Chemical Thinking / 4.1← Back to lesson

From a ratio to a rate.

Shrink the interval and watch two quantities vanish while their ratio does not. The secant pivots into the tangent, and the quotient settles on the derivative.

First-order kinetics

How fast is A disappearing at one moment?

[A](t) = 0.80 e−0.139t
Secant line approaching a tangentThe upper graph shows a chemical function with a secant line through two points whose spacing h shrinks; the lower graph shows the secant slope as a function of h approaching the derivative.
functionsecant (average rate)tangent / limit (derivative)
0.0 min
A long interval: the secant misses the tangent.

Model note: the decay curve is the lesson's first-order example, [A](t) = 0.80 e−0.139t (M, minutes). The reactor curve is the batch-reactor example n(t) = 3.0t − 0.25t² (mol, hours). Both are teaching models.