To determine the average rate of the reaction over the first 450 seconds, follow these detailed steps:
1. Identify the initial concentration and the concentration at 450 seconds:
- Initial concentration at [tex]\( t = 0 \)[/tex] seconds is [tex]\( 1.8 \, M \)[/tex].
- Concentration at [tex]\( t = 450 \)[/tex] seconds is [tex]\( 0.8 \, M \)[/tex].
2. Calculate the change in concentration ([tex]\(\Delta[\text{Reactant}]\)[/tex]) over the time period:
[tex]\[
\Delta[\text{Reactant}] = [\text{Concentration at } 450 \text{ s}] - [\text{Concentration at } 0 \text{ s}]
\][/tex]
[tex]\[
\Delta[\text{Reactant}] = 0.8 \, M - 1.8 \, M = -1.0 \, M
\][/tex]
3. Determine the time interval ([tex]\(\Delta t\)[/tex]) over which this change occurs:
[tex]\[
\Delta t = 450 \, \text{seconds} - 0 \, \text{seconds} = 450 \, \text{seconds}
\][/tex]
4. Calculate the average rate of the reaction using the formula:
[tex]\[
\text{Average rate} = \frac{\Delta[\text{Reactant}]}{\Delta t}
\][/tex]
[tex]\[
\text{Average rate} = \frac{-1.0 \, M}{450 \, \text{seconds}} = -\frac{1.0}{450} \, M/\text{second}
\approx -0.002222 \, M/\text{second}
\][/tex]
Since we often give the rate of the reaction in positive terms (despite the fact that the concentration is decreasing), we can simply express the magnitude:
[tex]\[
\text{Average rate} = 0.002222 \, M/\text{second}
\][/tex]
5. Compare our result with the given options:
[tex]\[
-0.002222 \approx -\frac{2.2}{1000} = -2.2 \times 10^{-3}
\][/tex]
So, the closest option to our calculation is:
[tex]\[
2.2 \times 10^{-3}
\][/tex]
Thus, the average rate of the reaction over the first 450 seconds is [tex]\( 2.2 \times 10^{-3} \, M/\text{second} \)[/tex].
