I looked up the original paper, and I think that there may be some pedagogical issues with respect to the steady-state approximation that might be worth exploring. For the reaction A
B
C, Sidney Miller (J. Chem Ed., p. 490) sets both the time derivative of B and [B ] itself to approximately zero in his definition of the steady-state for the reaction above.* However, suppose that we are looking at the steady state of a simple enzyme reaction mechanism, E + S
E•S
E + P. Typically, E and E•S are orders of magnitude lower in concentration than S, but not necessarily lower than the concentration of P. In other words, I wonder whether Miller's definition of the steady-state is too restrictive.
Miller wrote, "The pedagogic message is that k
ss may vary depending on how certain key concentrations are treated. That is, after one sets the conditions of concentration and time dependence under which the steady state applies to a mechanism ( 1 , 7 , 9 ) one may obtain apparently different rate laws." This may be pertinent to the homework problem given to the OP.
*As an aside, I recall an old textbook treatment (Atkins, Physical Chemistry, p. 870) of the steady-state approximation applied to a simpler system, A
B
C . In Atkins' treatment, the concentration of B remains low and does not change much with time (for most of the course of the reaction) when the second rate constant when the rate constant for the second step is much larger that the rate constant for the first step, and he defines this as the steady state.
(mod edit by sjb to fix format)