To determine the order of the reaction with respect to chlorine gas ([tex]\( Cl_2 \)[/tex]), we can use the data given and apply the method of initial rates.
From the given data table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Initial [NO], M} & \text{Initial } \left[ Cl_2 \right]\text{, M} & \text{Initial Reaction Rate } [NOCl]/s \\
\hline
0.2 & 0.3 & 2.2 \times 10^{-5} \\
\hline
0.2 & 0.6 & 4.4 \times 10^{-5} \\
\hline
0.4 & 0.3 & 8.8 \times 10^{-5} \\
\hline
\end{array}
\][/tex]
We will compare the first two rows of the data where the concentration of [tex]\( \text{NO} \)[/tex] remains constant at 0.2 M. This allows us to isolate the effect of changing [tex]\( \left[ Cl_2 \right] \)[/tex] on the reaction rate.
1. Compare Trials 1 and 2:
- In Trial 1, the initial concentration of [tex]\( \left[ Cl_2 \right] \)[/tex] is 0.3 M and the initial rate is [tex]\( 2.2 \times 10^{-5} \, \text{s}^{-1} \)[/tex].
- In Trial 2, the initial concentration of [tex]\( \left[ Cl_2 \right] \)[/tex] is 0.6 M and the initial rate is [tex]\( 4.4 \times 10^{-5} \, \text{s}^{-1} \)[/tex].
2. Determine the ratio of the rates and concentrations:
- The ratio of the rates is:
[tex]\[
\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{4.4 \times 10^{-5}}{2.2 \times 10^{-5}} = 2
\][/tex]
- The ratio of the concentrations of [tex]\( \text{Cl}_2 \)[/tex] is:
[tex]\[
\frac{\left[ Cl_2 \right]_2}{\left[ Cl_2 \right]_1} = \frac{0.6}{0.3} = 2
\][/tex]
3. Calculate the reaction order (n) with respect to [tex]\( \text{Cl}_2 \)[/tex]:
[tex]\[
\frac{\text{Rate}_2}{\text{Rate}_1} = \left( \frac{\left[ Cl_2 \right]_2}{\left[ Cl_2 \right]_1} \right)^n
\][/tex]
[tex]\[
2 = 2^n
\][/tex]
By inspection, we can see that:
[tex]\[
n = 1
\][/tex]
Therefore, the order of the reaction with respect to [tex]\( \text{Cl}_2 \)[/tex] is 1.
