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

Question

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.

Answer 1

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

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