A study of the rate of the reaction represented as 2A⟶ B gave the following data:

1. Determine the average rate of disappearance of A between 0.0 s and 10.0 s, and between 10.0 s and 20.0 s.
2. Estimate the instantaneous rate of disappearance of A at 15.0 s from a graph of time versus (A). What are the units of this rate?
3. Use the rates found in parts (a) and (b) to determine the average rate of formation of B between 0.00 s and 10.0 s, and the instantaneous rate of formation of B at 15.0 s.

\(\frac{{{\bf{ - \Delta A}}}}{{{\bf{2\Delta t}}}}{\bf{ = }}\frac{{{\bf{\Delta B}}}}{{{\bf{\Delta t}}}}\)

Average rate of disappearance from 0s to 10s

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{ - \Delta A}}}}{{{\bf{\Delta t}}}}\\{\bf{ = }}\frac{{{\bf{0}}{\bf{.370 - 0}}{\bf{.625}}}}{{{\bf{20 - 10}}}}\\{\bf{ = 0}}{\bf{.0255M/s}}\end{aligned}\)

Average rate of disappearance from 10s to 20s

\(\begin{aligned}{}{\bf{ = }}\frac{{{\bf{ - \Delta A}}}}{{{\bf{\Delta t}}}}\\{\bf{ = }}\frac{{{\bf{0}}{\bf{.370 - 0}}{\bf{.625}}}}{{{\bf{20 - 10}}}}\\{\bf{ = 0}}{\bf{.0255M/s}}\end{aligned}\)

Linear from of each order to plot point A y-axis, ln A y-axis and 1/A y-axis

First order:

\(\begin{aligned}{}\left( {\bf{A}} \right){\bf{ = - kt + (}}{{\bf{A}}_{\bf{0}}}{\bf{)}}\\{\bf{In}}\,\,\left( {\bf{A}} \right){\bf{ = - kt + In(}}{{\bf{A}}_{\bf{0}}}{\bf{)}}\\\frac{{\bf{1}}}{{{\bf{(A)}}}}{\bf{ = kt + }}\frac{{\bf{1}}}{{{\bf{(}}{{\bf{A}}_{\bf{0}}}{\bf{)}}}}\end{aligned}\)

Here notice that second order is most linear

Rate = \(\frac{{{\bf{ - d(A)}}}}{{{\bf{2dt}}}}{\bf{ = k(A}}{{\bf{)}}^{\bf{2}}}\,\,\,\,\,\,\,\,\,\, - - - - {\bf{(1)}}\)

we find k is 0.116 from slope of graph and at 15s (A)=0.465 to put all value in equation (1)

rate of disappearance of A is 0.05M/s

\(\frac{{{\bf{ - \Delta A}}}}{{{\bf{2\Delta t}}}}{\bf{ = }}\frac{{{\bf{\Delta B}}}}{{{\bf{\Delta t}}}}\,\,\,\,\,...{\rm{ }}\left( {\bf{2}} \right)\)

We divided the rate in part (a) and (b) by 2 to get 0.0188M/s and 0.25M/s respectively.

The average rate of formation of B between 0.00 s and 10.0 s is get 0.0188M/s and the instantaneous rate of formation of B at 15.0 s is 0.025.