Question #44701

The activation energy ($E_a$) of a second-order gas-phase dimerization \ce{2X(g) -> X2(g)} is exactly equal to the energy of one mole of photons emitted during the electronic transition from $n=3$ to $n=2$ in a \ce{He+} ion. A student attempts to calculate the rate constant $k$ at temperature $T$ and pressure $P$. The student incorrectly assumes that the Arrhenius pre-exponential factor $A$ is simply the frequency ($\nu$) of the emitted photon, which yields a rate constant with incorrect units of $\text{s}^{-1}$. According to a modified collision theory model for this specific reaction, the true pre-exponential factor $A_{true}$ is obtained by multiplying the photon frequency $\nu$ by the molar volume of an ideal gas at the given $T$ and $P$. Assuming ideal gas behavior, which of the following represents the correct rate constant $k_{true}$ for the reaction? (Given: $R_H$ = Rydberg constant, $h$ = Planck's constant, $c$ = speed of light, $N_A$ = Avogadro's number, $R$ = universal gas constant)