358 



NATURE 



[Afarc/i 4, 1875 



pressed by the double sum 2 2 [^/^r), which indicates that the 

 value of 4 v? ;- is to be found for every pair of particles, and the 

 results added together. 



Clausius has established this equation by a very simple mathe- 

 matical process, with which I need not trouble you, as we are not 

 studying mathematics to-night. We may see, however, that it 

 indicates two causes which may affect the pressure of the fluid 

 on the vessel which contains it : the motion of its particles, which 

 tends to increase tl;e pressure, and the attraction of its particles, 

 which tends to diminish the pressure. 



We may therefore attribute the pressure of a fluid either to 

 the motion of its particles or to a repulsion between them. 



Let us test by means of this result of Clausius the theory that 

 the pressure of a gas arises entirely from the repulsion which one 

 particle exerts on another, these particles, in the case of gas in 

 a fixed vessel, being really at rest. ^ 



In this case the virial must be negative, and since by Boyle s 

 Law the product of pressure and volume is constant, the vhial 

 also must be constant, whatever the volume, in the same quantity 

 of gas at constant temperature. It follows from this that Jir, 

 the product of the repulsion of two particles into the distance 

 between them, must be constant, or in other words that the 

 repulsion must be inversely as the distance, a law which Newton 

 has shown to be inadmissible in the case of molecular forces, as it 

 would make the action of the distant parts of bodies greater than 

 that of contiguous parts. In fact, we have only to observe that 

 ii RrU consiaut, the virial of every pair of particles must be the 

 same, so that the virial of the system must be proportional to 

 the number of pairs of particles in the system— that is, to the 

 square of the number of particles, or in other words to the square 

 of the quantity of gas in the vessel. The pressure, according to 

 this law, would not be the same in different vessels of gas at the 

 same density, but would be greater in a large vessel than in a 

 fraall one, and greater in the open air than in any ordinary 

 vessel. 



The pressure of a gas cannot therefore be explamed by assum- 

 ing repulsive forces between the particles. 



It must therefore depend, in whole or in part, on the motion 

 of the particles. 



If we suppose the particles not to act on each other at all, 

 there will be no viiial, and the equation will be reduced to the 

 form 



yp = ^r. 



If M is the mass of the whole quantity of gas, and <r is the 

 mean square of the velocity of a particle, we may write the 

 equation — 

 ^ f'/ = iMc'^ 



or in words, the product of the volume and the pressure is one- 

 third of the mass multiplied by the mean square of the velocity. 

 If we now assume, what we shall afterwards prove by an inde- 

 pendent process, that the mean square of the velocity depends 

 only on the temperature, this equation exactly represents Boyle's 



Law. T^ 1 > 



But we know that most ordinary gases deviate from Boyle s 

 Law, especially at low temperatures and great densities. Let us 

 see whether the hypothesis of forces between the particles, which 

 we rejected when brought forward as the sole cause of gaseous 

 pressure, may not be consistent with experiment when considered 

 as the cause of this deviation from Boyle's Law. 



When a gas is in an extremely rarefied condition, the number 

 of particles within a given distance of any one particle will be 

 proportional to the density of the gas. Hence the virial arising 

 from the action of one particle on the rest will vary as the 

 density, and the whole virial in unit of volume will vary as the 

 square of the density. 



Calling the density p, and dividing the equation by F, we 



J> = HP'' - t^r 

 where /i is a quantity which is nearly constant for small den- 

 sities. 



Now, the experiments of Regnault show that in most gases, 

 as the density increases the pressure falls below the value calcu- 

 lated by Boyle's Law. Hence the virial must be positive ; that 

 is to say, the mutual action of the particles must be in the main 

 attractive, and the effect of this action in diminishing the pres- 

 sure must be at first very nearly as the square of the density. 



On the other hand, when the pressure is made stiU greater 

 the substance at length reaches a state in which an enormous 

 increase of pressure produces but a very small increase of density. 



This indicates that the virial is now negative, or, in other words, 

 the action between the particles is now, in the main, repulsive. 

 We may therefore conclude that the action between two particles 

 at any sensible distance is quite insensible. As the particles 

 approach each other the action first shows itself as an attraction, 

 which reaches a maximum, then diminishes, and at length 

 becomes a repulsion so great that no attainable force can reduce 

 the distance of the particles to zero. 



The relation between pressure and densiiy arising from such 

 an action between the particles is of this kind. 



As the density increases from zero, the pressure at first depends 

 almost entirely on the motion of the particles, and therefore varies 

 almost exactly as the pressure, according to Boyle's Law. As the 

 density continues to increase, the effect of the mutual attraction 

 of the particles becomes sensible, and this causes the rise of 

 pressure to be less than that given by Boyle's Law. If the tem- 

 perature is low, the effect of attraction may become so large in 

 proportion to the effect of motion that the pressure, instead of 

 always rising as the density increases, may reach a maximum, 

 and then begin to diminish. 



At length, however, as the average distance of the particles 

 is still further diminished, the effect of repulsion will prevail over 

 that of attraction, and the pressure will increase so as not only 

 to be greater than that given by Boyle's Law, but so that an 

 exceedingly small increase of density will produce an enormous 

 increase of pres;ure. 



Hence the relation between pressure and volume may Ije 

 represented by the curve A B C D E F G, where the horizontal 

 ordinate represents the volume, and the vertical ordinate repre- 

 sents the pressure. 



As the volume diminishes, the pressure increases up to the 

 point C, then diminishes to the point E, and finally increases 

 without limit as the volume diminishes. 



We have hitherto supposed the experiment to be conducted in 

 such a way that the density is the same in every part of the 

 medium. This, however, is impossible in practice, as the only 

 condition we can impose on the medium from without is that 

 the whole of the medium shall be contained within a certain 

 vessel. Hence, if it is possible for the medium to arrange itself 

 so that part has one density and part another, we cannot prevent 

 it from doing so. 



Nov the points B and E represent two states of the medium 

 in which the pressure is the same but the density very different. 

 The whole of the medium may pass from the state B to the 

 state F, not through the intermediate states C D E, but by small 

 successive portions passing direct'y from the state B to the state 

 F. In this way the successive states of the medium as a whole 

 will be represented by points on the straight line B F, the point 

 B rt presenting it when entirely in the rarefied state, and i^ repre- 

 senting it when entirely condensed. This is what takes place 

 when a gas or vapour is liq\iefied. 



Under ordinary circumstances, therefore, the relation between 

 pressure and volume at constant temperature is represented by 

 the broken line A B F G. If, however, the medium when lique- 

 fied is carefully kept from contact with vapour, it may be pre- 

 served in the liquid condition and brought into states represented 

 by the portion of the curve between j^ and £. It is also pos- 

 sible that methods may be devised whereby the vapour may be 

 prevented from condensing, and brought into states represented 

 by points in B C. 



