Consider a system consisting of argon atoms adsorbed on a graphite surface. at time t=0, the fractional surface coverage of argon is r=ro, where r=1 represents complete surface coverage. the argon atoms desorb from the surface via a first-order mechanism, where the probability for a given atom to desorb in the time interval dt is given by k dt, where k is a rate constant for desorption. as suggested by this first order law, you may assume that all binding sites are equivalent and that argon atoms are non-interacting. you may also assume that desorbed atoms do not re-adsorb. (a) define r(t) to be the fraction of molecules that remain adsorbed after time t. write and expression for r(t+dt) where dt is an infinitesimal time interval (b) using the results from part (a) and the definition of a derivative as the limit of a finite difference, show that dr/dt = -kr(t), and then solve the equation using the appropriate initial condition to obtain an expression for the time dependence of . (c) using your work from part (b), write down an expression for the probability pt) that a particular atom which is on the surface at t=0 will desorb at time t. normalize this distribution over the time interval zero to infinity. (d) what is mean surface residence time of an argon atom, i. e. the average amount of time that an atom remains adsorbed? (e) what is the standard deviation of the surface residence time?