3.2 Instantaneous Rate

An instantaneous rate of reaction is the rate of a reaction at an instant in time such that Δt becomes infinitesimal.

\[ \require{color} \mathrm{avg.~rate} = \dfrac{1}{a}\textcolor{red}{\dfrac{-\Delta [\mathrm{A}]}{\Delta t}} ~~\longrightarrow~~ \mathrm{rate} = \dfrac{1}{a}\textcolor{red}{\dfrac{-d [\mathrm{A}]}{dt}} \]

This is the slope of a line that is tangent to any one point along the reaction curve. For example, let us consider the tangent line to e^–x at x = 1.

The tangent line was given by WolframAlpha. We can solve for y when x = 1.

\[\begin{align*} y &= \dfrac{2}{e^2} - \dfrac{x}{e^2} \\[1.5ex] &= \dfrac{2}{e} - \dfrac{1}{e} \\[1.5ex] &= 0.3679 \end{align*}\]

The blue point shown below (x, y) is (1, 0.3679) and the blue line is the tangent line.

Figure 3.4: Determining the instant rate of reaction by determining the slope a tangent line on a concentration vs. time plot

The slope of the line is

\[m = -0.3679\] Therefore, the instantaneous rate of reaction at t = 1 is 0.3679 M s^–1.

Let us determine the instantaneous rate of reaction when t = 2. The tangent line was given by WolframAlpha. We can solve for y when x = 2.

\[\begin{align*} y &= \dfrac{3}{e^2} - \dfrac{x}{e^2} \\[1.5ex] &= \dfrac{3}{e^2} - \dfrac{2}{e^2} \\[1.5ex] &= 0.13534 \end{align*}\]

The blue point shown below (x, y) is (2, 0.1353) and the blue line is the tangent line.

Figure 3.5: Determining the instant rate of reaction by determining the slope a tangent line on a concentration vs. time plot

The slope of the line is

\[m = -0.1353\]

Therefore, the instantaneous rate of reaction at t = 2 is 0.1353 M s^–1. Notice how the rate of reaction is slower later in the reaction (t = 2) than it was earlier in the reaction (t = 1).