Question

The reaction $\mathbf{2}\mathbf{NO}\left(g\right)\mathbf{+}{\mathbf{Cl}}_{\mathbf{2}}\left(g\right)\mathbf{\to }\mathbf{2}\mathbf{NOCl}\left(g\right)$was studied at$\mathbf{-}{\mathbf{10}}^{\mathbf{o}}\mathbf{C}$. The following results were obtained, where$\mathbf{Rate}\mathbf{=}\mathbf{-}\frac{\mathbf{d}\left[{\mathrm{Cl}}_2\right]}{\mathbf{dt}}$.

${\left[\mathrm{NO}\right]}_{\mathbf{0}}\left(\mathrm{mol}/L\right)$	${\left[{\mathrm{Cl}}_2\right]}_{\mathbf{0}}\left(\mathrm{mol}/L\right)$	Initial Rate$\left({\mathrm{molL}}^{-1}{\min }^{-1}\right)$
0.10	0.10	0.18
0.10	0.20	0.36
0.20	0.20	1.45

a. What is the rate law?

b. What is the value of the rate constant?

Short answer

1. The rate law for the given reaction is: $\mathrm{Rate}=\mathrm{k}{\left[\mathrm{NO}\right]}^2\left[{\mathrm{Cl}}_2\right]$
2. The value of the rate constants for a given reaction is:$\mathrm{k}=1.8\times {10}^2{\mathrm{L}}^2/{\mathrm{mol}}^2.\min$

Step-by-step solution

Determine the rate law for the given reaction

(a)

The expression of the rate law for the given reaction is:

$\mathrm{Rate}=\mathrm{k}{\left[\mathrm{NO}\right]}^{\mathrm{x}}{\left[{\mathrm{Cl}}_2\right]}^{\mathrm{y}}$

Using the given table data to determine the values of x and y:

For y,

$\begin{array}{rcl}\frac{{\mathrm{rate}}_2}{{\mathrm{rate}}_1}& =& \frac{\mathrm{k}{\left[\mathrm{NO}\right]}^{\mathrm{x}}{\left[{\mathrm{Cl}}_2\right]}^{\mathrm{y}}}{\mathrm{k}{\left[\mathrm{NO}\right]}^{\mathrm{x}}{\left[{\mathrm{Cl}}_2\right]}^{\mathrm{y}}}\\ \frac{0.36}{0.18}& =& \frac{\mathrm{k}{\left[0.10\right]}^{\mathrm{x}}{\left[0.20\right]}^{\mathrm{y}}}{\mathrm{k}{\left[0.10\right]}^{\mathrm{x}}{\left[0.20\right]}^{\mathrm{y}}}\\ 2.0& =& {\left(\frac{0.20}{0.10}\right)}^{\mathrm{y}}\\ 2.0& =& 2.0^{\mathrm{y}}\\ \mathrm{y}& =& 1\end{array}$

For x,

$\begin{array}{rcl}\frac{{\mathrm{rate}}_2}{{\mathrm{rate}}_1}& =& \frac{\mathrm{k}{\left[\mathrm{NO}\right]}^{\mathrm{x}}{\left[{\mathrm{Cl}}_2\right]}^{\mathrm{y}}}{\mathrm{k}{\left[\mathrm{NO}\right]}^{\mathrm{x}}{\left[{\mathrm{Cl}}_2\right]}^{\mathrm{y}}}\\ \frac{1.45}{0.36}& =& \frac{\mathrm{k}{\left[0.10\right]}^{\mathrm{x}}{\left[0.20\right]}^{\mathrm{y}}}{\mathrm{k}{\left[0.10\right]}^{\mathrm{x}}{\left[0.20\right]}^{\mathrm{y}}}\\ 4.0& =& {\left(\frac{0.20}{0.10}\right)}^{\mathrm{x}}\\ 4.0& =& 2.0^{\mathrm{x}}\\ \mathrm{x}& =& 2\end{array}$

Hence the rate law for the reaction is:

$\mathrm{Rate}=\mathrm{k}{\left[\mathrm{NO}\right]}^2\left[{\mathrm{Cl}}_2\right]$

Determine the value of the rate constant for the given reaction

(b)

Using the given table data to determine the values of k(rate constant).

$\begin{array}{rcl}\mathrm{k}& =& \frac{0.18\mathrm{mol}/\left(\mathrm{L}.\min \right)}{{\left(0.10\mathrm{mol}/\mathrm{L}\right)}^2\left(0.10\mathrm{mol}/\mathrm{L}\right)}\\ \mathrm{k}& =& 1.8\times {10}^2{\mathrm{L}}^2/{\mathrm{mol}}^2.\min \end{array}$