To determine the average rate of the reaction over the first 580 seconds, we need to follow these steps:
1. Identify the initial concentration and the concentration at 580 seconds from the table.
- At time [tex]\( t = 0 \)[/tex] seconds, the concentration is [tex]\( 1.8 \)[/tex] M.
- At time [tex]\( t = 580 \)[/tex] seconds, the concentration is [tex]\( 0.6 \)[/tex] M.
2. Calculate the change in concentration ([tex]\( \Delta [A] \)[/tex]):
[tex]\[
\Delta [A] = \text{Initial concentration} - \text{Concentration at 580 s} = 1.8 \, \text{M} - 0.6 \, \text{M} = 1.2 \, \text{M}
\][/tex]
3. Determine the time interval ([tex]\( \Delta t \)[/tex]):
[tex]\[
\Delta t = 580 \, \text{seconds} - 0 \, \text{seconds} = 580 \, \text{seconds}
\][/tex]
4. Calculate the average rate of the reaction using the formula for the rate of reaction:
[tex]\[
\text{Average rate} = \frac{\Delta [A]}{\Delta t}
\][/tex]
Substituting the values we have:
[tex]\[
\text{Average rate} = \frac{1.2 \, \text{M}}{580 \, \text{seconds}} = 0.00206896551724138 \, \text{M/s}
\][/tex]
5. Convert this value into scientific notation to match the given options:
[tex]\[
0.00206896551724138 \, \text{M/s} \approx 2.069 \times 10^{-3} \, \text{M/s}
\][/tex]
From the provided answer choices:
- [tex]\( 1.6 \times 10^{-3} \)[/tex]
- [tex]\( 1.9 \times 10^{-3} \)[/tex]
- [tex]\( 2.0 \times 10^{-3} \)[/tex]
- [tex]\( 2.2 \times 10^{-3} \)[/tex]
The closest value to [tex]\( 2.069 \times 10^{-3} \)[/tex] M/s is [tex]\( 2.0 \times 10^{-3} \)[/tex] M/s. Therefore, the average rate of the reaction over the first 580 seconds is:
[tex]\[ \boxed{2.0 \times 10^{-3}} \][/tex]
