March ii, 1875] 



NA TV RE 



!75 



shall have the same temperalure is that the average kinetic 

 energy of a single molecule shall be the same in the two gases. 



Now, we have already shown that llie pressure of a gas is two- 

 thirds of the kinetic energy in unit of volume. Hence, if the 

 pressure as well as the temperature be the same in tlie two 

 gases, the kinetic energy per unit of volume is the same, as well 

 as the kinetic tneigy per molecule. There must, therefore, be 

 the same number of molecules in unit of volume in the two 

 gases. 



This result coincides with the law of equivalent volumes esta- 

 blished by Gay Lussac. This law, however, has hitherto rested 

 on purely chemical evidence, the relative masses of the mole- 

 cules of difl'erent substances having been deduced from the 

 proportions in which the substances enter into chemical com- 

 bination. It is now demonstrated on dynamical principles. The 

 molecule is defined as that small portion of the substance which 

 moves as one lump during the motion of agitation. This is a 

 purely dynamical definition, independent of any experiments on 

 combination. 



The density of a gaseous medium, at standard temperature 

 and pressure, is proportional to the mass of one of its molecules 

 as thus defined. 



We have thus a safe method of estimating the relative masses of 

 molecules of different substances when in the gaseous state. This 

 method is more to be depended on than those founded on elec- 

 trolysis or on specific heat, because our knowledge of the condi- 

 tions of the motion of agitation is more complete than our 

 knowledge of electrolysis, or of the internal motions of the con- 

 stituents of a molecule. 



I must now say something about these internal motions, be- 

 cause the greatest difficulty which the kinetic theory of gases has 

 yet encountered belongs to this part of the subject. 



We have hltheito considered only the motion of the centre of 

 mass of the molecule. We have now to consider the motion of 

 the constituents of the molecule relative to the centre of mass. 



If we suppose that the constituents of a molecule are atoms, 

 and that each atom is what is called a maleiial point, then each 

 atom may move in tlrree different and independent ways, corre- 

 sponding to the three dimensions of space, so that the number of 

 variables required to determine the position and configuration of 

 all the atoms of the molecule is three times the number of 

 atoms. 



It is not essential, however, to the mathematical investigation 

 to assume that the molecule is made up of atoms. All that is 

 assumed is that the position and configuration of the molecule 

 can be completely expressed by a certain number of variables. 

 Let us call this number n. 



Of these vatiables, three are required to determine the position 

 of the centre of mass of the molecule, and the remaining « — 3 

 to determine its confi^juration relative to its centre of mas-. 



To each of the n variables corresponds a different kind of 

 motion. 



The motion of translation of the centre of mass has three 

 components. 



The motions of the paits relative to the centre of miss have 

 » - 3 components. 



The kinetic energy of the raoleciJe may be regarded as made 

 up of two parts — that of the mass of the molecule supposed to 

 be concentrated at its centre of mass, and that of the motions of 

 the ports relative to the centre of mass. The first pait is called 

 the energy of translation, the second that of rotation and vib/a- 

 tion. The sum of these is the whole energy of motion of the 

 molecule. 



The pressure of the gas depends, as we have seen, on the 

 energy of translation alone. The specific heat depends on the 

 rate at which the whole energy, kinetic and potential, increases 

 as the temperature rises. 



Clausius had long ago pointed out that the ratio of the incre- 

 ment of the whole energy to that of the {nergy of translation 

 may be determined if w-e know by experiment the ratio of the 

 specific heat at constant pressure to that at constant volume. 



He did not, however, attempt to determine « frioti the ratio 

 of the two parts of the energy, though he suggested, as an 

 extremely probable hypothesis, that the average values of the 

 two parts of the energy in a given substar.ce always adjust them- 

 selves to the same ratio. He left the numerical value of this 

 ratio to be determined by experiment. 



In iS6o 1 investigated the ratio of the two parts of the energy 

 on the hypothesis that the molecules are elastic bodies of invari- 

 able form. I found, to my great surprise, that vphatever be 

 the shape of the molecules, provided they are not perfectly 



smooth and spherical, the ratio of the two parts of the energy 

 must be always the same, the two parts being in fact equal. 



This result is confirmed by the researches of Boltzmann, who 

 has worked out the general caseol a molecule having « variables. 



He finds that while the average energy of translation is the 

 same for molecules of all kinds at the same temperature, the 

 whole energy of motion is to the energy of translation as n to 3. 



For a rigid body 71 = 6, which makes the whole energy of 

 motion twice the energy of translation. 



But if the molecule is capable of changing its form under the 

 action of impressed forces, it must be capable of storing up 

 potential energ)', and if the forces are such as to ensure the 

 stability of the molecule, the average potential energy will in- 

 crease when the average energy of internal motion increases. 



Hence, as the temperature rises, the increments of the energy 

 of translation, the energy of internal motion, and the potential 

 energy are as 3, (// - 3), and e respectively, where f is a positive 

 quantity of unknown value depending on the law of the force 

 which binds together the constituents of the molecule. 



When the volume of the substance is maintained constant, 

 the eflect of the application of heat is to increase the whole 

 energy. We thus find for the specific heat of a gas at constant 

 volume — 



21 



'±T^{" + '■) 



where /j, and Va are the pressure and volume of unit of mass at 

 zero centigrade, or 273° absolute temperature, and J is the dyna- 

 mical equivalent of heat. The specific heat at constant pres- 

 sure is 



2j 273° 



In gases whose molecules have the same degree of complexity 

 the value of 11 is the same, and that of t may be the same. 



If this is the case, the specific heat is inversely as the specific 

 gravity, according to the law ofDulongand Petit, which is, to 

 a certain degree of approximation, verified by experiment. 



Hut if we take the actual values of the specific heat as found 

 by Regnault and compare them with this formula, we find that 

 « + (• for air and several other gases cannot be more than 4*9. 

 For carbonic acid and steam it is greater. We obtain the same 

 result if we compare the ratio of the calculated specific heats 

 2 + n + e 

 H + e 

 with the ratio as determined by experiment for various gases, 

 namely, i '408. 



And here we are brought face to face with the greatest dilTi- 

 culty which the molecular theory has yet encountered, namely, 

 the interpretation of the equation n + e = 4'9. 



If we suppose that the molecules are atoms — mere material 

 points, incapable of rotatory energy or internal motion — then k 

 is 3 and e is zero, and the ratio of the specific heats is I '66, 

 which is too gi eat for any real gas. 



But we learn from the spectroscope that a molecule can exe- 

 cute vibrations of constant period. It cannot therefore be a mere 

 material point, but a system capable of changing its form. Such 

 a system cannot have less than six variables. This would make 

 the greatest value of the ratio of the specific heats i 33, which 

 is too small for hydrogen, oxygen, nitrogen, carbonic oxide, 

 nitrous oxide, and hydrochloric acid. 



But the spectroscope tells us that some molecules can execute 

 a great many different kinds of vibrations. They must therefore 

 be systems of a very considerable degree of complexity, having 

 far more than six variables. Now, every additional variable in- 

 troduces an additional amount of capacity for internal motion 

 without affecting the external pressure. Every additional vari- 

 able, therefore, increases the specific heat, whether reckoned at 

 constant pressure or at constant volume. 



So does any capacity which the molecule may have for storing 

 up energy in the potential form. But the calculated specific heat 

 is already too great when we suppose the molecule to consistof two 

 .atcms only. Hence eveiy additional degree of complexity which 

 we attribute to the molecule can only increase the difficulty of 

 reconciUng the observed with the calculated value of the specific 

 heat. 



1 have now put before you what I consider to be the greatest 

 difilculty yet encountered by the molecular theory. Boltzmann 

 has suggested that we are to look for the explanation in the 

 mutual action between the molecules and the retherial medium 

 which surrounds them. I am afraid, however, that if we call in 



