# How do you find the average rate and instantaneous rate given a kinetics plot?

What is average rate? What is instantaneous rate? How do you find these? My teacher described finding the tangent to a graph. I am not sure what is meant by this.

This is more of a question of calculus than chemistry, but in a first order reaction $\ce{A -> B}$, the average rate is:

$$\frac{\Delta A}{\Delta t} = \frac{A_2 - A_1}{t_2 - t_1}$$

Graciously borrowed from here

So, the average rate is the slope of a secant line between the two points,

$$\frac {\Delta A}{\Delta t} = \frac{1 - 0.5}{400 - 200}$$

The instantaneous rate is the derivative of the plot:

$$\frac{dA(t)}{dt}$$ or $$\lim_{t\rightarrow 0} \frac{\Delta A}{\Delta t}$$

For the above first order rate law, the equation is in the form:

$$A(t) = e^{-kt + ln(A_0)}$$ so the derivative is simply

$$\frac{dA(t)}{dt} = -ke^{-kt + ln(A_0)}$$

• Doubtless I've mangled something, so please correct my math! – jonsca Mar 13 '14 at 4:13