# 1911 Encyclopædia Britannica/Molecule

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​MOLECULE (from mod. Lat. molecula, the diminutive of moles, a mass), in chemistry and physics, the minutest particle of matter capable of separate existence. The word appears to have been invented during the 17th century, and remained synonymous with “atom” (Gr. ἄτομος from ἀ-, privative, and τέμνειν, to cut) until the middle of the 19th century, when a differentiation was established. “Atom” has mainly a chemical import, being defined as the smallest particle of matter which can take part in a chemical reaction; a “molecule” is composed of atoms, generally two or more. For the detailed chemical significance of these terms, see Chemistry; and for the atomic theory of the chemist (as distinguished from the atomic or molecular theory of the physicist) see Atom; reference may also be made to the article Matter.
The doctrine that matter can be divided into, or regarded as composed of, discrete particles (termed “atoms” by early writers, and “molecules” by modern ones) has at all times played an important part in metaphysics and natural science. The leading historical stages in the evolution of the modern conception of the molecular structure of matter are treated in the following passage from James Clerk Maxwell’s article Atom in the 9th edition of the Ency. Brit.
“Atom (ἄτομος) is a body which cannot be cut in two. The atomic theory is a theory of the constitution of bodies which asserts that they are made up of atoms. The opposite theory is that of the homogeneity and continuity of bodies, and asserts, at least in the case of bodies having no apparent organization, such, for instance, as water, that as we can divide a drop of water into two parts which are each of them drops of water, so we have reason to believe that these smaller drops can be divided again, and the theory goes on to assert that there is nothing in the nature of things to hinder this process of division from being repeated over and over again, times without end. This is the doctrine of the infinite divisibility of bodies, and it is in direct contradiction with the theory of atoms.
“The atomists assert that after a certain number of such divisions the parts would be no longer divisible, because each of them would be an atom. The advocates of the continuity of matter assert that the smallest conceivable body has parts, and that Whatever has parts may be divided.
“In ancient times Democritus was the founder of the atomic theory, while Anaxagoras propounded that of continuity, under the name of the doctrine of homoeomeria (Ὁμοιομέρια), or of the similarity of the parts of a body to the whole. The arguments of the atomists, and their replies to the objections of Anaxagoras, are to be found in Lucretius.
“In modern times the study of nature has brought to light many properties of bodies which appear to depend on the magnitude and motions of their ultimate constituents, and the question of the existence of atoms has once more become conspicuous among scientific inquiries.
“We shall begin by stating the opposing doctrines of atoms and of continuity. The most ancient philosophers whose speculations are known to us seem to have discussed the ideas of number and of continuous magnitude, of space and time, of matter and motion, with a native power of thought which has probably never been surpassed. Their actual knowledge, however, and their scientific experience were necessarily limited, because in their days the records of human thought were only beginning to accumulate. It is probable that the first exact. notions of quantity were founded on the consideration of number. It is by the help of numbers that concrete quantities are practically measured and calculated. Now, number is discontinuous. We pass from one number to the next per saltum. The magnitudes, on the other hand, which we meet with in geometry, are essentially continuous. The attempt to apply numerical methods to the comparison of geometrical quantities led to the doctrine of incommensurables, and to that of the infinite divisibility of space. Meanwhile, the same considerations had not been applied to time, so that in the days of Zeno of Elea time was still regarded as made up of a finite number of ‘moments,’ while space was confessed to be divisible without limit. This was the state of opinion when the celebrated arguments against the possibility of motion, of which that of Achilles and the tortoise is a specimen, were propounded by Zeno, and such, apparently, continued to be the state of opinion till Aristotle pointed out that time is divisible without limit, in precisely the same sense that space is. And the slowness of the development of scientific ideas may be estimated from the fact that Bayle does not see any force in this statement of Aristotle, but continues to admire the paradox of Zeno (Bayle’s Dictionary, art. ‘Zeno’). Thus the direction of true scientific progress was for many ages towards the recognition of the infinite divisibility of space and time.
“It was easy to attempt to apply similar arguments to matter. If matter is extended and fills space, the same mental operation by which we recognize the divisibility of space may be applied, in imagination at least, to the matter which occupies space. From this point of view the atomic doctrine might be regarded as a relic of the old numerical way of conceiving magnitude, and the opposite doctrine of the infinite divisibility of matter might appear for a time the most scientific. The atomists, on the other hand, asserted very strongly the distinction between matter and space. The atoms, they said, do not fill up the universe; there are void spaces between them. If it were not so, Lucretius tells us, there could be no motion, for the atom which gives way first must have some empty place to move into.
‘Quapropter locus est intactus, inane, vacansque Quod si non esset, nulla ratione moveri Res possent; namque, officium quod corporis exstat, Officere atque obstare, id in omni tempore adesset Omnibus: haud igitur quicquam procedere posset, Principium quoniam cedendi nulla daret res.’ De rerum natura, i. 335.
“The opposite school maintained then, as they have always done,
​ that there is no vacuum—that every part of space is full of matter, that there is a universal plenum, and that all motion is like that of a fish in the water, which yields in front of the fish because the fish leaves room for it behind.
‘Cedere squamigeris latices nitentibus aiunt Et liquidas aperire vias, quia post loca pisces Linquant, quo possint cedentes confluere undae.’ Ibid. i. 373.
“In modern times Descartes held that, as it is of the essence of matter to be extended in length, breadth and thickness, so it is of the essence of extension to be occupied by matter, for extension cannot be an extension of nothing.
“ ‘Ac proinde si quaeratur quid fiet, si Deus auferat omne corpus quod in aliquo vase continetur, et nullum aliud in ablati locum venire permittat? respondendum est, vasis latera sibi invicem hoc ipso fore contigua. Cum enim inter duo corpora nihil interjacet, necesse est ut se mutuo tangant, ac manifeste repugnat ut distent, sive ut inter ipsa sit distantia, et tamen ut ista distantia sit nihil; quia omnis distantia est modus extensionis, et ideo sine substantia extensa esse non potest.’—Principia, ii. 18.
“This identification of extension with substance runs through the whole of Descartes’s works, and it forms one of the ultimate foundations of the system of Spinoza. Descartes, consistently with this doctrine, denies the existence of atoms as parts of matter, which by their own nature are indivisible. He seems to admit, however, that the Deity might make certain particles of matter indivisible in this sense, that no creature should be able to divide them. These particles, however, would be still divisible by their own nature, because the Deity cannot diminish his own power, and therefore must retain his power of dividing them. Leibniz, on the other hand, regarded his monad as the ultimate element of everything.
“There are thus two modes of thinking about the constitution of bodies, which have had their adherents both in ancient and in modern times. They correspond to the two methods of regarding quantity—the arithmetical and the geometrical. To the atomist the true method of estimating the quantity of matter in a body is to count the atoms of it. The void spaces between the atoms count for nothing. To those who identify matter with extension, the volume of space occupied by a body is the only measure of the quantity of matter in it.
“Of the different forms of the atomic theory that of R. J. Boscovich may be taken as an example of the purest monadism. According to Boscovich matter is made up of atoms. Each atom is an indivisible point, having position in space, capable of motion in a continuous path, and possessing a certain mass, whereby a certain amount of force is required to produce a given change of motion. Besides this the atom is endowed with potential force, that is to say, that any two atoms attract or repel each other with a force depending on their distance apart. The law of this force, for all distances greater than say the thousandth of an inch, is an attraction varying as the inverse square of the distance. For smaller distances the force is an attraction for one distance and a repulsion for another, according to some law not yet discovered. Boscovich himself, in order to obviate the possibility of two atoms ever being in the same place, asserts that the ultimate force is a repulsion which increases without limit as the distance diminishes without limit, so that two atoms can never coincide. But this seems an unwarrantable concession to the vulgar opinion that two bodies cannot co-exist in the same place. This opinion is deduced from our experience of the behaviour of bodies of sensible size, but we have no experimental evidence that two atoms may not sometimes coincide. For instance, if oxygen and hydrogen combine to form water, we have no experimental evidence that the molecule of oxygen is not in the very same place with the two molecules of hydrogen. Many persons cannot get rid of the opinion that all matter is extended in length, breadth and depth. This is a prejudice of the same kind with the last, arising from our experience of bodies consisting of immense multitudes of atoms. The system of atoms, according to Boscovich, occupies a certain region of space in virtue of the forces acting between the component atoms of the system and any other atoms when brought near them. No other system of atoms can occupy the same region of space at the same time, because before it could do so the mutual action of the atoms would have caused a repulsion between the two systems insuperable by any force which we can command. Thus, a number of soldiers with firearms may occupy an extensive region to the exclusion of the enemy’s armies, though the space filled by their bodies is but small. In this way Boscovich explained the apparent extension of bodies consisting of atoms, each of which is devoid of extension. According to Boscovich’s theory, all action between bodies is action at a distance. There is no such thing in nature as actual contact between two bodies. When two bodies are said in ordinary language to be in contact, all that is meant is that they are so near together that the repulsion between the nearest pairs of atoms belonging to the two bodies is very great.
“Thus, in Boscovich’s theory, the atom has continuity of existence in time and space. At any instant of time it is at some point of space, and it is never in more than one place at a time. It passes from one place to another along a continuous path. It has a definite mass which cannot be increased or diminished. Atoms are endowed with the power of acting on one another by attraction or repulsion, the amount of the force depending on the distance between them. On the other hand, the atom itself has no parts or dimensions. In its geometrical aspect it is a mere geometrical point. It has no extension in space. It has not the so-called property, of impenetrability, for two atoms may exist in the same place. This we may regard as one extreme of the various opinions about the constitution of bodies.
“The opposite extreme, that of Anaxagoras—the theory that bodies apparently homogeneous and continuous are so in reality—is, in its extreme form, a theory incapable of development. To explain the properties of any substance by this theory is impossible. We can only admit the observed properties of such substance as ultimate facts. There is a certain stage, however, of scientific progress in which a method corresponding to this theory is of service. In hydrostatics, for instance, we define a fluid by means of one of its known properties, and from this definition we make the system of deductions which constitutes the science of hydrostatics. In this way the science of hydrostatics may be built upon an experimental basis, without any consideration of the constitution of a fluid as to whether it is molecular or continuous. In like manner, after the French mathematicians had attempted, with more or less ingenuity, to construct a theory of elastic solids from the hypothesis that they consist of atoms in equilibrium under the action of their mutual forces, Stokes and others showed that all the results of this hypothesis, so far at least as they agreed with facts, might be deduced from the postulate that elastic bodies exist, and from the hypothesis that the smallest portions into which we can divide them are sensibly homogeneous. In this way the principle of continuity, which is the basis of the method of Fluxions and the whole of modern mathematics, may be applied to the analysis of problems connected with material bodies by assuming them, for the purpose of this analysis, to be homogeneous. All that is required to make the results applicable to the real case is that the smallest portions of the substance of which we take any notice shall be sensibly of the same kind. Thus, if a railway contractor has to make a tunnel through a hill of gravel, and if one cubic yard of the gravel is so like another cubic yard that for the purposes of the contract they may be taken as equivalent, then, in estimating the work required to remove the gravel from the tunnel, he may, without fear of error, make his calculations as if the gravel were a continuous substance. But if a worm has to make his way through the gravel, it makes the greatest possible difference to him whether he tries to push right against a piece of gravel, or directs his course through one of the intervals between the pieces; to him, therefore, the gravel is by no means a homogeneous and continuous substance.
“In the same way, a theory that some particular substance, say water, is homogeneous and continuous may be a good working theory up to a certain point, but may fail when we come to deal with quantities so minute or so attenuated that their heterogeneity of structure comes into prominence. Whether this heterogeneity of structure is or is not consistent with homogeneity and continuity of substance is another question.
“The extreme form of the doctrine of continuity is that stated by Descartes, who maintains that the whole universe is equally full of matter, and that this matter is all of one kind, having no essential property besides that of extension. All the properties which we perceive in matter he reduces to its parts being movable among one another, and so capable of all the varieties which we can perceive to follow from the motion of its parts (Principia, ii. 23). Descartes’s own attempts to deduce the different qualities and actions of bodies in this way are not of much value. More than a century was required to invent methods of investigating the conditions of the motion of systems of bodies such as Descartes imagined. But the hydrodynamical discovery of Helmholtz that a vortex in a perfect liquid possesses certain permanent characteristics has been applied by Sir W. Thomson (Lord Kelvin) to form a theory of vortex atoms in a homogeneous, incompressible and frictionless liquid.”
The Molecular Structure of Matter
An enormous mass of experimental evidence now shows quite conclusively that matter cannot be regarded as having a continuous structure, but that it is ultimately composed of discrete parts. The smallest unit of matter with which physical phenomena are concerned is the molecule. When chemical phenomena occur the molecule may be divided into atoms, and these atoms, in the presence of electrical phenomena, may themselves be further divided into electrons or corpuscles. It ought accordingly to be possible to explain all the non-electrical and non-chemical properties of matter by treating matter as an aggregation of molecules. In point of fact it is found that the properties which are most easily explained are those connected with the gaseous state; the explanation of these properties in terms of the molecular structure of matter is the aim of the “Kinetic Theory of Gases.” The results of this theory have placed the molecular ​ conception of matter in an indisputable position, but even without this theory there is such an accumulation of electrical and optical evidence in favour of the molecular conception of matter that the tenability of this conception could not be regarded as open to question.
The Scale of Molecular Structure.—Apart from speculation, the first definite evidence for the molecular structure of matter occurs when it is found that certain physical phenomena change their whole nature as soon as we deal with matter of which the linear dimensions are less than a certain amount. As a single instance of this may be mentioned some experiments of Lord Rayleigh (Proc. Roy. Soc., 1890, 47, p. 364), who found that a film of olive oil spread over the surface of water produced a perceptible effect on small floating pieces of camphor, at places at which the thickness of the film was 10·6×10$^{−8}$ cms., but produced no perceptible effect at all at places where the thickness of the film was 8·1×10$^{−8}$ cms. Thus a certain phenomenon, of the nature of capillary action, is seen to depend for its existence on the linear dimensions of the film of oil; the physical properties of a film of thickness 10·6×10$^{−8}$ cms. are found to be in some way qualitatively different from those of a film of thickness 8·1×10$^{−8}$ cms. Here is proof that the film of oil is not a continuous homogeneous structure, and we are led to suspect that the scale on which the structure is formed has a unit of length comparable with 8×10$^{−8}$ cms. The probability of this conjecture is strengthened when it is discovered that in all phenomena of this type the critical length connected with the stage at which the phenomenon changes its nature is of the order of magnitude of 10$^{−8}$ cms.
Lord Rayleigh (Phil. Mag. 1890 [5], 30, p. 474) has pointed out that the earliest known attempt to estimate the size of molecules, made by Thomas Young in 1805, was based upon the consideration of phenomena of the kind just mentioned. Discussing the theory of capillary attractions, Young found that at a rough estimate “the extent of the cohesive force must be limited to about the 250-millionth of an inch” (＝10$^{−8}$ cms.), and then argues that “within similar limits of uncertainty we may obtain something like a conjectural estimate of the mutual distance of the particles of vapours, and even of the actual magnitude of the elementary atoms of liquids. . . . It appears tolerably safe to conclude that, whatever errors may have affected the determination, the diameter or distance of the particles of water is between the two thousand and the ten thousand millionth of an inch” (＝between ·125×10$^{−8}$ and ·025×10$^{−8}$ cms.).
The best estimates which we now possess of the sizes of molecules are provided by calculations based upon the kinetic theory of gases. In the following table are given the values of the diameters of the molecules of six substances with which it is easy to experiment in the gaseous state, these values being calculated in different ways from formulae supplied by the kinetic theory.
The agreement of the values obtained for the same quantity by different methods provides valuable confirmation of the truth of the molecular theory and of the validity of the methods of the kinetic theory of gases. That. the results do not agree even better need not cause surprise when it is stated that the quantities are calculated on the hypothesis that the molecules are spherical in shape. This hypothesis is introduced for the sake of simplicity, but is known to be unjustifiable in fact. What is given by the formulae is accordingly the mean radius of an irregularly shaped solid (or, more probably, of the region in which the field of force surrounding such a solid is above a certain intensity), and the mean has to be taken in different ways in the different phenomena. This and the difficulty of obtaining accurate experimental results fully account for the differences inter se in the values of the quantities calculated.
Heat a Manifestation of Molecular Motion.—An essential feature of the modern view of the structure of matter is that the molecules are supposed to be in rapid motion relatively to one another. We are led to this conception by a number of experimental results, some of which will be mentioned later. We are compelled also to suppose that the motion assumes different forms in different substances. Roughly speaking, it is found that there are three main types of molecular motion corresponding to the three states of matter—solid, liquid and gaseous. That the distances traversed by the molecules of a solid are very small in extent is shown by innumerable facts of everyday observation, as for instance, the fact that the surface of a finely-carved metal (such as a plate used for steel engraving) will retain its exact shape for centuries, or again, the fact that when a metal body is coated with gold-leaf the molecules of the gold remain on its surface indefinitely: if they moved through any but the smallest distances they would soon become mixed with the molecules of the baser metal and diffused through its interior. Thus the molecules of a solid must make only small excursions about their mean positions. In a gas the state of things is very different; an odour is known to spread rapidly through great distances, even in the stillest air, and a gaseous poison or corrosive will attack not only those objects which are in contact with its source but also all those which can be reached by the motion of its molecules.
As a preliminary to examining further into the nature of molecular motion and the differences of character of this motion, let us try to picture the state of things which would exist in a mass of solid matter in which all the molecules are imagined to be at rest relatively to one another. The fact that a solid body in its natural state is capable both of compression and of dilatation indicates that the molecules of the body must not be supposed to be fixed rigidly in position relative to one another; the further fact that a motion of either compression or of dilatation is opposed by forces which are brought into play in the interior of the solid suggests that the position of rest is one in which the molecules are in stable equilibrium under their mutual forces. Such a mass of imaginary matter as we are now considering may be compared to a collection of heavy particles held in position relatively to one another by a system of light spiral springs, one spring being supposed to connect each pair of adjacent particles. Let two such masses of matter be suspended by strings from the same point, and then let one mass be drawn aside, pendulum-wise, and allowed to impinge on the other. After impact the two masses will rebound, and the process may be repeated any number of times, but ultimately the two masses will be found again hanging in contact side by side. At the first impact each layer of surface molecules which takes the shock of the impact will be thrust back upon the layer behind it: this layer will in this way be set into motion and so influence the layer still further behind; and so on indefinitely. The impact will accordingly result in all the molecules being set into motion, and by the time that the masses have ceased impinging on one another the molecules of which they are composed will be performing oscillations about their positions of equilibrium. The kinetic energy with which the moving mass originally impinged on that at rest is now represented by the energy, kinetic and, potential, of the small motions of the ​ individual molecules. It is known, however, that when two bodies impinge, the kinetic energy which appears to be lost from the mass-motion of the bodies is in reality transformed into heat-energy. Thus the molecular theory of matter, as we have now pictured it, leads us to identify heat-energy in a body with the energy of motion of the molecules of the body relatively to one another. A body in which all the molecules were at rest relatively to one another would be a body devoid of heat. This conception of the nature of heat leads at once to an absolute zero of temperature—a temperature of no heat-motion—which is identical, as will be seen later, with that reached in other ways, namely, about −273° C.
The point of view which has now been gained enables us to interpret most of the thermal properties of solids in terms of molecular theory. Suppose for instance that two bodies, both devoid of heat, are placed in contact with one another, and that the surface of the one is then rubbed over that of the other. The molecules of the two surface-layers will exert forces upon one another, so that, when the rubbing takes place, each layer will set the molecules 'of the other into motion, and the energy of rubbing will be used in establishing this heat-motion. In this we see the explanation of the phenomenon of the generation of heat by friction. At first the heat-motion will be confined to molecules near the rubbing surfaces of the two bodies, but, as already explained, these will in time set the interior molecules into motion, so that ultimately the heat-motion will become spread throughout the whole mass. Here we have an instance of the conduction of heat. When the molecules are oscillating about their equilibrium positions, there is no reason why their mean distance apart should be the same as when they are at rest. This leads to an interpretation of the fact that a change of dimensions usually attends a change in the temperature of a substance. Suppose for instance that two molecules, when at rest in equilibrium, are at a distance a apart. It is very possible that the repulsive force they exert when at a distance a−ε may be greater than the attractive force they exert when at a distance a + ε. If so, it is clear that their mean distance apart, averaged through a sufficiently long interval of their motion, will be greater than a. A body made up of molecules of this kind will expand on heating.
As the temperature of a body increases the average energy of the molecules will increase, and therefore the range of their excursions from their positions of equilibrium will increase also. At a certain temperature a. stage will be reached in which it is a frequent occurrence for a molecule to wander so far from its position of equilibrium, that it does not return but falls into a new position of equilibrium and oscillates about this. When the body is in this state the relative positions of the molecules are not permanently fixed, so that the body is no longer of unalterable shape: it has assumed a plastic or molten condition. The substance attains to a perfectly liquid state as soon as the energy of motion of the molecules is such that there is a constant rearrangement of position among them.
A molecule escaping from its original position in a body will usually fall into a new position in which it will be held in equilibrium by the forces from a new set of neighbouring molecules. But if the wandering molecule was originally close to the surface of the body, and if it also happens to start off in the right direction, it may escape from the body altogether and describe a free path in space until it is checked by meeting a second wandering molecule or other obstacle. The body is continually losing mass by the loss of individual molecules in this way, and this explains the process of evaporation. Moreover, the molecules which escape are, on the whole, those with the greatest energy. The average energy of the molecules of the liquid is accordingly lowered by evaporation. In this we see the explanation of the fall of temperature which accompanies evaporation.
When a liquid undergoing evaporation is contained in a closed vessel, a molecule which has left the liquid will, after a certain number of collisions with other free molecules and with the sides of the vessel, fall back again into the liquid. Thus the process of evaporation is necessarily accompanied by a process of recondensation. When a stage is reached such that the number of molecules lost to the liquid by evaporation is exactly equal to that regained by condensation, we have a liquid in equilibrium with its own vapour. If the whole liquid becomes vaporized before this stage is attained, a state will exist in which the vessel is occupied solely by free molecules, describing paths which are disturbed only by encounters with other free molecules or the sides of the vessel. This is the conception which the molecular theory compels us to form of the gaseous state.
At normal temperature and pressure the density of a substance in the gaseous state is of the order of one-thousandth of the density of the same substance in the solid or liquid state. It follows that the average distance apart of the molecules in the gaseous state is roughly ten times as great as in the solid or liquid state, and hence that in the gaseous state the molecules are at distances apart which are large compared with their linear dimensions. (If the molecules of air at normal temperature and pressure were arranged in cubical order, the edge of each cube would be about 2·9×10$^{−7}$ cms.; the average diameter of a molecule in air is 2·8×10$^{−8}$ cms.) Further and very important evidence as to the nature of the gaseous state of matter is provided by the experiments of Joule and Kelvin. These experiments showed that the change in the temperature of a gas, consequent on its being allowed to stream out into a vacuum, is in general very slight. In terms of the molecular theory this indicates that the total energy of the gas is the sum of the separate energies of its different molecules: the potential energy arising from intermolecular forces between pairs of molecules may be treated as negligible when the matter is in the gaseous state.
These two simplifying facts bring the properties of the gaseous state of matter within the range of mathematical treatment. The kinetic theory of gases attempts to give a mathematical account, in terms of the molecular structure of matter, of all the non-chemical and non-electrical properties of gases. The remainder of this article is devoted to a brief statement of the methods and results of the kinetic theory. No attempt will be made to follow the historic order of development, but the present theory will be set out in its most logical form and order.
The Kinetic Theory of Gases.
A number of molecules moving in obedience to dynamical laws will pass through a series of configurations which can be, theoretically determined as soon as the structure of each molecule and the initial position and velocity of every part of it are known. The determination of the series of configurations developing out of given initial conditions is not, however, the problem of the kinetic theory: the object of this theory is to explain the general properties of all gases in terms only of their molecular structure. We are therefore called upon, not to trace the series of configurations of any single gas, starting from definite initial conditions, but to search for features and properties common to all series of configurations, independently of the particular initial conditions from which the gas may have started.
We begin with a general dynamical theorem, whose special application, when the dynamical system is identified with a gas, will appear later. Let q$_1$, q$_2$, . . . q$_n$, . be the generalized co-ordinates of any dynamical system, and let p$_1$, p$_2$, . . . p$_n$Dynamical Basis. be the corresponding momenta. If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations
| q̇$_r$＝∂E/∂p$_r$, ṗ$_r$＝−∂E/∂q$_r$, | (1) |
where q̇$_r$ denotes dq$_r$ /dt, &c., and E is the total energy expressed as function of p$_1$, q$_1$, . . . p$_n$, q$_n$, . When the initial values of p$_1$, q$_1$ . . . p$_n$, q$_n$, are given, the motion can be traced completely from these equations.
Let us suppose that an infinite number of exactly similar systems start simultaneously from all possible values of p$_1$, q$_1$, . . . p$_n$, q$_n$, each moving solely under its own internal forces, and therefore in accordance with equations (1). Let us confine our attention to those ​ systems for which the initial values of p$_1$, q$_1$, . . . p$_n$, q$_n$ lie within a range such that
p$_1$ is between p$_1$ and p$_1$ +dp$_1$
q$_1$ is between q$_1$ and q$_1$ +dq$_1$ and so on.
Let the product dp$_1$ dq$_1$ . . . dp$_n$, dq$_n$ be spoken of as the “extension” of this range of values.
After a time dt the value of p$_1$ will have increased to p$_1$+p$_1$dt, where p$_1$ is given by equations (1), and there will be similar changes in q$_1$, p$_2$, q$_2$, . . . q$_n$. Thus after a time dt the values of the co-ordinates and momenta of the small group of systems under consideration will lie within a range such that
p$_1$ is between p$_1$ + ṗ$_1$dt and p$_1$+dp$_1$ +(ṗ$_1$+∂ṗ$_1$/∂p$_1$dp$_1$)dt
q$_1$ is between q$_1$ + q̇$_1$dt and q$_1$+dq$_1$ +(q̇$_1$+∂q̇$_1$/∂q$_1$dq$_1$)dt
and so on. Thus the extension of the range after the interval dt is
dp$_1$ (1+∂ṗ$_1$/∂p$_1$dt)dq$_1$(q̇$_1$+∂q̇$_1$/∂q$_1$dt) . . .
or, expanding as far as first powers of dt,
dp$_1$dq$_1$ . . . dp$_n$dq$_n$ ${\Big \$1+ Σ$_{1n}$${\Big$∂ṗ$_1$/∂p$_1$ + ∂q̇$_1$/∂q$_1$${\Big )}$dt${\Big \}$.
From equations (1), we find that
∂ṗ$_1$/∂p$_1$ + ∂q̇$_1$/∂q$_1$＝0.
so that the extension of the new range is seen to be dp$_1$dq$_1$ . . . dp$_n$dq$_n$, and therefore equal to the initial extension. Since the values of the co-ordinates and momenta at any instant during the motion may be treated as “initial” values, it is clear that the “extension” of the range must remain constant throughout the whole motion.
This result at once disposes of the possibility of all the systems acquiring any common characteristic in the course of their motion through a tendency for their co-ordinates or momenta to concentrate about any particular set, or series of sets, of values. But the result goes further than this. Let us imagine that the systems had the initial values of their co-ordinates and momenta so arranged that the number of systems for which the co-ordinates and momenta were within a given range was proportional simply to the extension of the range. Then the result proves that the values of the coordinates and momenta remain distributed in this way throughout the whole motion of the systems. Thus, if there is any characteristic which is common to all the systems after the motion has been in progress for any interval of time, this same characteristic must equally have been common to all the systems initially. It must, in fact, be a characteristic of all possible states of the systems.
It is accordingly clear that there can be no property common to all systems, but it can be shown that when the system contains a gas (or any other aggregation of similar molecules) as part of it there are properties which are common to all possible states, except for a number which form an insignificant fraction of the whole. These properties are found to account for the physical properties of gases.
Let the whole energy E of the system be supposed equal to E$_1$+E$_2$, where E$_2$ is of the form
| E$_2$＝1/2$\sum _{s}$(mu$^2$+mv$^2$+mw$^2$+α$_1$θ$_1$$^2$+ . . . +α$_n$θ$_n$$^2$)+1/2$\sum _{s}'$(m′u′+m′v′$^2$+mw′$^2$+β$_1$φ$_1$$^2$+β$_2$φ$_2$$^2$+ . . . +β$_{n′}$φ$_{n′}$$^2$) | (2) |
where θ$_1$,θ$_2$, . . . θ$_n$ and similarly φ$_1$,φ$_2$, . . . φ$_{n′}$ are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the coefficients α$_1$, α$_2$, &c.
In this expression the first line may be supposed to represent the energy (or part of the energy) of s similar molecules of a kind which we shall call the first kind, the terms 1/2(mu$^2$+mv$^2$+mw$^2$) being the kinetic energy of translation, and the remaining terms arising from energy of rotation or of internal motion, or from the energy, kinetic and potential, of small vibrations. The second line in E$_2$ will represent the energy (or part of the energy) of s′ similar molecules of the second kind, and so on. It is not at present necessary to suppose that the molecules are those of substances in the gaseous state. Considering only those states of the system which have a given Value of E$_2$, it can be proved, as a theorem in pure mathematics that when s, s′, . are very large, then, for all states except an infinitesimal fraction of the whole number, the values of u, 1/, w lie within ranges such that
(i) the values of u (and similarly of v, w) are distributed among the s molecules of the first kind according to the law of trial and error; and similarly of course for the molecules of other kinds:
$\sum _{s}$
$\sum _{s}$
$\sum _{s}$
$\sum _{s}$
| ＝$\sum _{s}$1/2 m′u′$^2$/s′＝$\sum _{s}$1/2 β$_1$φ$_1$$^2$/s′＝. . . | (3) |
A state of the system in which these two properties are true will be called a “normal state”; other states will be spoken of as “abnormal.” Let all possible states of the system be divided into small ranges of equal extension, and of these let a number P correspond to normal, and a numberThe Normal State. p to abnormal, states. What is proved is that, as s, s′, . . . become very great, the ratio P/p becomes infinite. Considering only systems starting in the p abnormal ranges, it is clear, from the fact that the extensions of the ranges do not change with the motion, that after a sufficient time most of these systems must have passed into the P normal ranges. Speaking loosely, we may say that there is a probability P/(P+p), amounting to certainty in the limit, that one of these systems, selected at random, will be in the normal state after a sufficient time has' elapsed. Again, considering the systems which start from the P normal ranges, We see that there is a probability p/(P+p) which vanishes in the limit, that a system selected at random from these will be in an abnormal state after a sufficient time. Thus, subject to a probability of error which is infinitesimal in the limit, we may state as general laws that—
A system starting from an abnormal state tends to assume the normal state; while
A system starting from the normal state will remain in the normal state.
It will now be found that the various properties of gases follow from the supposition that the gas is in the normal state.
If each of the fractions (3) is put equal to 1/4h, it is readily found, from the first property of the normal state, that, of the s molecules of the first kind, a numberLaw of Distribution of Velocities.
| $s{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hm(u^{2}+v^{2}+w^{2})}dudvdw$ | (4) |
have velocities of which the components lie between u and u+du, v and v+dv, w and w+dw, while the corresponding number of molecules of the second kind is, similarly,
| $s'{\sqrt {(h^{3}m'^{3}{\mbox{/}}\pi ^{3})}}e^{-hm'(u^{2}+v^{2}+w^{2})}dudvdw$ | (5) |
If c is the resultant velocity of a molecule, so that c$^2$=u$^2$+v$^2$+w$^2$, it is readily found from formula (4) that the number of molecules of the first kind of which the resultant velocity lies between c and c+dc is
| $4\pi s{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hmc^{2})}c^{2}dc$ | (6) |
These formulae express the “law of distribution of velocities” in the normal state: the law is often called Maxwell’s Law of Distribution.
If ${\tfrac {1}{2}}{\overline {mu^{2}}}$ denote the mean value of ${\tfrac {1}{2}}mu^{2}$ averaged over the s molecules of the first kind, equations (3) may be written in the form
| ${\tfrac {1}{2}}{\overline {mu^{2}}}={\tfrac {1}{2}}{\overline {mv^{2}}}={\tfrac {1}{2}}{\overline {mw^{2}}}={\tfrac {1}{2}}{\overline {\alpha _{1}\theta _{1}^{2}}}\cdots =1/4h$ | (7) |
Equipartition of Energy. showing that the mean energy represented by each term in E$_2$ (formula 2) is the same. These equations express the “law equipartition of energy,” commonly spoken of as the Maxwell-Boltzmann Law.
The law of equipartition shows that the various mean energies of different kinds are all equal, each being measured by the quantity 1/4h. We have already seen that the mean energy increases with the temperature: it will now be supposed that the mean energy is exactly proportional to theTemperature. temperature. The complete justification for this supposition will appear later: a partial justification is obtained as soon as it is seen how many physical laws can be explained by it. We accordingly put 1/2h=RT, where T denotes the temperature on the absolute scale, and then have equations (7) in the form
| ${\overline {mu^{2}}}={\overline {mv^{2}}}{\mbox{RT.}}$ | (8) |
When a system is composed of a mixture of different kinds of molecules, the fact that h is the same for each constituent [cf. formulae. (5) and (6)] shows that in the normal state the different substances are all at the same temperature. For instance, if the system is composed of a gas and a solid boundary, some of the terms in expression (2) may be supposed to represent the kinetic energy of the molecules of the boundary, so that equations (7) show that in the normal state the gas has the same temperature as the boundary. The process of equalization of temperature is now seen to be a special form of the process of motion towards the normal state: the general laws which have been stated above in connexion with the normal state are seen to include as special cases the following laws:—
Matter originally at non-uniform temperature tends to assume a uniform temperature; while
Matter at uniform temperature will remain at uniform temperature.
It will at once be apparent that the kinetic theory of matter enables us to place the second law of thermodynamics upon a purely dynamical basis. So far it has not been necessary to suppose the matter to be in the gaseous state. We now pass to the consideration of laws and properties which are peculiar to the gaseous state.
A simple approximate calculation of the pressure exerted by a gas on its containing vessel can be made by supposing that the molecules are so small in comparison with their distances apart that they may be treated as of infinitesimal size. Let a mixture of gases contain per unit volume v moleculesPressure of a Gas. of the first kind, v′ of the second kind, and so on. Let us fix our attention on a small area dS of the boundary of the ​ vessel, and let co-ordinate axes be taken such that the origin is in dS, and the axis of x is the normal at the origin into the gas. The number of molecules of the first kind of gas, whose components of velocity lie within the ranges between u and u+du, v and v+dv, w and w+dw, will, by formula (5), be
| $v{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hm(u^{2}+v^{2}+w^{2})}dudvdw$ | (9) |
per unit volume. Construct a small cylinder inside the gas, having dS as base and edges such that the projections of each on the coordinate axes are udt, vdt, wdt. Each of the molecules enumerated in expression (9) will move parallel to the edge of this cylinder, and each will describe a length equal to its edge in time dt. Thus each of these molecules which is initially inside the cylinder, will impinge on the area dS within an interval dt. The cylinder is of volume u dt dS, so that the product of this and expression (9) must give the number of impacts between the area dS and molecules of the kind under consideration within the interval dt. Each impinging molecule exerts an impulsive pressure equal to mu on the boundary before the component of velocity of its centre of gravity normal to the boundary is reduced to zero. Thus the contribution to the total impulsive pressure exerted on the area dS in time dt from this cause is
| $mu\times udtd{\mbox{S}}\times v{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hm(u^{2}+v^{2}+w^{2})}dudvdw$ | (10) |
The total pressure exerted in bringing the centres of gravity of all the colliding molecules to rest normally to the boundary is obtained by first integrating this expression with respect to u, v, w, the limits being all values for which collisions are possible (namely from −∞ to 0 for u, and from −∞ to +∞ for v and w), and then summing for all kinds of molecules in the gas. Further impulsive pressures are required to restart into motion all the molecules which have undergone collision. The aggregate amount of these pressures is clearly the sum of the momenta, normal to the boundary, of all molecules which have left dS within a time dt, and this will be given by expression (10), integrated with respect to u from 0 to ∞, and with respect to v and w from −∞ to +∞, and then summed for all kinds of molecules in the gas. On combining the two parts of the pressure which have been calculated, the aggregate impulsive pressure on dS in time dt is found to be
$\Sigma dt\ d{\mbox{S}}\iiint _{-\infty }^{+\infty }v{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hm(u^{2}+v^{2}+w^{2})}mu^{2}dudvdw,$
where Σ denotes summation over all kinds of molecules. This is equivalent to a steady pressure p$_1$ per unit area where
$p_{1}-\Sigma dtd{\mbox{S}}\iiint _{-\infty }^{+\infty }v{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hm(u^{2}+v^{2}+w^{2})}mu^{2}dudvdw$
Clearly the integral is the sum of the values of mu$^2$ for all the molecules of the first kind in unit volume, thus
| $p=v{\overline {mu^{2}}}+v'{\overline {m'u^{2}}}+\cdots$ | (11) |
On substituting from equations (7) and (8), this expression assumes the forms
| p＝(v+v ′+...)/2h | (12) |
| ＝(v+v ′+...)RT | (13) |
The number of molecules per unit volume in a gas at normal temperature and pressure is known to be about 2·75 X 10$^{19}$. If in formula (13) we put p＝1·013×10$^6$, (v+v ′+ ...)2·75×10$^{19}$ T＝273, we obtain R＝1·35×10$^{−16}$ and this enables us to determine the mean velocities produced by heatMolecular Velocities. motion in molecules of any given mass. For molecules of known gases the calculation is still easier. If ρ is the density corresponding to pressure p, we find that formula (11) assumes the form
p＝1/3ρC$^2$,
where C is a velocity such that the gas would have its actual translational energy if each molecule moved with the same velocity C. By substituting experimentally determined pairs of values of p and ρ we can calculate C for different gases, and so obtain a knowledge of the magnitudes of the molecular velocities. For instance, it is found that
| for hydrogen at | 0° Cent. C＝183,900 cms. per sec. |
| for air ,, | 15° Cent. C＝ 49,800 cms. per sec. |
| for mercury vapour at | 0° Cent. C＝ 18,500 cms. per sec. |
and other velocities can readily be calculated.
From the value R＝1·35×10$^{−16}$ it is readily calculated that a molecule, or aggregation of molecules, of mass 10$^{−12}$ grammes, ought to have a mean velocity of about 2 millimetres a second at 0° C. Such a velocity ought accordingly to be set up in a particle of 10$^{−12}$ grammes mass immersed in air or liquid“Brownian Movements.” at 0° C., by the continual jostling of the surrounding molecules or particles. A particle of this mass is easily visible microscopically, and a velocity of 2 mm. per second would of course be visible if continued for a sufficient length of time. Each bombardment will, however, change the motion of the particle, so that changes are too frequent for the separate motions to be individually visible. But it can be shown that from the aggregation of these separate short motions the particle ought to have a resultant motion, described . with an average velocity which, although much smaller than 2 mm. a second, ought still to be microscopically visible. It has been shown by R. von. S. Smoluchowski (Ann. d. Phys., 1906, 21, p. 756) that this theoretically predicted motion is simply that seen in the “Brownian movements” first observed by the botanist Robert Brown in 1827. Thus the “Brownian movements” provide visual demonstration of the reality of the heat-motion postulated by the kinetic theory.
Dalton’s Law.—The pressure as given by formula (12) can be written as the sum of a number of separate terms, one for each gas in the mixture. Hence we have Dalton’s law: The pressure of a mixture of gases is the sum of the pressures which would be exerted separately by the severalPressure, Volume and Temperature Relations. constituents if each alone were present.
Avogadro’s Law.—From formula (13) it appears that v+v ′+ . . ., the total number of molecules per unit volume, is determined when p, T and the constant R are given. Hence we have Avogadro’s law: Different gases, at the same temperature and pressure, contain equal numbers of molecules per unit volume.
Boyle’s and Charles’ Laws.—If v is the volume of a homogeneous mass of gas, and N the total number of its molecules, N＝v(v+v ′+ . . .), so that
| pv＝RNT. | (14) |
In this equation we have the combined laws of Boyle and Charles: When the temperature of a gas is kept constant the pressure varies inversely as the volume, and when the volume is kept constant the pressure varies as the temperature.
Since the volume at constant pressure is exactly proportional to the absolute temperature, it follows that the coefficients of expansion of all gases ought, to within the limits of error introduced by the assumptions on which we are working, to have the same value 1/273.
Van der Waals’s Equation.—The laws which have just been stated are obeyed very approximately, but not with perfect accuracy, by all gases of which the density is not too great or the temperature too low. Van der Waals, in a famous monograph, On the Continuity of the Liquid and Gaseous States (Leiden, 1873), has shown that the imperfections of equation (14) may be traced to two causes:—
(i.) The calculation has not allowed for the finite size of the molecules, and their consequent interference with one another’s motion, and
(ii.) The calculation has not allowed for the field of inter-molecular force between the molecules, which, although small, is known to have a real existence. The presence of this field of force results in the molecules, when they reach the boundary, being acted on by forces in addition to those originating in their impact with the boundary.
To allow for the first of these two factors, Van der Waals finds that v in equation (14) must be replaced by v−b, where b is four times the aggregate space occupied by all the molecules, while to allow for the second factor, p must be replaced by p+a/v$^2$. Thus the pressure is given by the equation
(p+a/v$^2$) (v−b)＝RNT,
which is known as Van der Waals’s equation. This equation is found experimentally to be capable of representing the relation between p, v, and T over large ranges of values. (See Condensation of Gases.)
Let us consider a single gas, consisting of N similar molecules in a volume v, and let the energy of each molecule, as in formula (2) be given byCalorimetry.
| E＝1/2$\sum _{\mbox{N}}$ (mu$^2$mv$^2$+mw$^2$+α$_1$θ$_1$$^2$+ . . . α$_n$θ$_n$$^2$) | (15) |
＝N(n+3)/4h by equation (7)
| ＝1/2(n+3)RNT | (16) |
Let a quantity dQ of energy, measured in work units, be absorbed by the gas from some external source, so that its pressure, volume and temperature change. The equation of energy is
| dQ＝dE+pdv, | (17) |
expressing that the total energy dQ is used partly in increasing the internal energy of the gas, and partly in expanding the gas against the pressure p. If we take p＝RNT/v from equation (14) and substitute for E from equation (16), this last equation becomes
| dQ＝1/2 (n+3)RNdT+RNTdv/v, | (18) |
which may be taken as the general equation of calorimetry, for a gas which accurately obeys equation (14).
Second Law of Thermodynamics.—If we divide throughout by T, we obtain
dQ/T ＝1/2(n+3)RNdT/T+RNdv/v.
showing that dQ/T is a perfect differential. This not only verifies that the second law of thermodynamics is obeyed, but enables us to identify T with the absolute thermodynamical temperature.
If the volume of the gas is kept constant, we put dv＝0 in equation (18) and dQ＝JC$_v$NmdT, where C$_v$, is the specific heat of the gas at constant volume and J is the mechanical equivalent of heat. We obtainSpecific Heats.
| C$_v$＝1/2(n+3)R/Jm | (19) |
On the other hand, if the pressure of the gas is kept constant ​throughout the motion, T/v is constant and dQ＝JC$_p$NmdT, whence
| C$_p$＝1/2(n+5) R/Jm. | (20) |
By division of the values of C$_p$ and C$_v$, , we find for γ, the ratio of the specific heats.
| γ＝1 +2/n+3). | (21) |
The comparison of this formula with experiment provides a striking confirmation of the truth of the kinetic theory but at the same time discloses the most formidable difficulty which the theory has so far had to encounter.
On giving different values to n in formula (21), we obtain the values for γ:
| n ＝ | 0, | 1, | 2, | 3, | 4, | 5, | |
| γ ＝ | 1·66, | 1·5, | 1·4, | 1·33, | 1·28, | 1·25, | &c. |
Thus, to within the degree of approximation to which our theory is accurate, the value of γ for every gas ought to be one of this series. The following are the values of γ for gases for which γ can be observed with some accuracy:—
| Mercury | 1·66 | Nitrogen | 1·40 |
| Krypton | 1·66 | Carbon monoxide | 1·41 |
| Helium | 1·65 | Hydrogen | 1·40 |
| Argon | 1·62 | Oxygen | 1·40 |
| Air | 1·40 | Hydrochloric acid | 1·39 |
It is clear that for the first four gases n＝0, while for the remainder n＝2. To examine what is meant by a zero value of n we refer to formula (15). The value of n is the number of terms in the energy of the molecule beyond that due to translation. Thus when n＝0, the whole energy must be translational: there can be no energy of rotation or of internal motion. The molecules of gases for which n＝0 must accordingly be spherical in shape and in internal structure, or at least must behave at collisions as though they were spherical, for they would otherwise be set into rotation by the forces experienced at collisions. In the light of these results it is of extreme significance that the four gases for which n＝0 are all believed to be monatomic: the molecules of these gases consist of single atoms. Moreover, these four are the only monatomic gases for which the value of γ is known, so that the only atoms of which the shape can be determined are found to be spherical. It is at least a plausible conjecture, until the contrary is proved, that the atoms of all elements are spherical.
The next value which occurs is n＝2. The kinetic energy of the molecules of these gases must contain two terms in addition to those representing translational energy. For a rigid body the kinetic energy will, in general, consist of three terms (Aω$_1$$^2$+Bω$_2$$^2$+Cω$_3$$^2$) in addition to the translational energy. The value n＝2 is appropriate to bodies of which the shape is that of a solid of revolution, so that there is no rotation about the axis of symmetry. We must accordingly suppose that the molecules of gases for which n＝2 are of this shape. Now this is exactly the shape which we should expect to find in molecules composed of two spherical atoms distorting one another by their mutual forces, and all gases for which n＝2 are diatomic.
No molecule could possibly be imagined for which n had a negative value or the value n＝1. The theory therefore passes a crucial test when it is discovered that no gases exist for which n is either negative or unity. On the other hand, the theory encounters a very serious difficulty in the fact that all molecules possess a great number of possibilities of internal motion, as is shown by the number of distinct lines in their spectra both of emission and of absorption. So far as is known, each line in the spectrum of, say, mercury, represents a possibility of a distinct vibration of the mercury atom, and accordingly provides two terms (say αφ$^2$+βφ$^2$, where φ is the normal co-ordinate of the vibration) in the expression for the energy of the molecule. There are many thousands of lines in the mercury spectrum, so that from this evidence it would appear that for mercury vapour n ought to be very great, and γ almost equal to unity. Instead of this we have n＝0, and γ=12/3 . As a step towards removing this difficulty we notice that the energy of a vibration such as is represented by a spectral line has the peculiarity of being unable to exist (so far as we know) without suffering dissipation into the ether. This energy, therefore, comes under a different category from the energy for which the law of equipartition was proved, for in proving this law conservation of energy was assumed. The difficulty is further diminished when it is proved, as it can be proved, that the modes of energy represented in the atomic spectrum acquire energy so slowly that the atom might undergo collisions with other atoms for centuries before being set into oscillations which would possess an appreciable amount of energy. In fact the proved tendency for the gas to pass into the “normal state” in which there is equipartition of energy, represents in this case nothing but the tendency for the translational energy to become dissipated into the energy of innumerable small vibrations. We find that this dissipation, although undoubtedly going on, proceeds with extreme slowness, so that the vibrations pass their energy on to the ether as rapidly as they acquire it, and the “normal state” is never established. These considerations suggest that the difficulty which has been pointed out may be apparent rather than real. At the same time this difficulty is only one aspect of a wider difficulty which cannot be lightly passed over; Maxwell himself regarded it as the principal obstacle in the way of the full acceptance of the theory of which he was so largely the author.  (J. H. Je.)
* ↑ It will be noted that Clerk Maxwell’s “atom” and “atomic theory” have the significance which we now attach to “molecule” and “molecular theory.”
* ↑ “On the Cohesions of Fluids,” Phil. Trans. (1805); Young’s Coll. Works, i. 461.
* ↑ Other processes also help in the conduction of heat, especially in substances which are conductors of electricity.
* ↑ See Jeans, Dynamical Theory of Gases (1904), ch. v.
* ↑ Very significant confirmation of this conjecture is obtained from a study of the specific heats of the elements in the solid state. If a solid body is regarded as an aggregation of similar atoms each of mass m, its specific heat C is given, as in formula (19) by C＝1/2(n+3)R/Jm. From Dulong and Petit’s law that Cm is the same for all elements, it follows that n+3 must be the same for all atoms. Moreover, the value of Cm shows that n+3 must be equal to six. Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form E＝1/2(mu$^2$+mv$^2$+mw$^2$+x$^2$∂$^2$V/∂x$^2$ + y$^2$∂$^2$V/∂y$^2$ + z$^2$∂$^2$V/∂z$^2$), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.
* ↑ J. H. Jeans, Dynamical Theory of Gases, ch. ix.