of temperature and pressure. The determination of the critical volume
was generally subordinated to that of the critical temperature, and it
was verified that experimental inaccuracies of the order of 0.1 ºC, for
example, produced significant greater alterations in the volume. About
one decade later the French physicist Luis Georges Gouy (1854-1926)
raised the influence of the gravitational field on matter and its capacity
for generating a sizeable density gradient in a supercritical fluid due
to the large compressibility as a second reason for the inaccuracies
in the determination of the critical volume.4
The search for an alternative method for the determination of
critical volume was increasingly turning into an evident need in the
concluding part of the nineteenth century. Although the early inves-
tigators on the critical phenomena didn’t establish this searching as
an explicit objective, it would be incidentally reached in the course
of the researches carried out in this field by two French scientists.
The purpose of this article is to show the more relevant facts related
with the formulation of the first analytical proposed method for the
determination of the critical volume of pure compounds, from then
known as the law of rectilinear diameter, as well as its first modifica-
tions. The fact that this method of determination is still nowadays cited
with comparative purposes in the evaluation of newer and obviously
more sophisticated analytical and experimental techniques for the
estimation of this critical parameter emphasizes its importance. Some
few modifications in the original nomenclature used by different
scientists over the years this story succeeded were done in order to
preserve the consistence of the full article.
The Andrew’s results about a symmetric description of what ha-
ppened to liquid and vapour at the critical point aroused the interest
of the scientific community on the subject. The initial interests of the
researchers, focused on the determination of physical and thermody-
namical properties in this region, were quickly involved in theoretical
controversy over the interpretation of the experiments on the nature
of the vapour-liquid transition. The polemic, which extended over
several decades, set proponents of two basic conceptions against each
other; one which defended the notion of continuity of states and the
indistinctness of the two states in the supercritical condition, and the
other which hold the theory of the differences between liquid and
vapour molecules and the explanation of the critical phenomena in
terms of the liquid dissolution in the vapour at the critical point.5 One
incidental figure in this whole research program and the mentioned
controversy was the French physicist Louis Paul Cailletet (1832-
1913) (Figure 1).6
From the beginning the Cailletet’s main interest was not focused
on the formulation and discussion of sophisticated theories, but in
the development and implementation of experimental techniques for
studying the fluids’s behaviour at high pressures. From the 1860’s he
centred the efforts of his lately scientific career especially in the study
of the pressure influence, initially on the chemical phenomena and the
spectral stripes,7,8 and later on the compressibility of gases and liquids,
Reif-Acherman2004 Quim. Nova
studies in which he combined a great deal of technological skill and
experimental curiosity.9 His most widely known achievement is the
first liquefaction - almost simultaneously with Raoul-Pierre Pictet
(1846-1929) - of oxygen in 1877, using a method that involved high
compression, mild cooling and finally a sudden decrease in pressure.10
It was precisely through his researches on liquefaction of this and
other gaseous compounds and their corresponding applications that
Cailletet was involved in the studies on critical points.11,12 Although
his description of the critical point phase transition was wrong,5 the
skills he acquired by analyzing different details involved in the design
not only of some apparatus but also of most of the instruments of
measuring required for guarantee the precision in the whole resear-
ch let clearly him for understanding the technical limitations of the
lectures in the critical zone.13
Cailletet carried out great part of his whole research on liquefac-
tion of gases at the École Normale Supérieure in the laboratory of
the French chemist Henri Etienne Sainte-Claire Deville (1818-1881).
It is there when he met the physicist of the Faculty of Sciences of
Paris, Émile Ovide Joseph Mathias (1861-1942) (Figure 2), who
became Préparateur in the Laboratory of Education of Physics at
the Sorbonne.14 As Cailletet, Mathias didn’t accept the notion of
continuity of states, and, consequently, the identity of gaseous and
liquid molecules.5 The Director at the Sorbonne was by then the also
physicist Edmond Marie Leopold Bouty (1846-1922), who knew
Cailletet in relation to the work they carried out together on the elec-
tric conductivity of metals.15 He was the person that very probably
recommended Mathias to the latter as assistant in his experiments.
Cailletet and Mathias began in 1886 their research on the densities
of liquefied gases and their saturated vapours. Their main interest
was the possibility for studying the principle of corresponding states
and its deviations. The definition of the more appropriate apparatus
to be used for the determination of both saturated vapour and liquid
densities, which showed then experimental difficulties at high pres-
sures, become the first objective. The Figure 3 shows the apparatus
designed by Cailletet and Mathias for determining liquid (left) and
gas (right) densities. 19
The traditional procedure of the glass floats used by early expe-
rimentalists, such as, for example, Michael Faraday (1791-1867),
for determining liquid densities as a part of his whole research on
liquefaction of different gases,16 or other proposed by the Polish
scientist Zygmunt Wroblewski (1845-1888) by comparing the ga-
seous and liquid volumes in equilibrium,17 led to only approximate
results because of the compression effects and their limited ranges
of application, and were consequently dismissed. The apparatus
Cailletet and Mathias designed, based on a by then recent simple
adaptation of the communicating vessels’s principle, consisted of a
reservoir of about 600 cm3 fixed to a big test tube in steel containing
mercury.18 A glass tube O-shaped, composed of two equal finely
graduated branches of about 0.5 m length and 1.5 mm internal dia-
meter, also including a definite amount of mercury, was welded to
the gas reservoir by other bent back twice tube. The compression
of the gas in the reservoir by using a pump previously designed by
Cailletet for his purpose,20 and the simultaneous refrigeration of
one of the two branches of the graduated tube led to some evapo-
ration of liquid and the condensation of other amount in the second
branch. The liquid density could then be easily determined with the
lectures of the mercury heights in both branches and the knowledge
of the density of mercury. This design, which allowed to always
working with the same weight of gaseous substance, was found to
be resistant to pressures up to 200 atmospheres, correcting so the
weakness of previous methods.
With regard to the vapour densities, the inapplicability of
some original previous methods at higher pressures allowed again
Cailletet and Mathias for designing different equipments to those
previously used.21 The apparatus consisted of a precisely gauged
thick tube welded to a cylindrical reservoir of about 60 cm3 opened
in its low extreme.19 An exact determination of the total volume
and the measurements of pressure and temperature allowed the
estimation of the weight of the gaseous substance. The reservoir
was screwed on to the test tube of the apparatus and this latter
one jacketed with concentric tube containing a liquid at constant
temperature. The saturation point was determined, following a
procedure previously reported for the liquefaction of hydrochlo-
ric acid,22 by observing the cessation of the rising or decreasing
of the lecture of a manometer with the gradual compression or
decompression of the gas under consideration. The exact determi-
nation of the total volume, as well the pressure and temperature
at saturation, allowed the estimation of the weight of the gaseous
substance, and hence its density.
Figure 1. Louis Paul Cailletet (Courtesy of François Darbois)
Figure 2. Émile Ovide Joseph Mathias (Courtesy of the Président de Clermont
Communauté and the Observatoire de Physique du Globe de Clermont-
Figure 3. Apparatus for the determination of the (a) liquid and (b) vapour
densities (from ref. 19)
The history of the rectilinear diameter law 2005
Vol. 33, No. 9
Cailletet and Mathias reported experimental results for only three
compounds in their first report. It was there where the authors did by
first time reference, although minimal and without consider it as a
law, to the possibility for determining the density in the critical point
based on the graphical representation of the density of liquid rl and of
saturated vapour rv in equilibrium with it in the ordinates versus the
temperature t in the abscissas. After observing the linear behaviour
of the locus of the points bisecting the joins of corresponding values
of these orthobaric densities with temperature,
where r0 is the mean density and a a constant, different for each
substance, they concluded that the intersection of this line with the
critical temperature tC (where “the two curves would seem to be con-
nected”) would allow them to estimate the critical density (Figure
4).19 According to it, the equation took the form of
The curves showed in Figure 4 indicate the values of critical
density they found of 0.46, 0.41 and 0.22 for nitrous oxide, carbonic
acid (carbon dioxide), and ethylene, respectively. While the two last
values don’t differ significantly of those nowadays accepted values
of 0.214 and 0.468; that corresponding to nitrous oxide presents a
relative error of about 10% regarding the actually used of 0.452.23
The successful results led Cailletet and Mathias to conclude that the
same experimental techniques could be applied to all the substances
whose critical temperature was higher than that corresponding to the
freezing of mercury.19 Nevertheless, and in order to generalize the
described method, they determined to also use it with substances with
higher critical points, and chose sulphurous acid (sulphur dioxide)
for that objective. These results were the subject of their second
report.24 The corresponding critical density they found by using this
original graphical method of 0.520 was very close to the nowadays
accepted value of 0.525.23
The proposed law was confirmed about five years later by the
French physicist Emile Hilaire Amagat (1841-1915) for the specific
case of carbon dioxide,25 with new results of saturated liquid and
vapour densities obtained by implementing different and remarkably
more accurate techniques for its experimental determination. The
new obtained critical density of 0.468 was far more closely to the
actual value.26 The fact that the range of critical temperatures of the
substances under study until the moment, with the only exception of
sulphur dioxide, was very narrow, didn’t allow still, however, a more
complete confirmation.
Improvements in the law
Once the two mentioned reports were published, Cailletet se-
emed to have lost interest on the subject and focused his research
work mainly on atmospheric phenomena and aeronautics. The only
exceptions were some researches particularly focused in the study of
water.27 Nevertheless, other person simultaneously appeared in the
scene: the British chemist Sidney Young (1857-1937) (Figure 5).28
His interest on critical constants arose few years after his enrolment
in 1882 as lecturer in Chemistry at University College, Bristol, as a
consequence of a fruit partnership with his compatriot Sir William
Ramsay (1852-1916), which lasted for 5 years.
When Young came to Bristol, he found Ramsay engaged in two main
investigations: the determinations of specific volumes of liquids at their
boiling points, and of vapour pressures and critical constants of benzene
and ether. Ramsay convinced Young to get involved in the searching
of possible relationships among the behaviour of different compounds
at equal values of reduced variables, according to deductions made by
Johan Diderik van der Waals at setting out his equation of state.29 The
verification process required the disposal of great amount of experimen-
tal data, such as liquid and vapour densities, vapour pressures, and, of
course, critical constants of the compounds under consideration. Young
was conscious of the technical difficulties for directly determining the
critical density and after to suggest different experimental procedures and
some analytical relations he finally decided to proceed with the method
of Cailletet and Mathias and verify its accuracy. Using a new method
for the determination of saturated densities in both states,30 he was able
to state the validity of the law, although with some few quantitative
differences.31 The compounds included in the study were benzene and
three of its halogen derivates, carbon tetrachloride, stannic chloride, acid
acetic, ether and three alcohols, with temperatures that went up until
325 ºC. Very quickly the verification process included five of the lower
esters (methyl formate, methyl acetate, methyl propionate, ethyl formate
and ethyl acetate). The constants (A = r0) are given in Table 1.32
Figure 4. First graphical representation of the rectilinear diameter for nitrous
oxide, ethylene, and carbon dioxide (from ref. 19)
Figure 5. Sydney Young (Reprinted with permission from J. Timmermans,
Endeavour 6, 11-14. Copyright 1947)
Reif-Acherman2006 Quim. Nova
The verification of the law showed not be, however, perfect. The
three alcohols, methanol, ethanol and propanol, and specially the first
one, showed a very decided curvature, too pronounced to be neglected.
Although Mathias initially considered these results within the range
of probable experimental errors, this first notice of exceptions to
the supposed universality of the proposed law rose his’ and Young’s
interests for deeper analysis.
In order to preserve the utilization of a so simple and useful law,
Young decided to undertake a careful experimental research that
involved 30 compounds. One objective was to try of correlating the
circumstances of the abnormal results and to classify the observed
deviations according to some new clearly defined theoretical para-
meters, which must to be characteristic for each substance. Although
separately exposed, the Mathias’s and Young’s approaches coincided
in their selection. The first parameter for studying was the ratio of
the actual to the theoretical (for an ideal gas) density, rC / rC , both
evaluated at the critical point. Mathias, and something later Young
and the Swiss chemist Philippe A. Guye (1862-1922) had separately
concluded that a requisite for the true observance of the van der
Waals’s equation (and others equations of state by then known too, as
that by Clausius for example) for normal substances (it meant those
whose molecules underwent no dissociation or polymerization) was
that this ratio should to be strictly constant.31,33,34 The value of this
parameter could to serve as a criterion for evaluating the degree of
complexity of molecules, which in their opinion should to be related
with the observed departures of the estimated values of the critical
density for alcohols. According to van der Waals, the value of this
constant was 8/3, or 2.6, which derived from the author’s observation
about that the volume correction depending on the molecular size in
the equation, b, was equal to 4 times the real volume occupied by
spherical molecules under normal conditions of pressure and tem-
perature. Young fulfilled the requisite at replacing the coefficient 4
by 42, previously suggested by the German physicist Oskar Emil
Meyer (1834-1909),35 which led him to a constant ratio of 3.77 for
all the substances included in the study.
A second parameter was related with the slope of the rectilinear
diameter for each substance. Mathias similarly showed that in order
to guaranty the obedience of these lines to the corresponding states
principle the angular coefficient a should to be directly proportional
to the critical densities of the respective substances and inversely
proportional to their absolute critical temperatures.33 In mathematical
terms, for each substance,
or (3)
The whole experimental program had, however, significant limita-
tions. In the great majority of cases the only densities experimentally
determined were those of the liquids at temperatures below the boi-
ling point. If the vapour pressures were known, the saturated vapour
densities at these low temperatures were then calculated by using
the van der Waals’s equation of state, and thus the mean densities of
both values could to be ascertained. The data of critical temperature
became the main obstacle. If it was known, the critical density was
then calculated by following the law of Cailletet and Mathias. But
as for most substances the critical temperature had not been by then
directly determined, Young must to use an alternative analytical
formula for its estimation. The selected empirical expression was
one deduced by Thorpe and Rücker,36 which combined the van der
Waals’s theory with an expression developed by the Russian chemist
Dmitri Ivanovich Mendeleev (1834-1907),37
where rl1 and rl2 were the densities of the substance in liquid state at
temperatures T1 and T2, respectively, and B an universal constant. Al-
though the authors had recommended a value for B of 1.995, Mathias
had showed that if it was equal to 2, then it followed that a = 1 and
it would be then possible to ascertain both the critical density and
temperature by using geometrical methods. It was later demonstrated
that not only the suggested value for B led to low accuracy, but that
it always differed sensibly from 2 as well as a from the unity, with
which the geometrical method became inapplicable.
In a detailed paper published some years later, Young reached
important conclusions.38 The first one arose as an unplanned conse-
quence of the incidental range of temperatures in which he worked.
According to the by then existing experimental limitations, he found
himself forced to use densities data at different intervals of tempe-
rature for determining the values of the mean densities for each one
of the 30 substances. While for some of them he used experimental
information between the respective normal boiling points and some
few degrees below to their critical points, for the others the densities
used for his conformation of the law corresponded to temperatures
between 0 ºC and the mentioned boiling points. After the analysis of
the full package of results he was able to conclude that while the errors
involved in the use of the simple right line proposed by Cailletet and
Mathias rarely exceeded 0.25%, and generally not exceeded 0.1% for
the first interval of temperatures, they were considerably higher for
the other range. As it could be easily imagined, Young found signi-
ficant differences between the pair of constants calculated by using
experimental data from one or other range This observation allowed
him to suggest a more reliable representation of the locus of the mean
value of densities by using a equation, which after intersecting the
critical temperature took the form:
where the constants r0, a, and b must to be calculated again for each
substance. For the alcohols, however, he proposed to add a fourth
term, γtC
3, in order to get a satisfactory agreement. The constants
for the thirty substances are shown in Table 2. The circumstance by
which all the values of a are negative in both formulas, the simplest
one and the extended, is consequence of the fact that the variation of
the saturated liquid density is always greater than that of saturated
vapour. The values of the constant b found by Young were positive
for some substances and negative for others.
The marked rapid increasing in the deviations Young observed
with regard to the using of the law of Cailletet and Mathias below
the boiling point was reinforced with the different types of curvature
showed by plotting the differences against temperature for several
substances. The examples shown in the Figure 6 cover different
possibilities (DC/DC = rC / rC ). Pentane, the only substance of those
under investigation that didn’t show the slightest deviation from the
law of Cailletet and Mathias and for which b is obviously equal to
0, could be presented as a reference point. Young also concluded that
Table 1. Young’s constants for the right line of the rectilinear diameter for
five lower esters
Substance A. a.
Methyl Formate . . . . . . . . . 5025 – . 0007155
Methyl Acetate . . . . . . . . . 4839 – . 0006740
Ethyl Formate . . . . . . . . . . 4759 – . 0006490
Ethyl Acetate . . . . . . . . . . . 4644 – . 0006250
Methyl Propionate . . . . . . 4721 – . 0006210
The history of the rectilinear diameter law 2007
Vol. 33, No. 9
the deviations from the simplest form of the law, or the curvatures
in geometrical terms, were generally smaller the nearer rC /rC ap-
proached the normal value of 3.77 and the nearer a approached the
value of 0.93. In nearly every case, the showed curvature was in op-
posite directions according to rC /rC was greater or less than 3.77, and
a was greater or less than 0.93. It was found too that b was positive
whe0n rC /rC was lower than 3.77 and negative when it was higher.
Among confirmations, limitations, extensions and alternative
The conclusions reached by Young put the law in its true place.
The so called law was not universal and the existing deviations were
mainly associated with the nature of substances and the range of
temperatures in which the required saturated densities data for its
application were estimated. As with Cailletet, the Young’s interests
began to slightly move to other fronts, such as the experimental
determination of vapour pressures and boiling points of specific
groups of substances and probable correlations between them, and
didn’t work more on the rectilinear diameter. Mathias unsuccessfully
tried to generalize the estimation of a by correlating it with different
variables in the framework of the principle of corresponding states.39
The experimental evidence on the variability of this parameter led
him to also conclude about the non-universality of the principle. This
fact could to be historically interpreted as one of the first suggestions
about the requirement for including in the principle new parameters
related with the nature of substances in addition to the reduced vari-
ables proposed by van der Waals.
Mathias continued insisting in the verification of the law. Af-
ter verifying the law with acetylene,40 his experimental work was
translated abroad. The full program of liquefaction of the so-called
permanent gases carried out by the Dutch physicist Heike Kamerlingh
Onnes (1853-1926) at his laboratory at the University of Leiden from
the last quarter of the XIX century became an excellent opportunity
for his purposes. The expectation that the diameter of the densities
curve would reveal a characteristic feature of the whole representation
of the different behaviours and their respective analytical reduced
equations of state for substances with very low critical temperatures
with regard to those of the compounds by then considered as normal,
put this parameter in the first front of whatever comparative study.41
Mathias received financial support from the French Academy of
Sciences for his journey and stay at The Netherlands in at least two
opportunities, 2000 francs in 1909,42 and 4000 francs in 1920.43 The
money came from some funds created in 1908 by the Prince Roland
Napoleon Bonaparte (1858-1924) for the promotion of the scientific
research.44 During the periods Mathias was summer guest at Leyden,
initially between 1910 and 1923 with Kamerlingh Onnes as host, and
until 1932 with the also Dutch physicist Claude August Crommelin
(1878-1965) as the new director of the laboratory, he collaborated
with the measurement of the coexistence curves and the estimation of
the parameters of the mathematical expression of the rectilinear diam-
eter of many fluids such as oxygen,45 argon,46 nitrogen,47 hydrogen,48
neon,49 helium,50 ethylene,51 carbon monoxide,52 and krypton.53 The
Figure 1S, supplementary material, shows the schematic arrangement
used in these researches for the specific case of oxygen.
As the experimental procedures for the measurement of densities
slowly became more precise,54 different reviews of critical constants
for several substances began to be published.54,55 The comparison of
the reported values with those calculated by assuming the rectilinear
behaviour of the diameter increased the possibilities for finding examples
of deviations to the proposed law. Unlike most papers published by the
Leyden Physical Laboratory, these deviations were well above the limits
of experimental errors usually estimated between 0.5 and 2.0%, and,
occasionally higher. These deviations, which were a clear consequence
of the discontinuous character of the densities curves, were specifically
apparent in the vicinity of the critical point and were related with the yet
mentioned extreme experimental difficulties for working in this zone. The
Figure 7 shows the results found in the beginnings of the XX century for
Figure 6. Observed deviations of the estimations by using the law of Cailletet
and Mathias below the boiling point of the substances (from ref. 38)
Table 2. Young’s constants for the extended formula of the rectilinear diameter
for thirty substances
Name D0a x 107b x 1010 γ x 1013
Carbon Tetrachloride .8165 – 9564 + 1480
Hexamethylene .3985 – 4685 + 791
Isopentane .3202 – 4658 + 463
Stannic Chloride 1.1387 –12760 + 977
Benzene .4501 – 5248 + 693
Di-isopropyl .3401 – 4445 + 413
Normal Pentane .3232 – 4610 0
Fluorbenzene .5236 – 6000 + 293
Chlorobenzene .5640 – 5337 – 509
Iodobenzene .9303 – 7556 – 519
Di-isobutyl .3550 – 4115 – 592
Ether .3685 – 5377 – 475
Bromobenzene .7609 – 6655 – 725
Normal Hexane .3388 – 4445 0
Normal Heptane .3504 – 4192 – 621
Methyl Isobutyrate .4558 – 5593 – 689
Normal Octane .3590 – 3954 – 1046
Propyl Formate .4647 – 5748 – 459
Ethyl Formate .4741 – 6251 – 694
Methyl Propionate .4696 – 5921 – 729
Methyl Butyrate .4601 – 5430 – 906
Ethyl Propionate .4564 – 5644 – 784
Methyl Formate .5020 – 7013 – 665
Propyl Acetate .4553 – 5469 – 1124
Methyl Acetate .4799 – 6280 – 1467
Ethyl Acetate .4624 – 5992 – 764
Ethyl Alcohol .4028 – 3827 – 5940 + 651
Propyl Alcohol .4095 – 3790 – 3750 – 5533
Methyl Alcohol .4050 – 4479 + 1330 – 23760
Acetic Acid .5355 – 5366 – 1191
Reif-Acherman2008 Quim. Nova
the specific case of sulfur dioxide, where accurate measurements of the
critical density indicated a difference of 1.36% between the experimental
value and that calculated with the relationship of Cailletet and Mathias.56
The evidence of deviations to the law raised as well the interest of
researchers for proposing alternative analytical expressions for a more
precise estimation of the critical density. The Dutch chemist Johannis
Jacobus van Laar (1860-1938) and the German professor of physical
chemistry Walther Hertz (1855-1930) modified the law by stating
the following expression for density as a function of temperature,
where r0 is the density at absolute zero and a = [δ/(1 + δ) TC].57 For
normal substances δ = 0.9, hence a = 1/2.1 Tc and r = r0 (1-T/2.1 TC),
from which the density at 0 K may be calculated, with only a pair of
experimental values. The critical density was then easily estimated
by evaluating the Equation 6 for TC.
Two other expressions that proclaimed to supply more satisfactory
results than those gotten by using the law by Cailletet and Mathias
didn’t receive, however, any additional repercussion. The first one,
arisen as a consequence of the initial development of an equation
for the calculation of the heat of vaporization of normal and non-
associated substances.58 It stated,
where rl and rv are the saturated liquid and vapour densities, respec-
tively, at the temperature T. The other one had a similar mathematical
simplicity than that of the rectilinear diameter but was verified only
for very few compounds.59 It was
Mathias himself, and the Austrian professor Hanns von Jüptner,
proposed the different expression,
where Tr is the reduced temperature and a a characteristic function of
the critical temperature.41,60 According to Mathias, a = bTC
n, where b
is a specific constant for each particular substance and n an universal
constant very close to 0.25.
Other work suggest the possibility for estimating the angular
coefficient a from characteristic parameters of each substance,
or (10)
where rb and ab are the liquid density and the coefficient of expan-
sion of the liquid at the boiling point Tb.61
A renewed interest in the subject of rectilinear diameter arose
from 1970’s. The initial very little or no one experimental support
for the theoretically predicted deviation from a linear diameter on the
liquid-vapour transition near the critical point strongly changed.62,63
The report of the coexistence curves of some metals (rubidium and
cesium) offers evidence of significant singularity in the diameter of
the predicted form.64 Several suggestions have been proposed based
on the analysis of these results, which include, as an example, a linear
dependence of the slope of the diameter with the acentric factor of
the substance and the possibility for relating it with the shape of the
pair potential.65 A more recent approach, based on the philosophical
structure of the law of rectilinear diameter, allows the estimation of
the critical density without the help of the critical temperature.66 The
mathematical relationship, so-called “the model of three densities”,
derived from two empirical equations for the surface tension and
with a general pattern suggested by the graphic image of the coexis-
tence system with the rectilinear diameter of Cailletet and Mathias,
states that
where A is a universal constant. If the only single data point of coex-
isting densities required for the method is taken at a pressure of one
atmosphere, for example, the relationship may be written as
where rlnbp and rvnbp are the coexisting densities. The values of the
two constants were determined on the statistical analysis of a data
base of 183 substances used in the research, which included elements
and inorganic and organic compounds. According to the author of
this research, the testing of the performance of this equation with the
data base gave a mean absolute difference of 1.24% and maximum
difference of 5.77%.
Although the rectilinear diameter rule has recently satisfactorily
verified for some specific substances,67 it is generally nowadays
considered incorrect for the full group of elements and compounds,
and non-analytic contributions determine the behaviour when ap-
proaching the critical point.68 The relative importance of the various
non-analytic terms determining the asymptotic behaviour is difficult
to assess and remains an actual topic of experimental and theoretical
The so-called law of Cailletet and Mathias becomes an interesting
case in the study of the volumetric and thermodynamic properties of
substances because of the continued actuality it had evidenced from
the initial formulation. Its original empirical character has not any
doubt in a historical context. Two elements support this assertion: the
fact that there is not any minimal suggestion in the available historical
accounts that allow us to affirm that Cailletet and Mathias knew intui-
tively the existence of a mathematical relation between the densities
of the liquefied gases and their saturated vapours in 1886, and the
publication, initially in a very specific lattice model (with a finite lattice
constant) in 1952,70 and later in a more general frame in 1973,71 of the
Figure 7. Representative graph of the deviations to the law of rectilinear
diameter for sulfur dioxide (from ref. 56)
The history of the rectilinear diameter law 2009
Vol. 33, No. 9
first trials for its theoretical derivation. As it is true that this expression
is not the only mathematical representation of a physical phenomena
that subsists almost a century and a quarter after his publication, it is
also true and unusual that a law with a so “beautiful simplicity” (as
Mathias himself referred to),72 has still validity according not only to
its highly comparative accuracy regarding more modern methods for
the estimation of critical densities, but also to the applications that have
been conformed or added to those initially established.73
The Figure 1 showing the schematic diagram of the arrangement
used for the measuring of saturated liquid and vapour densities of
oxygen at the Leyden Physical Laboratory is available on http://, in PDF format, with free access.
I am indebted with F. Darbois and S. Smith from Elsevier’s Global
Rights Department for permission to reproduce the photographs of
Louis Paul Cailletet and Sydney Young, respectively, in my paper. I
thank to the Président de Clermont Communauté and the Observatoire
de Physique du Globe de Clermont-Ferrand for the photograph and
biographical information of Émile Ovide Joseph Mathias. I am also
grateful for the helpful comments did by the anonymous reviewers
on a draft of this paper.
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Quim. Nova, Vol. 33, No. 9, S1, 2010
Material Suplementar
Simón Reif-Acherman
Escuela de Ingeniería Química, Universidad del Valle, A. A. 25360 Unicentro, Cali, Colombia
Figure 1S. Schematic diagram of the arrangement used in Leiden for the measuring of saturated liquid and vapour densities of oxygen (from ref. 45)
... Hence, the question arises if the uniaxial anisotropy of MBBA molecules, which can support the similar uniaxial symmetry of pretransitional fluctuations, can lead to the emergence of mean-field behavior hallmarks near in the tested MBBA + isooctane mixture? Figure 6 shows the evolution of the diameter of the MBBA-isooctane binodal coexistence curve. Obtaining unequivocal evidence for the precritical anomaly of the diameter, finally related to Equation (3), constituted the long-standing challenge terminated only in the mid-eighties [12,[117][118][119][120]. It can be associated with the relative weakness of the anomaly and the necessity of reliable multi-parameter fitting when using Equation (3) [12,85,86,109]. ...
... It can be associated with the relative weakness of the anomaly and the necessity of reliable multi-parameter fitting when using Equation (3) [12,85,86,109]. Earlier, the binodal diameter was considered in the frame of the classical empirical Cailletet-Mather (CM) 'law' of rectilinear diameter [12,117,119,121], which can also be derived with mean-field related models: ...
... First, it is the anomalous increase of the order parameter exponent in the immediate vicinity of the critical consolute temperature. Second, the diameter of the binodal is relatively close to the Cailletet-Mathias 'law' of rectilinear diameter [117][118][119][120][121], showing a very weak precritical anomaly. Both phenomena can be explained as the consequence of the mean-field type behavior emerging close to , which can be associated with the rod-like structure of MBBA molecules. ...
Full-text available
The transition from the isotropic (I) liquid to the nematic-type (N) uniaxial phase appearing as the consequence of the elongated geometry of elements seems to be a universal phenomenon for many types of suspensions, from solid nano-rods to biological particles based colloids. Rod-like thermotropic nematogenic liquid crystalline (LC) compounds and their mixtures with a molecular solvent (Sol) can be a significant reference for this category, enabling insights into universal features. The report presents studies in 4′-methoxybenzylidene-4-n-butylaniline (MBBA) and isooctane (Sol) mixtures, for which the monotectic-type phase diagram was found. There are two biphasic regions (i) for the low (TP1, isotropic liquid-nematic coexistence), and (ii) high (TP2, liquid-liquid coexistence) concentrations of isooctane. For both domains, biphasic coexistence curves’ have been discussed and parameterized. For TP2 it is related to the order parameter and diameter tests. Notable is the anomalous mean-field type behavior near the critical consolute temperature. Regarding the isotropic liquid phase, critical opalescence has been detected above both biphasic regions. For TP2 it starts ca. 20 K above the critical consolute temperature. The nature of pretransitional fluctuations in the isotropic liquid phase was tested via nonlinear dielectric effect (NDE) measurements. It is classic (mean-field) above TP1 and non-classic above the TP2 domain. The long-standing problem regarding the non-critical background effect was solved to reach this result.
... For decades, the CM law was used to determine critical concentration or density. However, the ultimate evidence for the diameter critical anomaly showed that it leads to strongly biased estimations [20,21]. The alternative way of testing binodal's properties is based on measurements of selected physical properties in coexisting phases. ...
... Both curves overlap. The green line terminated by the arrow shows the hypothetical Cailletet -Mathias law of rectilinear diameter [19][20][21], illustrating its essential inadequacy Figure 7 shows the focused insight into the coexistence curve diameter, only roughly visible in Fig. 4. It can be portrayed both by Eqs. (4a) and (4b) as indistinguishable. ...
... However, it causes a considerable problem in determining the critical concentration, which explains the popularity of the Caillatet-Mathias 'law' (Eq. 5) [19][20][21]. It was invalidated by the clear evidence for the precritical anomaly of the diameter (Eq. ...
Full-text available
Coexistence curves in mixtures of limited miscibility with the lower critical consolute temperature (LCT), of 3-picoline with deuterium oxide (D 2 O), and D 2 O/H 2 O have been determined. They were tested with respect to the order parameter, the diameter of the binodal, and coordinates of the critical consolute point (critical temperature $${T}_{\mathrm{C}}$$ T C and critical concentration $${\phi }_{\mathrm{C}}$$ ϕ C ). Studies were carried out using the innovative method based on the analysis of relative volumes occupied by coexisting phases, yielding high-resolution data. The clear violation of the Cailletet-Mathew law of rectilinear diameter for the LCT mixtures of limited miscibility is evidenced. For the order parameter, the new distortion-sensitive analysis method yielded the evidence for the model-value of the order parameter critical exponent $$\beta =0.326\pm 0.003$$ β = 0.326 ± 0.003 , up to ca. 1 K from $${T}_{\mathrm{C}}$$ T C . Finally, the simple & easy method for determining the critical concentration by testing relative volumes of coexisting phases (or alternatively fractional meniscus heights, h ) is presented. The significance of the invariant value $${h}_{\mathrm{C}}=h\left({T}_{\mathrm{C}},{\phi }_{\mathrm{C}}\right)=1/2$$ h C = h T C , ϕ C = 1 / 2 is highlighted. The appearance of the milky & bluish critical opalescence is also shown. Graphical abstract
... Nobody in the history of laboratory p-V-T measurements along critical isotherms, from Andrews in 1869 [2] to NIST 200-fluid thermophysical properties in 2022, has ever measured any liquid or gas density within ~ 10% either side of the LRD "mean critical density"! [6,38,39]. ...
... The two most striking experimental phenomena in gas-liquid phase diagrams are the straight lines for which there are no explanatory theories, yet. The law of rectilinear diameters (LRD) has a long history of investigation [38] but no theory. The LRD has been observed ever since the original experimental measurements of Andrews because it connects the mean coexisting gas and liquid densities, by definition. ...
Full-text available
HIGHLIGHTS • The Andrews 1869 'continuity of liquid and gas' hypothesis (van der Waals, Nobel lecture 1910), has never agreed with experiment: a supercritical mesophase bounded by percolation lines delineates gas and liquid states above a 1 st order transition at Tc. • Experimental heat capacities confirm the 'universality' hypothesis in Wilson's RG theory (Nobel lecture1982), is inapplicable to the liquid-gas critical line at Tc. • The KTHNY 'hexatic mesophase' hypothesis (Kosterlitz: Nobel lecture 2016) predicts a 2 nd order 2-stage 2D melting mechanism; the theory is invalidated by1st-order phase separation in computer experiments. • Besides the Boyle line, that delineates dense states as cohesive liquids and crystals, a rigidity symmetry line, RT = (dp/d)T, interpolates to an amorphous RCP (random close packing) ground state that is a starting point for theory of simple liquids. • 'Liquid' can be defined phenomenologically as an area of thermodynamic p-T-states for which the shear modulus is zero and the solidity, (d2p/d 2)T, is positive to distinguish it from cohesive (i.e. p < RT) solid and gas phases below the Boyle line.
... As is known, the diameter of the coexistence curve ( ) in van der Waals-type systems (binodal midline) is linearly dependent on the temperature (the law of rectilinear diameter). This property is often used to determine the critical point coordinates [14,15]. However, there are many cases of violation of this law [9]. ...
Вперше у широкому температурному дiапазонi виконано експериментальне дослiдження парорiдинної рiвноваги у двовимiрнiй (2D) адсорбованiй плiвцi Сu на гранi (112)W. Визначено критичнi характеристики. Встановлено область спiвiснування фаз. Показано, що особливостi фазового переходу “рiдина–газ” у 2D Cu–(112)W є аналогiчними переходам у тривимiрних (3D) системах Rb i Cs. Обговорено причини термодинамiчної подiбностi 2D i 3D металiчних систем. Запропоновано аналiтичнi вирази для апроксимацiї експериментальних даних. Вони визначають межi iснування парорiдинної рiвноваги та дозволяють з високою точнiстю здiйснювати екстраполяцiю в областi критичної та потрiйної точок. Координати точок на рiдиннiй i газовiй вiтках бiнодалей було визначено на основi аналiзу дифузiйно сформованих концентрацiйних профiлiв.
... The well-known example of such regularity is the principle of corresponding state, discovered together with the famous van der Waals equation of state. 1 One more example is the law of rectilinear diameter. 4 Both these laws can be explained within statistical mechanics if the interaction potentials between the particles of considered substances have the same functional form, 5 like, for instance, for noble gases. However, both these regularities can be violated if the functional forms of interactions are different. ...
The behavior of ideal lines for pressure and enthalpy and the related similarity laws on the density–temperature phase diagram for various liquids, which were previously confirmed for many real and model systems, are considered. On the basis of new data and numerical simulation, it is shown that these relations are also valid for complex organic substances and two-dimensional systems, respectively, with the exception of the Timmermans relation.
Full-text available
A fundamental, hitherto unanswered, question in liquid-state physics is: "What is the minimum requirement of a molecular interaction Hamiltonian for the existence of a stable liquid that can coexist with its vapor phase?". It has been the subject of speculation in the thermophysical property literature since Hagen et al. (Nature 1993) reported 'no liquid phase' in a computer site-site pairwise model Hamiltonian for C 60. In more recent reports we have found that for simple fluids, with spherical, pairwise model Hamiltonians there exists a supercritical mesophase colloidal description of gas-liquid coexistence with a T-p density-surface critical divide being defined thermodynamically by the intersection of percolation loci. We have also reported compelling experimental evidence for the existence of a pre-freezing percolation transition whence hetero-phase fluctuations of micro-crystallites percolate equilibrium liquid-state phase volume. These percolation phenomena can explain the apparent disappearance of the boiling line at finite range of attraction. As the attractive range shortens, the interception of the percolation line that define the critical line between two-phase coexistence, and one-phase supercritical mesophase, shifts to lower T. It then intercepts with the pre-freezing percolation line, to trigger a triple point of gas, liquid and solid states, all at the same T,p-state hence also the same chemical potential. Consequently, all model pairwise classical molecular Hamiltonians with a finite size, plus attractive term, however short-range, or however weak, exhibit a triple point with a liquid-vapor coexisting state at a sufficient low temperature.
Full-text available
The amplitude of the rectilinear diameter is examined for a large number of normal fluids and a few other fluids. For the normal fluids, the slope of the diameter shows a linear dependence on the acentric factor, as do other fluid properties on a reduced basis. Thus, we conclude that the shape of the pair potential is the primary factor in determining the slope of the diameter rather than the relative strength of three-body forces, as has been suggested by analogy to recent results for rubidium and cesium. The situation for the near-critical singularity in the diameter remains ambiguous for normal fluids and a suggestion is made for its resolution.
Compiled by an expert in the field, the book provides an engineer with data they can trust. Spanning gases, liquids, and solids, all critical properties (including viscosity, thermal conductivity, and diffusion coefficient) are covered. From C1 to C100 organics and Ac to Zr inorganics, the data in this handbook is a perfect quick reference for field, lab or classroom usage. By collecting a large - but relevant - amount of information in one source, the handbook enables engineers to spend more time developing new designs and processes, and less time collecting vital properties data. This is not a theoretical treatise, but an aid to the practicing engineer in the field, on day-to-day operations and long range projects. Simplifies research and significantly reduces the amount of time spent collecting properties data Compiled by an expert in the field, the book provides an engineer with data they can trust in design, research, development and manufacturing. A single, easy reference for critical temperature dependent properties for a wide range of hydrocarbons, including C1 to ClOO organics and Ac to Zr inorganics.
We have obtained expressions including corrections to scaling terms for a number of thermodynamic properties of fluids near the critical point by specializing the appropriate derivatives of the logarithm of the grand partition function to trajectories of experimental interest. Our justification for applying Wegner's general predictions, for the functional form of the free energy, to this thermodynamic potential is that it is the potential for a fluid most closely analogous to the Helmholtz free energy for the Ising model. It is found that the average of the coexisting densities, the so-called rectilinear diameter, is a nonlinear function of the temperature with a temperature derivative which diverges like the constant volume specific heat, at the critical point, with a system-dependent coefficient. The second-temperature derivative of the chemical potential, along either the coexistence curve or the critical isochore, is found to be nondivergent at the critical point and its value is the same above and below Tc. The corrections due to the irrelevant scaling fields are found to be as important as those due to higher-order terms in the expansion of the scaling fields around the critical point. Using the parametric representation of the linear model, we have been able to obtain expressions for the elements of the matrix relating relevant scaling fields and physical variables in linear form, in terms of experimentally measurable quantities.
Two basic ideas about the nature of the gas-liquid transition were voiced around 1870: that in the supercritical state, vapor and liquid are indistinguishable (Andrews), and that condensation and critical behavior can be understood on the basis of a simple assumption about molecular interaction (Van der Waals). For many scientists at that time, however, the notion of continuity of states was almost inconceivable. The older ideas, that liquid and gas molecules were different from each other, and that the liquid “dissolves” in the vapor at the critical point, survived for a long time. Unfamiliarity with the implications of Gibbs' thermodynamics in Europe helped to keep the older ideas alive. This paper describes the controversies that raged between proponents and opponents of the Andrews-Van der Waals view between 1880 and 1907, and how they were fanned by erroneous or incorrectly interpreted experiments carried out all over Europe. Kamerlingh Onnes and his staff in Leiden repeated a number of the controversial experiments in the period 1892–1907 and discovered that impurity was by far the largest source of error. By 1907 the controversy was unequivocally decided by the Leiden group in favor of the Andrews-Van der Waals view. A replay of the old controversy took place in the period 1933–1952.
A new approach to the problem of the gas-liquid phase transition, based on the Mayer cluster expansion of the partition function, is proposed. It is shown that the necessary and sufficient condition for phase transition to occur is that there exist a temperatureT= Tc > 0 such that forT ⩽T c, all theb l (except perhaps a finite number of them) are positive, where theb l, are the cluster integrals (as defined by Mayer) in the thermodynamic limit. Explicit expressions for the isotherms for gas-saturated vapor and liquid phases are given.
A rigorous semi-microscopic derivation of the law of rectilinear diameter for the liquid-gas phase transition has been provided.
The coexistence curves of CO2, N2O, and CClF3 are analyzed in the critical region. The curves were obtained by refractive index measurements which are virtually free of gravity effects and contain much detail near Tc. After proper weight assignment, it is established that the top of the coexistence curve is asymptotically symmetric: &rgr;± = &rgr;c ± Btβ; that the exponent β is independent of the range, varies little from substance to substance, and is insensitive to impurities; and that the data are in agreement with the law of the rectilinear diameter. “Best” values for β, B, and for the slope of the diameter are presented. An analysis of earlier coexistence curves for CO2 and N2O, including a weight assignment, is presented; there is agreement between the older and newer data.
The critical density ${$\rho${}}_{c}$ of Xe has been determined by two independent methods. The first value, 1.1128 \ifmmode\pm\else\textpm\fi{} 0.0003 g/${\mathrm{cm}}^{3}$, is based on coexistence-curve data and extrapolation to the critical temperature using the "law of rectilinear diameter." The second method, based on critical-isotherm data and an analysis of gravitational effects, reveals no deviation of the diameter from linearity within the limits of error. If ${$\rho${}}_{c}$ differs from the extrapolated value, the difference is less than 0.5%.