Lecture 29 : Temperature Dependence of Reaction Rates
Equations (29.21) and (29.17), now become
(29.28)
(29.29)
This is the Maxwell - Boltzmann distribution of molecular speeds. F (x, y, z) dx, dy, dz gives the probability of finding an arbitrary molecule with a velocity (x, y, z) in the corresponding volume element. A more appealing interpretation of the same is that it is the fraction (of the total molecules) of molecules having velocities ( x, y, z).
Analogous to the radial probability distribution, we can now estimate the probability of finding a particle in a spherical shell of volume 42 d. This probability in such a spherical shell is given by
(29.30)
The plot of this function at two temperature is given in Fig 29.2.
Fig 29.2 The Maxwell - Boltzmann velocity distribution at low and high temperatures.
The ordinate gives the probability of finding molecules with velocity in a spherical shell of radius in a unit interval d.
Having described the velocity distribution, we can now interpret Arrhenius's equation for temperature dependence of reaction rates. Only a fraction of collisions are with molecular velocities that correspond to sufficient energy Ea (activation energy). When the temperature is raised, the fraction of molecules with high velocities (or large energies) is raised, and so we get a larger value of the rate constant.
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