To find the average rate of the reaction over the entire course of the reaction, we need to use the following formula:
[tex]\[
\text{Average Rate} = \frac{\Delta \text{Concentration}}{\Delta \text{Time}}
\][/tex]
Given the data from the table, we can identify the initial and final concentrations as well as the corresponding times:
- Initial time ([tex]\(t_0\)[/tex]): 0 seconds
- Final time ([tex]\(t_f\)[/tex]): 720 seconds
- Initial concentration ([tex]\([A]_0\)[/tex]): 1.8 M
- Final concentration ([tex]\([A]_f\)[/tex]): 0.4 M
Now, we need to calculate the change in concentration ([tex]\(\Delta \text{Concentration}\)[/tex]) and the change in time ([tex]\(\Delta \text{Time}\)[/tex]):
[tex]\[
\Delta \text{Concentration} = [A]_0 - [A]_f = 1.8 \, \text{M} - 0.4 \, \text{M} = 1.4 \, \text{M}
\][/tex]
[tex]\[
\Delta \text{Time} = t_f - t_0 = 720 \, \text{s} - 0 \, \text{s} = 720 \, \text{s}
\][/tex]
Using these values, we can calculate the average rate of the reaction:
[tex]\[
\text{Average Rate} = \frac{1.4 \, \text{M}}{720 \, \text{s}} = 0.0019444444444444444 \, \text{M/s}
\][/tex]
To express this in scientific notation:
[tex]\[
0.0019444444444444444 \, \text{M/s} = 1.9444444444444444 \times 10^{-3} \, \text{M/s}
\][/tex]
Looking at the given options, the closest value to our calculation is:
[tex]\[
1.9 \times 10^{-3} \, \text{M/s}
\][/tex]
Therefore, the average rate of the reaction over the entire course of the reaction is:
[tex]\[
1.9 \times 10^{-3} \, \text{M/s}
\][/tex]
So, the correct answer is [tex]\(1.9 \times 10^{-3}\)[/tex].
