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Note that we could have assumed the second molecule to have any value of a velocity component u2 other than zero orthogonal to < previous page page_11 next page > < previous page page_12 next page > Page 12 the line of centers, but this component would have no effect on the final velocity of the first molecule as given by Eq. 46). If we consider only orthogonal collisions in which the first molecule is repeatedly struck in successive collisions by one of the other molecules with a velocity component v2 which is a constant equal to the most probable velocity, vm, and assume that the initial u1 = vm also, then N successive orthogonal collisions would give a final speed By a process of reasoning similar to that for calculating the rate of absorbtion of a molecular beam by scattering in a gas , it can be shown that the probability of N + 1 or more successive orthogonal collisions, without collisions of another type, is which is the probability that the speed of m1 is equal to or greater than W1(0 < W1 < ∞), where b is a constant to be determined.
Previous page page_13 next page > < previous page page_14 next page > Page 14 Fig. 2 Plots illustrating Maxwell-Boltzmann distribution laws. Plot fc shows distribution function for random velocity, c expressed in terms of the most probable velocity α; plot fx shows distribution function for energy, E, in terms of x = E/(kT); y corresponds to the fraction of the total number of molecules for which the random velocity (expressed in terms of α) is less than or equal to a given value c. gives the fraction of the total number of molecules which have a random velocity equal to or less than that corresponding to the value c, or to v = αc.
11) it follows that the number of molecules per cubic centimeter is given by where Pµb is the pressure in microbars and PPa is the pressure in pascal units, while where Pτ is the pressure in Torr. 1 gives values of n for a series of values of T, Pµb, PPa and Pτ (Torr). 687 × 1019 cm3 is known as the Loschmidt number. 1. 3 Molecular Collisions; Mean Free Path; MaxwellBoltzmann Distribution Laws It is evident that there must be a nonuniform distribution of velocities among all the molecules in a given volume because of the constant occurrence of collisions.