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Chemistry · Ch 7 — Chemical Kinetics

Average and Instantaneous Rate

7.1.2

Average and Instantaneous Rate

Two different meanings of 'the rate'. Take the isomerisation of cyclopropane to propene at 780 K, followed by measuring [cyclopropane][\text{cyclopropane}] at 5-minute intervals: 2.00, 1.67, 1.40, 1.17, 0.98, 0.82, 0.69 mol L−1^{-1} at t = 0, 5, 10, ..., 30 min.

Average rate over any chosen time window is simply −Δ[cyclopropane]/Δt-\Delta[\text{cyclopropane}]/\Delta t computed between the window's endpoints. Over the FULL 30 minutes: −(0.69−2.00)30−0=4.36×10−2 mol L−1min−1-\dfrac{(0.69-2.00)}{30-0}=4.36\times10^{-2}\ \text{mol L}^{-1}\text{min}^{-1}. But narrow the window to the FIRST 10 minutes (2.00 to 1.40 M) and you get a faster 6×10−2 mol L−1min−16\times10^{-2}\ \text{mol L}^{-1}\text{min}^{-1}; narrow it instead to the LAST 10 minutes (20-30 min, 0.98 to 0.69 M) and you get a slower 2.9×10−2 mol L−1min−12.9\times10^{-2}\ \text{mol L}^{-1}\text{min}^{-1}. So the 'average rate' you quote depends entirely on WHICH window you pick, and none of these numbers tells you the true rate at any single instant. …

Table 7.1Concentration of cyclopropane at various times during its isomerisation at 780 K
Time (min)[cyclopropane] (mol L−1^{-1})
02.00
51.67
101.40
151.17
200.98
Table 7.2Rate of isomerisation of cyclopropane at three concentrations
[cyclopropane] (mol L−1^{-1})Rate (mol L−1^{-1} min−1^{-1})
26.92×10−26.92\times10^{-2}
13.46×10−23.46\times10^{-2}
Figure 7.2Concentration of cyclopropane vs time -- graph

What this figure shows. A smoothly falling curve of [cyclopropane] in M (y-axis, 0 to 2.00) against time in minutes (x-axis, 0 to 100), matching the data of Table 7.1 extended further in time. Tangent lines are conceptually drawn at different points on the curve (at [cyclopropane] = 2, 1 and 0.5 M) to read off the instantaneous rate at each concentration, tabulated in Table 7.2 -- the steeper the tangent, the faster the instantaneous rate; the curve flattens as concentration falls, showing the rate itself …

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