Oct. 15, 1874J 



NATURE 



479 



tionary motion. The motion of the solar system satisfies 

 this condition, and so does the motion of the molecules 

 of a gas contained in a vessel. 



The kinetic energy of a particle is half the product of 

 its mass into the square of its velocity, and the kinetic 

 energy of a system is the sum of the kinetic energy of its 

 parts. 



When an attraction or repulsion exists between two 

 points, half the product of this stress into the distance 

 laetween the two points is called the Virial of the stress, 

 and is reckoned positive when the stress is an attraction, 

 and negative when it is a repulsion. The virial of a 

 system is the sum of the virial of the stresses which exist 

 in it. 



If the system is subjected to the external stress of the 

 pressure of the sides of a vessel in which it is contained, 

 the amount of virial due to this external stress is three 

 halves of the product of the pressure into the volume of 

 the vessel. 



The virial due to internal stresses must be added to 

 this. 



The theorem of Clausius may now be written — 



\ ^{jn 2^^) = 3 / F+ .1 2 2 {R r) 

 The left-hand member denotes the kinetic energy. 

 On the right hand, in the first term, p is the external 

 pressure on unit of area, and V is the volume of the 

 vessel. 



The second term represents the virial arising from the 

 action between every pair of particles, whether belonging 

 to different molecules or to the same molecule. R is the 

 attraction between the particles, and r is the distance 

 between them. The double symbol of summation is used 

 because every pair of points must be taken into account, 

 those between which there is no stress contributing, of 

 course, nothing to the virial. 



As an example of the generality of this theorem, we 

 may mention that in any framed structure consisting of 

 struts and ties, the sum of the products of the pressure 

 in each strut into its length, exceeds the sum of the pro- 

 ducts of the tension of each tie into its length, by the 

 product of the weight of the whole structure into the 

 height of its centre of gravity above the foundations. 

 (See a paper on " Reciprocal Figures, &c." Trans. R. S. 

 Edin., vol. xxvi. p. 14. 1870.) 



In gases the virial is very small compared with the 

 kinetic energy. Hence, if the kinetic energy is constant, 

 the product of the pressure and the volume remains con- 

 stant. This is the case for a gas at constant temperature. 

 Hence we might be justified in conjecturing that the tem- 

 perature of any one gas is determined by the kinetic 

 energy of unit of mass. 



The theory of the exchange of the energy of agitation 

 from one body to another is one of the most difficult 

 parts of molecular science. If it were fully understood, 

 the physical theory of temperature would be perfect. At 

 present we know the conditions of thermal equilibrium 

 only in the case of gases in which encounters take place be- 

 tween only a pair of molecules at once. In this case the 

 condition of thermal equilibrium is that the mean kinetic 

 energy due to the agitation of the centre of mass of a 

 molecule is the same, whatever be the mass of the mole- 

 cule, the mean velocity being consequently less for the 

 more massive molecules. 



With respect to substances of more complicated con- 

 stitution, we know, as yet, nothing of the physical con- 

 dition on which their temperature depends, though the 

 researches of BoUzmann on this subject are likely to 

 result in some valuable discoveries. 



M. Van der Waals seems, therefore, to be somewhat too 

 hasty in assuming that the temperature of a substance is 

 in every case measured by the energy of agitation of its 

 individual molecules, though this is undoubtedly the case 

 with substances in the gaseous state. 



Assuming, however, for the present that the tempera- 

 ture is measured by the mean kinetic energy of a mole- 

 cule, we obtain the means of determining the virial by 

 observing the deviation of the product of the pressure 

 and volume from the constant value given by Boyle's 

 law. 



It appears by Dr. Andrews' experiments that when the 

 volume of carbonic acid is diminished, the temperature 

 remaining constant, the product of the volume and pres- 

 sure at first diminishes, the rate of diminution becoming 

 more and more rapid as the density increases. Now, the 

 virial depends on the number of pairs of molecules which 

 are at a given instant acting on one another, and this 

 number in unit of volume is proportional to the square 

 of the density. Hence the part of the pressure depend- 

 ing on the virial increases as the square of the density, 

 and since, in the case of carbonic acid, it diminishes the 

 pressure, it must be of the positive sign, that is, it must 

 arise from attraction between the molecules. 



But if the volume is still further diminished, at a certain 

 point liquefaction begins, and from this point till the gas 

 is all liquefied no increase of pressure takes place. As 

 soon, however, as the whole substance is in the liquid 

 condition, any further diminution of volume produces a 

 great rise of pressure, so that the product of pressure and 

 volume increases rapidly. This indicates negative virial, 

 and shows that the molecules are now acting on each 

 other by repulsion. 



This is what takes place in carbonic acid below the 

 temperature of 30-92" C. Above that temperature there is 

 first a positive and then a negative virial, but no sudden 

 liquefaction. 



Similar phenomena occur in all the hquefiable gases. 

 In other gases we are able to trace the existence of attrac- 

 tive force at ordinary pressures, though the compression 

 has not yet been carried so far as to show any repulsive 

 force. In hydrogen the repulsive force seems to prevail 

 even at ordinary pressures. This gas has never been 

 liquefied, and it is probable that it never will be liquefied, 

 as the attractive force is so weak. 



We have thus evidence that the molecules of gases 

 attract each other at a certain small distance, but when 

 they are brought still nearer they repel each other. This 

 is quite in accordance with Boscovich's theory of atoms as 

 massive centres of force, the force being a function of the 

 distance, and changing from attractive to repulsive, and 

 back again several times, as the distance diminishes. If 

 we suppose that when the force begins to be repulsive it 

 increases very rapidly as the distance diminishes, so as to 

 become enormous if the distance is less by a very small 

 quantity than that at which the force first begins to be 

 repulsive, the phenomena will be precisely the same as 

 those of smooth elastic spheres. 



