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In the early 1850s Joule and Thomson measured the cooling experienced by a flowing gas on passing an obstacle that caused a decrease in pressure. The mythical ‘perfect gas’, which conforms exactly to Boyle's and Charles's laws, would show no such cooling. They used their results to put the new theory of thermodynamics on a more secure foundation and to establish a practical route for converting the measurement of temperature on a gas scale to an absolute temperature based on the second law of thermodynamics. Their experiments were sound but their calculations were in error. Later in the century William Hampson and Carl von Linde independently devised a simple method of liquefying air based on Joule–Thomson cooling, but whereas Linde understood the theory, Hampson, and many chemists, confused the process with the cooling of a gas doing external work, which is an effect that would occur also with a perfect gas. James Dewar copied Hampson's process, without acknowledgement, in his work on the liquefaction of air and hydrogen at the Royal Institution. He, too, did not at first understand the theoretical basis of the apparatus. In the twentieth century the effect was exploited to investigate intermolecular forces, but it is now rarely used.

Statistical mechanics has two foundations; first, the early attempts from the late 17th century onwards to explain the bulk properties of matter in terms of the forces between the constituent particles, and secondly, the attempts in the early 19th century to explain the properties of gases in terms of the rapid motion of free particles. The combination of these two approaches was a task of the late 19th and early 20th centuries. The advent of quantum theory clarified many of the difficulties and showed how appropriate was Willard Gibbs's treatment of the subject. The paper shows how modern work has evolved from these foundations.

The Maxwell–Boltzmann distribution of molecular velocities and densities was derived first in the 1870s, but the derivations were difficult to follow and led to predictions of the heat capacities of gases that did not agree with experiment. The systems discussed were molecules or assemblies of molecules to which a definable energy could be ascribed, the equivalent of the microcanonical ensemble of Gibbs. This led to complications when dense gases and liquids were considered, and so, for the next thirty years, to arguments about the validity of the derivations. In principle these complications can be removed most simply by considering a set of systems of fixed temperature, or a canonical ensemble. It was, however, many years before most textbooks followed Gibbs down this route.

Le Sage's Essay was an ingenious eighteenth–century attempt to explain both the cohesive forces between particles of matter and gravitational forces, without recourse to Newtonian attraction at a distance. His explanation of cohesive forces never gained acceptance but the application to gravity drew some attention in the nineteenth century, and some recent physical work has a curious similarity to his ideas on cohesion.

The polite world took little interest in the Alps before the 18th century. The local inhabitants had ventured far enough to shoot chamois and to search for crystal (i.e. quartz), but few of the educated took any notice. The earliest natural philosophers to study the botany and mineralogy of these regions were from the German–speaking towns of Switzerland. Their first conclusions have not always stood the test of time; one of them, J.J. Scheuchzer (F.R.S., 1703) even reported the presence of dragons. Such frivolities did not appeal to the Calvinists of the city republic of Geneva. They could see the ‘montagnes maudites’ of Savoy to the south–east, but they never went to them. The first serious attempts to describe this world of snow and ice were made in 1741 and 1742.

In the 1840s Kincardine was a small port on the river Forth above Edinburgh where the local inn, the Unicorn, was kept by Thomas Dewar and his wife Ann. They had seven sons, of whom six survived infancy, and the last of whom, James, was born in 1842. He was educated first at the local school and then, after the death of both parents, at the nearby Dollar Institution (now Dollar Academy). In 1859 he went to Edinburgh University where he studied under the physicist Guthrie Tait and the chemist Lyon Playfair. As was usual for students from financially modest backgrounds, he took no degree but served as an assistant to Playfair and later to his successor, Crum Brown. His earliest research was in organic chemistry and physiology but he spread himself widely and published steadily, so creating a reputation as an active and coming young man. He applied unsuccessfully for the Regius Chair of Chemistry at Glasgow in 1874, but was then a candidate for the Jacksonian Professorship of Natural Experimental Philosophy at Cambridge where, in 1875, the electors were looking primarily for someone who could teach chemistry to medical students, but where the wide but obsolete conditions of the endowment allowed the occupant considerable freedom to choose his own field of teaching and research. The resources immediately available to the new professor were, however, even less than Dewar had been accustomed to in Edinburgh, so when the Fullerian Professorship of Chemistry at the Royal Institution in London was advertised in March 1877 he applied for that also. He was again successful and held both chairs for the rest of his life. The resources of the Royal Institution, or the RI as it was commonly known, proved over the years to be greater than anything that he could probably have mustered in Cambridge in the last quarter of the nineteenth century. He gave his statutory lectures at Cambridge, and cooperated with George Liveing, the Professor of Chemistry, in a long series of papers on visible and ultraviolet spectroscopy. He was assisted there also by a series of Demonstrators with whom he maintained his interest in organic chemistry but, from 1877, the centre of his research was firmly established in London at the Royal Institution.

The structure and thermodynamic properties of the interface between a drop of liquid and its vapour have long been matters of controversy. The system can be described in terms of its macroscopic thermodynamic properties, in terms of local thermodynamic properties or in the language of statistical mechanics. It is shown that the second of these routes leads to difficulties but that there is good evidence that the first and third are concordant, at least as far as the term in R-1 in an expansion in powers of the reciprocal of the drop radius, R. There are, however, serious problems if the expansion is taken to higher powers of R-1.

It is shown that, for systems with repulsive (non-negative) interactions, the Yang-Lee zeros of the grand canonical partition function lie on the real z axis. As a result, a series of successively improving lower and upper bounds for the thermodynamic ratio of density to activity is obtained. In addition bounds on the transition activity and density are given in terms of the cluster coefficients.

C. N. Hinshelwood and his physical and inorganic chemical colleagues in Oxford worked throughout World War II on the improvement of charcoal for use in respirators and on other physicochemical problems. The surviving reports and correspondence give a detailed picture of what they accomplished and on the way in which extramural research contracts were then handled.

Why does matter stick together? Why do gases condense to liquids, and liquids to solids? This book provides a detailed historical account of how some of the leading scientists of the past three centuries have tried to answer these questions. The topic of cohesion and the study of intermolecular forces has been an important component of physical science research for hundreds of years. This book is organised into four broad periods of advances in our understanding. The first three are associated with Newton, Laplace and van der Waals. The final section gives an account of the successful use in the twentieth century of quantum mechanics and statistical mechanics to resolve most of the remaining problems. The book will be of primary interest to physical chemists and physicists, as well as historians of science interested in the historical origins of our modern day understanding of cohesion.

This paper reviews the present position of statistical or molecular theory of simple liquids and their mixtures, with particular emphasis on the quantitative success of perturbation theories. It is shown that the limits to our present understanding are provided not so much by imperfections in the statistical mechanics as by our ignorance of the precise form and strength of the intermolecular potentials for all molecules except the noble gases and their mixtures.

Van der Waals justifies the choice of minimization of the (Helmholtz) free energy as the criterion of equilibrium in a liquid-gas system (Sections 1–4). If densityρ is a function of heighth then the local free energy density differs from that of a homogeneous fluid by a term proportional to (d 2 ρ/dh 2); the extra term arises from the energy not from the entropy (Section 5). He uses this result to show howρ varies withh (Section 6), how this variation leads to a stable minimum free energy (Section 7), and to calculate the capillary energy or surface tensionσ (Section 9). Near the critical pointσ varies as (τ k -τ)3/2, whereτ k is the critical temperature (Section 11). The paper closes with short discussions of the thickness of the surface layer (Section 12), of the difficulty of assuming thatρ varies discontinuously with height (Section 14), and of the possible effect of derivatives of higher order than (d 2 ρ/dh 2) on the free energy and surface tension (Section 15).

The change of surface tension with curvature is governed by a microscopic length, delta , introduced by Tolman. It is shown that this length diverges weakly at the liquid-gas critical point of the penetrable-sphere model, with an exponent ( mu - beta -1), where mu is the exponent of the surface tension, and beta of the orthobaric densities. This result is compared with those derived from Landau-Ginzburg-Wilson Hamiltonians.

We have investigated the thermodynamic properties of a binary mixture of hard spheres (with special reference to the existence of a phase transition) by using the recently obtained exact solution of the generalized equations of Percus and Yevick for the radial distribution functions of such a mixture. The distribution function obtained from the equations of Percus and Yevick is only an approximation and so yields two different pressures, pc and pv, when used, respectively, in the compressibility equation of Ornstein and Zernike and in the equation of state obtained from the virial theorem. Comparisons with machine calculations show that pc is slightly above and pv slightly below the true pressure but that both are close to it. Our results show that the volume change on mixing at constant pressure is negative at all densities and compositions within the fluid phase when it is calculated from pc, but that it becomes positive at high densities when calculated from pv. In neither case is there a separation in...

This report opens with a brief statement of our experimental knowledge of the equation of state of dense fluids and with an account of those parts of statistical mechanics that relate this knowledge to the intermolecular potential. The main part of the report falls into two sections. The first is a review of the progress made in the last ten years in the direct calculation of the properties of systems with model intermolecular potentials. The second is a review of recent approximations by which the statistical equations can be solved, and a comparison of these solutions with the direct calculations for model systems and with the properties of real systems. Most emphasis is placed upon the hyper-netted chain approximation and on the Percus-Yevick approximation. The prospects of improving upon these solutions are reviewed in the final section.

A new model is proposed for the study of the liquid–vapor phase transition. The potential energy of a given configuration of N molecules is defined by U(r1, ···, rN) = [W(r1, ···, rN) − Nυ0]e/υ0≤0, where W is the volume covered by N interpenetrating spheres each of volume υ0 and each centered on one molecule, and where e is an arbitrary positive energy. This continuum model proves to have a line of symmetry comparable with those found hitherto only in lattice models. The line is that of the diameter, or mean orthobaric density ρ = 12ρ1 + 12ρg, below the critical point, and continues through the one‐phase region to infinite temperature. The existence of this line allows some of the properties to be obtained explicitly, the most unusual of which is that the diameter has a singularity comparable with that in Cυ; hence the law of the rectilinear diameter is not obeyed. An exact solution of the model is obtained in one dimension, in which there is no phase transition, and a mean‐field solution in three dimensi...

At low temperatures gas—liquid and liquid—liquid phase equilibria are determined by the balance between the like and unlike intermolecular potentials in a mixture For molecularly simple systems it is now clear that the unlike characteristic energies are weaker than the geometric means of the like and there is evidence that this difference increases with difference of molecular size In mixtures of aliphatic fluorocarbons and hydrocarbons the energy difference is large enough to lead to liquid—liquid equilibria and to upper critical solution temperatures (UCST). If the molecules are more complicated, by virtue of either their polarity or their ability to form charge-transfer complexes, then the effective unlike interactions can be smaller or larger than the geometric mean. The former case is understood, at least seniquantitatively, and can lead again to UCST. The latter can lead in special circumstances to lower critical points (LCST) Here our understanding is less satisfactory since only lattice models have been studied theoretically. At higher temperatures and pressures both simple and complicated systems have critical lines some of which can unambiguously be called gas—liquid lines and some liquid—liquid but many of which defy such simple classification The exploration of the kinds of critical line which can occur has recently been an active field of experimental research and It iS probable that there is now little really new to be discovered for binary mixtures. However, the precise behaviour of the therm dynamic functions on critical lines is a field which is ripe for further study since Griffiths and Wheeler have set up a classical thermodynamic scheme of classification This leads to many precise predictions which now await con firation or disproof. Our attempts to understand the critical behaviour of mixtures in terms of the intermolecular forces has proceeded along two lines First the great variety of critical lines has proved surprisingly easy of at least semiquantitative interpretation. Even van der Waals' equation, with suitable choice of the parameters, can represent most of the phenomena, and one can do better by using more accurate equations On the other hand, the nature of the singu laritics tt critical points in both one and two-component systems has proved one of the most formidable problems in statistical thermodynamics. Such progress as has been made is reviewed.

The classical description of the critical point leads to infinities in such properties as the compressibility and the heat capacity at constant pressure, Cp, as the critical point is approached along a path in the homogeneous fluid. However, this description does not require a singularity in the heat capacity at constant volume, Cv, and supposes that the Helmholtz free-energy, A, is everywhere an analytic function of volume, V, and temperature, T. It is now clear that this description is inadequate, that Cv is infinite at the critical point and that A is a non-analytic function of V and T. The classical description must, therefore, be abandoned and it is shown that we have, in its place, a set of inequalities between the degrees of the singularities in such thermodynamic functions as Cv, compressibility, coefficient of thermal expansion along the orthobaric curve, and the curvature of the vapour pressure line. These inequalities provide powerful tests of even the best modern measurements and enable us to give a fairly complete thermodynamic description of the singularities of the critical region. The transport properties are not so well understood. It is probable that the shear viscosity is non-singular whilst the thermal conductivity is known to be highly singular. This last singularity is probably related to the infinity in Cv. The dimensionless functions—Rayleigh number, Prandtl number, and Grashof number—all become infinite at the critical point.

An equation of state is obtained in closed form for a gas composed of molecules with Lennard-Jones (n,1/2n) potentials. It is useful at temperatures above about 12 (e/k), where -e is the minimum energy of interaction and at all densities at which the equilibrium state is a fluid. It is derived by summing over all the cluster integrals of the virial expansion that occur in the approximation for the pair distribution function proposed by Percus and Yevick. Each cluster integral is represented correctly to terms of the order of n -1. The equation agrees well with machine calculations of the pressure of dense gases for n = ∞ and n = 12, with static measurements of the compression of gases at high reduced temperatures, and with some preliminary measurements of the density of argon compressed by shockwaves to pressures in the range 100–200 kb.

An apparatus of the Burnett type has been built with which to measure accurately the second virial coefficients of gases at low temperatures. The principal feature is a mercury manometer with which pressures up to 1 bar can be measured with an accuracy of 2 × 10–6 bar. The second virial coefficients B of argon (80–190°K) and of krypton (110-225°K) have been measured with absolute accuracies of from 0.001 to 0.01 m3 kmol–1, and are discussed in relation to the pair potential of the intermolecular forces. The rapid fall of B at low temperatures implies, in each case, a deeper well in the potential than has been proposed heretofore.

The third virial coefficient is calculated for a gas composed of spherical molecules whose intermolecular potential has a repulsive term proportional to r−12, an attractive term proportional to r−6, and which have point‐dipoles at their centers. The calculated values are compared with the experimental results for steam and ammonia, using parameters previously calculated from the second virial coefficient. The agreement is good for ammonia but poor for steam. It is suggested that this is because of the neglect of the higher multipoles in the interaction potential.

Equations are derived for the number of collisions per unit time in which the centres of the molecules approach to some arbitrary distance, d, or less, for molecules with spherical intermolecular potentials. The collision rates based upon this criterion of closeness-of-approach have a different dependence on temperature from those based upon cross sections determined from the viscosity. It is suggested that the rates calculated in this paper are the more appropriate for the discussion of experiments on the exchange of molecular energy and on the kinetics of chemical reactions.

Summary No rigorous calculation of the properties of a liquid has yet been made from a knowledge of the forces between its molecules. This review describes the method by which most solutions have been sought and the progress that has been made in the last five years. The equilibrium properties of a simple liquid can be expressed in terms of the pair distribution function g(r), which measures the probability of finding a second molecule at distance r from a first. Kirkwood, and Born and Green showed that an approximate calculation of g(r) can be made by solving an integral equation. This equation, its solutions and their imperfections are described briefly. More recently it has been found convenient to write the correlation function, h(r)=g(r) -1, as the sum of two terms, a direct term, c(r), and an indirect term. Four recent approximations can be described and compared by the form they assume for the direct correlation function, c(r). The most promising of these is the approximation of Percus and Yevick a...