The hydrogen exchange reaction is shown as below:

H$+$H$\_2$H$\_2+$H

The barrier height is $38\mathrm{.}2$kJ$$mol$^-1$ at $300$K$.$ The activated complex structure is a symmetric linear arrangement of the three atoms with the internuclear distance of $0\mathrm{.}90\backslash$times $10^-10$m$.$ The vibrational wavenumbers for H$\_3$ are $2184$cm$^-1($symmetric stretch$)$ and $979$cm$^-1($doubly degenerate bend$).$ The H$\_2$ has only one vibrational mode and its vibrational wavenumber is $4395\mathrm{.}2$cm$^-1.$ The H$\_2$ equilibrium distance is $0\mathrm{.}7416\backslash$times $10^-10$m$.$ The electronic partition functions, q$\_$e$,$ of H and H$\_3$ are $2.$ Calculate the rate constant for the hydrogen exchange reaction using the simple transition$-$state theory. Given: q$\_$t$=(2\backslash$pi mkT$/$h$^2)^3/2$v ; q$\_$t$=8\backslash$pi $^2$IkT$/\backslash$sigma h$^2$ ; q$\_$v$=1/1-$e$^-$hv$/$kT