Distribution in Spherical Coordinates

PivxyVyyV dvx dvy dug represents the fraction of all molecules that have velocities in the range y to + dvxy Vy to Vy + dvy, and v to Vg + dvg. For many purposes we are interested in expressing this in spherical coordinates. That is, we would like to know the fraction of molecules with velocity vectors in the range c to c + dc, to + dSj and t to l + d l where c, and are the spherical coordinates of the velocity vector. The relation between these systems is illustrated in Fig. VII. 1. Algebraically we have 

From this we can obtain the distribution function P(c,0) by integrating over 6 and the function P(c) by integrating again over (f) (Sec. VI.4)  

P(c) will be of particular interest to us because it represents the fraction of molecules that have speeds in the range c to c + dc. It is no longer a Gaussian distribution but instead has the shape shown in Fig. VII.2, being zero at the origin. The function P(vz) is shown plotted for comparison. The most probable speed Cp is obtained by solving 

The median speed Cm is that speed which is exceeded by half of the molecules. It can be obtained from the integral equation 

From tables of the error function and exponentials this can be solved to give Cm = 1.088a. 

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The relative velocity between the molecules not only determines whether A and B collide, but also if the kinetic energy involved in the collision is sufficient to surmount the reaction barrier. Velocities in a mixture of particles in equilibrium are distributed according to the Maxwell-Boltzmann distribution in spherical coordinates ... 

Taking the squared absolute value of Eq. (6.12) and using the definition of the parabolic coordinates given in Eq. (6.4), we can write an expression for the electron probability distribution in spherical coordinates,1... 

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