Question #44252

In the standard kinetic theory of gases, the collision frequency $Z_{11}$ is calculated assuming atoms are hard spheres with a fixed collision diameter $d = 2r_{max}$, where $r_{max}$ is the radius at the peak of the radial probability distribution function. However, the Heisenberg Uncertainty Principle dictates that the electron cloud extends diffusely, making the effective collision diameter $d_{eff}$ energy-dependent. At high temperatures, $d_{eff}$ is defined as the distance of closest approach where the average relative kinetic energy of the colliding atoms equals their repulsive potential energy.

If the repulsive potential due to the overlap of the diffuse electron clouds is modeled as $V(r) = \frac{\alpha}{r^4}$ (where $\alpha$ is a positive constant), let $Z_{eff}$ be the true effective collision frequency and $Z_{hard}$ be the classical hard-sphere collision frequency. How does the ratio $\frac{Z_{eff}}{Z_{hard}}$ scale with the absolute temperature $T$?