144 HEAT. 



section of each is increased, evidently the chance of hitting one within a 

 short distance is increased. We shall show below that, as we might 

 perhaps expect, the collision frequency with a given velocity is propor- 

 tional to the cross section of the molecular systems. If the cross section 

 remains the same while the number of molecules is increased, the fre- 

 quency of collision increases in the same proportion. This again might 

 perhaps be expected. For if we put into a given space a second equal 

 number of molecules, we might expect that the projected molecule 

 would collide with the added molecules as often as with those pre- 

 viously in the space if the packing was so open that the second set 

 were not appreciably screened by the first, and thus it would have 

 double the collision frequency. These results may be obtained as 

 follows. Imagine a straight line AF drawn from a point occupied by 

 a molecule A, and passing in succession through the spheres of action 



-B C D E 



of molecules at B, C, D, &c. Then the average of the lengths AB, BO, 

 CD, &c., is the M.F.P. For it is the average distance which a molecule 

 projected in the direction AF will travel before it collides with another 

 molecule, and this average distance will be the same whether we project 

 always in one direction or whether we project in all directions in turn. 

 Now let us suppose that round the molecule at A is a sphere of action of 

 radius s and cross section Trs 2 , and let us represent the other molecules 

 at B, C, D, <fec., by points. Let Trs 2 sweep forward in the direction AF 



through I cm. If it impinges on n molecular points in this distance the 

 M.F.P. , which we shall denote by L, is equal to l/n. But the volume 

 swept out by ?rs 2 is Trs 2 Z. Let the number of molecules per c.c. be N. 

 Then if irsH is large enough the number of molecules in it will be Trs 2 ZN. 

 Equating this to n or Z/L, we have Trs 2 ZN = l/L, or L = l/irs 2 N = m/irs^p 

 where Nm p, m being the mass of one molecule and p the gas-density. 

 L, then, is inversely as the molecular cross section and inversely as the 

 density, if Trs 2 is constant, and we may probably assume that it is constant, 

 at a given temperature. 



The Mean Free Path calculated from the Coefficient of 



Viscosity Of a Gas. If a gas is moving in a given direction, but 

 faster on one side of a given plane containing that direction than on the 

 other side, then the slower-moving gas exerts a dragging action on the 

 faster-moving gas, which in turn tends to hurry on the slower-moving 

 gas. This tangential force is termed the force of viscosity. It is ex- 

 plained in the kinetic theory as due to the interchange of molecules 

 between the two portions of gas across the plane. If we think of the 

 plane as horizontal and the upper part of the gas as moving the faster, 

 then the molecules moving downwards through the plane have on the 

 average a greater momentum in the given direction than those moving 

 upwards through the plane to replace them, and therefore the upper 

 portion tends to lose momentum in the given direction and the lower 

 portion tends to gain it. In other words, there is a force on the lowei 



