The mercury$-$photosensitized hydrogenation of ethylene in presence of mercury vapor is first$-$order with respect to ethylene and half$-$order with respect to $H_2.$ Its rate is proportional to the square root of the intensity of the light absorbed$.$ The following mechanism has been suggested to account for these observations.

$Hg+hvH{g}^{**}$

$H{g}^{**}+H_2{k}_{1}Hg+2H$

$H+C_2{H}_4{k}_2{C}_2{H}_5$

$C_2{H}_5+H_2{k}_3{C}_2{H}_6+H$

$H+H{?}^{k_4}{H}_2$

Applying the steady$-$state approximation to $H$ and $C_2{H}_5,$ verify that the mechanism indeed supports the observations. What is the observed rate constant in terms of the rate constants of the elementary reactions? $($Denote the observed rate constant with $k_1,k_2,k_3,$ or $k_4,$ if needed$)$