Velocity space spherical shell

Then we may ask, what fraction of the molecules have a speed between v and v + dv, independent of direction To answer this question we consider the distribution of points in momentum space. The volume of momentum space corresponding to velocities between v and v + dv is the volume of a spherical shell, of radii mv and m(v + dv). Thus it is 4w(mv)2 d(mv). We must substitute this volume for the volume dp9 dpy dpz of Eq. (2.4). Then we find that the fraction of molecules for which the magnitude of the velocity is between v and v + dv, is... 

Figure 4.8 (a) Three-dimensional velocity space, (b) Spherical shell. 

Consider a spherical shell of radius v and thickness dv as shown in Figure 9.11. This shell contains all of the points in velocity space that represent speeds between v and V + (in. To find the probability of speeds in this shell we integrate the probability shown in Eq. (9.4-2) over all values of 9 and for fixed values of v and dv ... 

Figure 9.11 A Spherical Shell of Thickness dv in Velocity Space.

The result in Eq. (9.4-3) is a product of three factors. The factor g(v) is the probability per unit volume in velocity space. The factor 47tv is the area of one surface of the spherical shell in Figure 9.11, and the factor dv is the thickness of the shell. The product 4nv dv is the volume of the shell. The product g(v) 47tv dv equals the probability per unit volume times the volume of the shell. 

Figure 9.12 shows this probability distribution of speeds for oxygen molecules at 298 K. The most probable speed, the mean speed, and the root-mean-square speed are labeled on the speed axis. Compare this figure with Figure 9.7. The most probable value of a velocity component is zero, while the most probable speed is nonzero and the probability of zero speed is zero. This difference is due to the fact that the speed probability density is equal to the area of the spherical shell in velocity space (equal to 4nxP-) times the probability density of the velocities lying in the spherical shell. Zero speed is improbable not because the velocity probability density is zero (it is at its maximum value), but because the area of the spherical shell vanishes at n = 0.

As shown in Fig. 4.1, we use a system of spherical polar coordinates in the velocity space with Cx, Cy, c -axes. The shaded spherical shell in Fig. 4.1 has a radius c and thickness dc. With spherical symmetry, the increment of volume in velocity space is ... 

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