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THE MOLECULAR STRUCTURE OF
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THE MOLECULAR STRUCTURE OF See also:MATTER An enormous See also:mass of experimental See also: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 See also:physical phenomena are concerned is the See also:molecule. When chemical phenomena occur the molecule may be divided into atoms, and these atoms, in the presence of See also: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 See also:aggregation of molecules. In point of fact it is found that the properties which are most easily explained are those connected with the gaseous See also: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 See also:accumulation of electrical and See also: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 See also:Scale of Molecular Structure.—Apart from See also:speculation, the first definite evidence for the molecular structure of matter occurs when it is found that certain physical phenomena See also:change their whole nature as soon as we See also: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 See also:Lord See also:Rayleigh (Prot. See also:Roy. See also:Soc., 1890, 47, p. 364), who found that a film of See also:olive oil spread over the See also:surface of See also:water produced a perceptible effect on small floating pieces of camphor, at places at which the thickness of the film was 1(2..6 X cms., but produced no perceptible effect at all at places where the thickness of the film was 8.1X ro 8 cms. Thus a certain phenomenon, of the nature of capillary See also: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 ro•6Xro 8 cms. are found to be in some way qualitatively different from those of a film of thickness 8.1X To' cms. Here is See also: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 X ro 8 cms. The See also:probability of this conjecture is strengthened when it is discovered that in all phenomena of this type the See also:critical length connected with the See also:stage at which the phenomenon changes its nature is of the See also:order of magnitude of ro 8 cms.
Lord Rayleigh (Phil. Mag. 1890 [51, 30, p. 474) has pointed out that the earliest known See also:attempt to estimate the See also:size of molecules, made by See also: Trans. (18o5); Young's See also:Coll. See also:Works, i. 461.better need not cause surprise when it is stated that the quantities are calculated on the See also:hypothesis that the molecules are spherical in shape. This hypothesis is introduced for the See also:sake of simplicity, but is known to be unjustifiable in fact. What is given by the formulae is accordingly the mean See also:radius of an irregularly shaped solid (or, more probably, of the region in which the See also: Thus the molecules of a solid must make only small excursions about their mean positions. In a See also:gas the state of things is very different; an odour is known to spread rapidly through See also:great distances, even in the stillest See also:air, and a gaseous See also:poison or corrosive will attack not only those See also: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 See also: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 See also:rest relatively to one another. The fact that a solid body in its natural state is capable both of See also:compression and of See also:dilatation indicates that the molecules of the body must not be supposed to be fixed 15gidly in position relative to one another; the further fact that a motion of either compression or of dilata- tion is opposed by forces which are brought into See also:play in the interior of the solid suggests that the position of rest is one in which the molecules are in See also:stable See also: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 See also:system of See also:light See also:spiral springs, one See also: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 See also:drawn aside, pendulum-See also:wise, and allowed to impinge on the other. After impact the two masses will rebound, and the See also:process may be repeated any number of times, but ultimately the two masses will be found again See also:hanging in See also:con- tact See also:side by side. At the first impact each layer of surface molecules which takes the See also: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 See also: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 See also: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 See also: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 Diameter calculated by the kinetic theory of gases. Gas. From devia- From co- efficient co- From co- Mean value. Lions from efficient of fficient of efficient of See also:Boyle's See also:law. viscosity. See also:conduction of See also:diffusion. heat. See also:Hydrogen 2.05X10-8 2.05X10-8 I.99XIO-8 2.02X10—8 2.03X10-8 See also:Carbon — 2.90X10-8 2•74XIO-8 2.92Xro—8 2'85)00-8 monoxide See also:Nitrogen 3.12 X10-8 2.90X10-8 2.74X10-8 — 2.92X10—8 Air 2.90 X I 0--e 2.86 X 10-8 2.72 X I 0--e — 2.83 X 10-8 See also:Oxygen — 2.81X10_8 2.58X10_8 2.70X10 8 2.70X10 e Carbon 3.00X10-8 3.47X10-8 3.58X10-8 3.28X10-8 3.33X10-8 dioxide . 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 See also: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 See also: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 See also:generation of heat by See also: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.l When the molecules are oscillating about their equilibrium positions, there is no See also:reason why their mean distance apart should be the same as when they are at rest. This leads to an See also: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 — e may be greater than the attractive force they exert when at a distance a + e. If so, it is clear that their mean distance apart, averaged through a sufficiently See also:long See also:interval of their motion, will be greater than a. A body made up of molecules of this kind will expand on See also:heating. As the temperature of a body increases the See also: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 See also: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 See also:constant rearrangement of position among them. A molecule escaping from its See also: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 See also:close to the surface of the body, and if it also happens to start off in the right direction, it may See also:escape from the body altogether and describe a See also:free path in space until it is checked by See also: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. More-over, 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 See also:vessel, a molecule which has See also:left the liquid will, after a certain t Other processes also help in the conduction of heat, especially in substances which are conductors of See also:electricity.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 See also:form of the gaseous state. At normal temperature and pressure the See also: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 See also:cube would be about 2.9XIo—' See also:ems.; the average diameter of a molecule in air is 2.8Xro-8 ems.) Further and very important evidence as to the nature of the gaseous state of matter is provided by the experiments of See also:Joule and See also: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 See also:general very slight. In terms of the molecular theory this indicates that the See also:total energy of the gas is the sum of the See also: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 See also:remainder of this See also: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 See also: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 See also:laws will pass through a See also:series of configurations which can be' theoretically determined as soon as the structure of each molecule and the initial position and velocity of every See also: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 See also: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 See also:search for features and properties See also: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 See also:special application, when the dynamical system is identified with a gas, will appear later. Let qh q2, .. qn be the generalized co- D,,namical ordinates of any dynamical system, and let p,, p2, . . . pm Basis. be the corresponding momenta. If the system is supposed to obey the conservation of energy and to move solely under its own See also:internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations aE aE q. =Fps,' Ps = —aq.' where q, denotes dq.ldt, &c., and E is the total energy expressed as a See also:function of p,, q,, . . p,., g,. When the initial values of PI, ql . . . p,,, q,,, are given, the motion can be traced completely from these equations. Let us suppose that an See also:infinite number of exactly similar systems start simultaneously from all possible values of pi, ql, . . . p,,, q,., each moving solely under its own internal forces, and therefore in accordance with equations (I). Let us confine our See also:attention to those (I) systems for which the initial values of q,, . . . Ps, q, See also:lie within a range such that pi is between pi and p,+dpi q, ,, „ q,+dq,, and so on. Let the product dpi dq, ... dpn dqn be spoken of as the " See also:extension " of this range of values. After a time dt the value of p, will have increased to p,+p,dt, where p, is given by equations (1), and there will be similar changes in q,, p2, q2, ... q,,. Thus after a time dt the values of the co-ordinates and momenta of the small See also:group of systems under consideration will lie within a range such that Pi is between p,+p,dt and p,+dpi+ (iii+-a—dp,) dt 41 ,, +4,dt „ + (4, +aldq,) dt, and so on. Thus the extension of the range after the interval dt is dpi (1+apldt) dg, (I+a—gidt) . . . or, expanding as far as first See also:powers of dt, dpidq, . . . dpndgn) i+See also:Min (a+) dt al From equations (I),.we find that api See also:a4,-o ap,+aq, so that the extension of the new range is seen to be dp,dq, ... dpndq„, 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 Foes 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 co-ordinates 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 See also: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,+E2, where E2 is of the form E2 = 2E(mu2+mv2+mw2+a,012+a2022+ . . . +ar0n2). +1z,(m'u'2+m'v'2+m'w'2+0,h,2+02o22+ . . . +/3n'tbn'2) (2) s where 01,02, . . 0,. and similarly Or, d2, ... 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 al, an &c. In this expression the first See also:line may be supposed to represent the energy (or part of the energy) of s similar molecules of a kind which we shall See also:call the first kind, the terms i(mu"2+mv2+mw2) being the kinetic energy of See also: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 E2 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 E2, it can be proved, as a theorem in pure See also:mathematics,' that when s, s', ... are very large, then, for all states except an infinitesimal fraction of the whole number, the values of u, v, 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 See also:error; and similarly of course for the molecules of other kinds: (ii) lzmu2 Ezmv2 1mw2 T2a,0,2 S S S = = _... s s s zm'u'2 1z00,2 s' s' s' 1 See Jeans, Dynamical Theory of Gases (1904), ch. v. 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 The Normal divided into small ranges of equal extension, and of State. these let a number P correspond to normal, and a number 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 See also:random, will be in the normal state after a sufficient time has elapsed. Again, considering the systems which start from the P normal r m s, 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 theLawofDiss molecules of the first kind, a number tribution of ss/ (h3ma/7ra)e h `("4v "2)dudedw (4) Velocities. have velocities of which the components lie between u and u+du, v and v+dv, w and w+See also:dw, while the corresponding number of molecules of the second kind is, similarly, s's (h3m"/ir')e hm'(v.2+v2+See also:w2)dudedw. (5) If c is the resultant velocity of a molecule, so that c2=u2+v2+w2, it is readily found from See also:formula (4) that the number of molecules of the first kind of which the resultant velocity lies between c and c+dc is 47s.4 (hams/zs)e-1molc2dc. (6) These formulae See also:express the " law of See also:distribution of velocities " in the normal state: the law is often called See also:Maxwell's Law of Distribution. If 2mu2 denote the mean value of Zmu2 averaged over the s molecules of the first kind, equations (3) may be written in the form 'mug = jnv2 = 1 mw2 = 1 a 82 - 2 2 2 2 1 1- ... =1/4h, (7) E9uipartl- showing that the mean energy represented by each tioa of See also:term in E2 (formula 2) is the same. These equations Energy. express the " law of 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 in-creases with the temperature: it will now be supposed See also:Tempera-that the mean energy is exactly proportional to the tune. temperature. The See also:complete See also: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 mug=mv2= ... =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-See also: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 See also: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 See also:peculiar to the gaseous state. A See also: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 Pressure of apart that they may be treated as of infinitesimal size. Eras. Let a mixture of gases contain per unit See also:volume s See also:mole- a cules of the first kind, v' of the second kind, and so on. Let us See also:fix our attention on a small See also:area dS of the boundary of the s (3) vessel, and let co-See also:ordinate axes be taken such that the origin is in dS, and the See also: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~(h2m2/71.3)e n''(ut+„2+,,,2)dudvdw (9) per unit volume. Construct a small See also:cylinder inside the gas, having dS as See also:base and edges such that the projections of each on the co-ordinate 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 muXudtdSXvJ(V1n3/r3)e )dudvdw (I0) 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 — so too for u, and from —°o to + °o 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 (io), integrated with respect to u from o to °o , and with respect to v and w from — °o to +co, 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 Edt dS f fl vs/ (h3m3/7r3)e-hm("±,'2+"2)mu2dudvdw, where denotes summation over all kinds of molecules. This is See also:equivalent to a steady pressure pi per unit area where +~ pi—Tfff vs/ (h3m3/.7r3)e-h'''( 2)mu2dudvdw. Clearly the integral is the sum of the values of mu' for all the molecules of the first kind in unit volume, thus p=vmu'+v'm'u2+... (II) On substituting from equations (7) and (8), this expression assumes the forms p=(v+Y +, , ,)/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 X10". If in formula (13) we put p=I.013X106, (v+v'+...)=2.75X10'9 Molecular T =273, we obtain R =1.35 X Io -16 and this enables us Velocities. to determine the mean velocities produced by heat motion in molecules of any given mass. For molecules of known gases the calculation is still easier. If p is the density corresponding to pressure p, we find that formula (II) assumes the form p = 3pC2, 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 p 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 ems. per sec. air 15° „ C = 49,800 „ „ See also:mercury vapour at o° „ C = 18,500 „ „ and other velocities can readily be calculated. From the value R =1.35 X 10-” it is readily calculated that a mole- cule, or aggregation of molecules, of mass so -12 grammes, ought to have a mean velocity of about 2 millimetres a second at "Brownian o° C. Such a velocity ought accordingly to be set up in a Movements." particle of to -12 grammes mass immersed in air or liquid at o° 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 See also: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 See also: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 (See also:Ann. d. Phys., 1906, 2I, p. 756) that this theoretically predicted motion is simply that seen in the " Brownian movements " first observed by the botanist See also:Robert See also: Boyle's and See also: If we take p=RNT/v from equation (14) and substitute for E from equation (16), this last equation becomes dQ = z (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 See also:divide throughout by T, we obtain dQ = (n+3)RNT+RN vv, showing that dQ/T is a perfect See also: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=o in equation (18) and dQ = JC0NmdT, where C„ is the specific Specific heat of the gas at constant volume and J is the mechani- Heats. cal equivalent of heat. We obtain C,, = z (n+3)R/Jm• (19) On the other See also:hand, if the pressure of the gas is kept constant throughout the motion, T/v is constant and dQ=JC,NmdT, whence Cp = i(n+5) R/Jm• (20) By See also:division of the values of Cp and C„ we find for y, the ratio of the specific heats. 7=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 7: n = o, 1, 2, 3, 4, 5, -y = 1.66, 1.5, 1.4, 1.33, 1.21.25, &C. Thus, to within the degree of approximation to which our theory is accurate, the value of -y for every gas ought to be one of this series. The following are the values of y for gases for which y can be observed with some accuracy: Mercury . . 1.66 Nitrogen . 1.40 Krypton . . 1.66 Carbon monoxide . 1.41 See also:Helium . . 1.65 Hydrogen . 1.40 See also:Argon . P62 Oxygen 1.40 . . . . . 1.40 Hydrochloric See also:acid . . 1.39 It is clear that for the first four gases n=o, 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 =o, the whole energy must be translational: there can be no energy of rotation or of internal motion. The molecules of gases for which n= o 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 y 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 (Awi2+Bw22+Cwa2) 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= i. The theory therefore passes a See also: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 See also:atom, and accordingly provides two terms (say a¢2+134,2, where 4 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 y almost equal to unity. Instead of this we have n=o, and y=i. As a step towards removing this difficulty we See also: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 See also:ether. This energy, therefore, comes under a different See also:category from the energy for which the law of equipartition was proved, for in proving this law conservation of ' 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(n+3)R/Jm. From See also:Dulong and See also: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 = I (mu'+mv2+mw2+x'a OW O'V (32V x + y2 ay + z2 a ), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes. energy was assumed. The difficulty is further diminished when it is proved, as it can be proved,2 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 See also: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 See also:principal obstacle in the way of the full See also:acceptance of the theory of which he was so largely the author. (J. H. JE.) MOLE-See also:RAT, the name of a group of See also:blind burrowing rodents, typified by the large See also:grey Spalax typhlus of eastern See also:Europe and See also:Egypt, which represents the Old See also:World See also:family Spalacidae. All the mole-rats of the genus Spalax are characterized by the want of distinct necks, small or rudimentary ears and eyes, and short limbs provided with powerful digging claws. There are three pairs of cheek-See also:teeth which are rooted, and show folds of See also:enamel on the See also:crown. Mole-rats are easily recognized by the peculiarly flattened See also:head, in which the See also:minute eyes are covered with skin, the See also:wart-like ears, and rudimentary tail; they make burrows in sandy See also:soil, and feed on bulbs and roots. See also:Bamboo-rats, of which one genus (Rhizomys) is See also:Indian and Burmese, and the other (Tachyoryctes) See also:East See also:African, differ by the See also:absence of skin over the eyes, the presence of short ears, and a short, sparsely-haired tail. They burrow either among tall grass, or at the roots of trees (see See also:RODENTIA). MOLE-See also:SHREW, any individual of the genera Urotrichus and Uropsilus (see INSECTIVORA). These animals, which are some-times called shrew-moles, are not moles with shrew-like habits, but shrews with the burrowing habits of moles and resembling them in See also:appearance. Additional information and CommentsThere are no comments yet for this article.
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