instantaneous rate
To obtain the tabulated results for this decomposition, the concentration of hydrogen peroxide was measured every 6 hours over the course of a day at a constant temperature of 40 C. Reaction rates were computed for each time interval by dividing the change in concentration by the corresponding time increment, as shown here for the first 6-hour period: [H 2 O 2] 0.500 mol/L 1.000 mol/L = = 0.0833 mol L 1 h 1 t 6.00 h 0.00 h Notice that the reaction rates vary with time, decreasing as the reaction proceeds. Results for the last 6-hour period yield a reaction rate of: [H 2 O 2] 0.0625 mol/L 0.125 mol/L = = 0.0104 mol L 1 h 1 t 24.00 h 18.00 h This behavior indicates the reaction continually slows with time. Using the concentrations at the beginning and end of a time period over which the reaction rate is changing results in the calculation of an average rate for the reaction over this time interval. At any specific time, the rate at which a reaction is proceeding is known as its instantaneous rate. The instantaneous rate of a reaction at time zero, when the reaction commences, is its initial rate. Consider the analogy of a car slowing down as it approaches a stop sign. The vehicles initial rateanalogous to the beginning of a chemical reactionwould be the speedometer reading at the moment the driver begins pressing the brakes t0 . A few moments later, the instantaneous rate at a specific momentcall it t1would be somewhat slower, as indicated by the speedometer reading at that point in time. As time passes, the instantaneous rate will continue to fall until it reaches zero, when the car or reaction stops. Unlike instantaneous speed, the cars average speed is not indicated by the speedometer; but it can be calculated as the ratio of the distance traveled to the time required to bring the vehicle to a complete stop t . Like the decelerating car, the average rate of a chemical reaction will fall somewhere between its initial and final rates. The instantaneous rate of a reaction may be determined one of two ways. If experimental conditions permit the measurement of concentration changes over very short time intervals, then average rates computed as described earlier provide reasonably good approximations of instantaneous rates. Alternatively, a graphical procedure may be used that, in effect, yields the results that would be obtained if short time interval measurements were possible. If we plot the concentration of hydrogen peroxide against time, the instantaneous rate of decomposition of H2O2 at any time t is given by the slope of a straight line that is tangent to the curve at that time Figure 12.3 . We can use calculus to evaluating the slopes of such tangent lines, but the procedure for doing so is beyond the scope of this chapter.