often presented via a rate (v) vs. substrate concentration ([S]) plot. This requires obtaining and combining both the [S] vs. t and the v vs. t plots described above (note that one may be obtained from the other by simple differentiation or integration.) The combination leads to a standard set of curves in which reaction progress is read from right to left along the x-axis and reaction rate is read from bottom to top along the y-axis. While these plots often provide a visually compelling demonstration of basic kinetic trends, differential methods are generally superior for extracting numerical rate constants. (see below) Catalytic kinetics and catalyst resting state In catalytic kinetics, two basic approximations are useful (in different circumstances) to describe the behavior of many systems. The situations in which the pre-equilibrium and steady-state approximations are valid can often be distinguished by reaction progress kinetic analysis, and the two situations are closely related to the resting state of the catalyst. The steady-state approximation Under steady-state conditions, the catalyst and substrate undergo reversible association followed by a relatively rapid consumption of the catalyst–substrate complex (by both forward reactions to product and reverse reactions to unbound catalyst.) The steady-state approximation holds that the concentration of the catalyst-substrate complex is not changing over time; the total concentration of this complex remains low as it is whisked away almost immediately after formation. A steady-state rate law contains all of the rate constants and species required to go from starting material to product, while the denominator consists of a sum of terms describing the relative rates of the forward and reverse reactions consuming the steady-state intermediate. For the simplest case where one substrate goes to one product through a single intermediate: = In a slightly more complex situation where two substrates bind in sequence followed by product release: = Increasingly complex systems can be described simply with the algorithm described in this reference. In the case of the steady-state conditions described above, the catalyst resting state is the unbound form (because the substrate-bound intermediate is, by definition, only present at a minimal concentration.) The pre-equilibrium approximation Under pre-equilibrium conditions, the catalyst and substrate undergo rapid and reversible association prior to a relatively slow step leading to product formation and release. Under these conditions, the system can be described by a "one-plus" rate law where the numerator consists of all rate constants and species required to