THE KINETIC THEORY OF MATTER. 



153 



will be simply an interchange of velocities, and the second molecule will 

 take its place on the journey across the cube, and we may now fix atten- 

 tion on the second molecule. If this comes into direct collision with, and 

 interchanges velocities with a third molecule, this third molecule takes the 

 place of the second, and so on. But at each collision the substituted 

 molecule starts with its centre a distance s farther on, where s is the radius 

 of molecular action or the diameter of a single molecular system. If, then, 

 there are v collisions per second of this kind, the total distance covered 

 will be, not Y but, V + vs, and the molecule, or its representative, will 

 return to the original starting-point oftener than if the molecules were 

 mere points in the ratio V + vs : V, and on this account the pressure will 

 be greater than that originally calculated in the same ratio. But this is 

 on the supposition that all the collisions are direct, whereas they must 

 be regarded as of all degrees of 

 obliquity. 



To find the average increase 

 of path, let us suppose one of the 

 pair of molecules in collision to 

 have its centre at O, Fig. 83. 

 Let ACB be a section of the 

 hemisphere of molecular action 

 with radius 00 = s. If the second 

 molecule comes along LO, in direct 

 collision, the path omitted is 

 CO = s ; but if the molecule comes 

 along MP, and rebounds along 

 PQ, producing MP to N, the path 

 omitted is PN. For, if the radius s were indefinitely small, the .second 

 molecule with the same obliquity of collision would move off along NE, 

 parallel to PQ. 



Now the number of molecules making collision of a given type over 

 a given area of the hemisphere will be proportional to the projection of 

 the area on the diametral plane through AOB, for we may suppose the 

 numbers moving up towards that plane evenly distributed over it. If 

 the number moving up in given time to unit area is n, the number moving 

 towards a given ring of radius ON = r and breadth dr is n x Zwdr, and 

 these each add path PN, or total path 



n x lirrdr x PN. 



Then all the molecules coming up in unit time add path riSlirrdr x PN, 



2 



where we sum up for the whole hemisphere. But ^irrdr x PN = ^m?, the 



o 



volume of the hemisphere, and the total path added by all the n x vrs 2 

 molecules coming to the hemisphere is 



27TS 3 



The average path added is, therefore, 



27TS 3 



W _^W = |< 



and we must use this instead of s. 



