Question

${\text{2 NO (g)}}\,{\text{ + }}\,{\text{C}}{{\text{l}}_{\text{2}}}{\text{(g)}} \to {\text{2}}\,{\text{NOCl(g)}}$
The following data were collected. All the measurements were taken at $263\,{\text{K}}$.

Experiment No.	Initial $\left[ {{\text{NO}}} \right]$ M	Initial $\left[ {{\text{C}}{{\text{l}}_2}} \right]$ M	Initial rate of disappearance of M/min.
1	0.15	0.15	0.60
2	0.15	0.30	1.20
3	0.30	0.15	2.40
4	0.25	0.25	?

(a) Write the expression for rate law.
(b) Calculate the value of rate constant and specify its units.
(c) What is the initial rate of disappearance of ${\text{C}}{{\text{l}}_{\text{2}}}$.

Answer 1

Hint: The rate law is the relationship between rate constant, rate, and concentration of reactants. The unit of rate constant can be determined by dividing the rate with concentrations of reactants.

Complete step by step answer:
(i) The relation between rate constant, rate and concentration is given by rate law.
The rate law is as follows:
\[{\text{rate = K}}{\left[ {{\text{conc}}{\text{.}}} \right]^{\text{n}}}\]
Where,
K is the rate constant.
n is the order of the reaction.
The reaction is as follows:
${\text{2 NO (g)}}\,{\text{ + }}\,{\text{C}}{{\text{l}}_{\text{2}}}{\text{(g)}} \to {\text{2}}\,{\text{NOCl(g)}}$
The rate constant for the given reaction is as follows:
\[{\text{rate = K}}{\left[ {{\text{NO}}} \right]^x}{\left[ {{\text{C}}{{\text{l}}_2}} \right]^y}\]…..\[\left( 1 \right)\]
Where,
x and y are the order of each reactant.
Write the rate law for each experiment as follows:
Experiment -1 \[{\text{rate = K}}{\left[ {{\text{0}}{\text{.15}}} \right]^x}{\left[ {0.15} \right]^y}\]
Experiment -2 \[{\text{rate = K}}{\left[ {{\text{0}}{\text{.15}}} \right]^x}{\left[ {0.30} \right]^y}\]
Experiment -3 \[{\text{rate = K}}{\left[ {{\text{0}}{\text{.30}}} \right]^x}{\left[ {0.15} \right]^y}\]
Experiment -4 \[{\text{rate = K}}{\left[ {{\text{0}}{\text{.25}}} \right]^x}{\left[ {0.25} \right]^y}\]
Compare the experiment-1 and 2 to determine the rate law as follows:

Substitute 0.60 for rate in Experiment -1 and 1.20 for rate in Experiment -2.
\[\dfrac{{0.60}}{{1.20}} = \dfrac{{{\text{K}}{{\left[ {{\text{0}}{\text{.15}}} \right]}^x}{{\left[ {0.15} \right]}^y}}}{{{\text{K}}{{\left[ {{\text{0}}{\text{.15}}} \right]}^x}{{\left[ {0.30} \right]}^y}}}\]
\[y = 1\]
Compare the experiment-1 and 3 to determine the rate law as follows:
Substitute 0.60 for rate in Experiment -1 and 2.40 for rate in Experiment -3.
\[\dfrac{{0.60}}{{2.40}} = \dfrac{{{\text{K}}{{\left[ {{\text{0}}{\text{.15}}} \right]}^x}{{\left[ {0.15} \right]}^y}}}{{{\text{K}}{{\left[ {{\text{0}}{\text{.30}}} \right]}^x}{{\left[ {0.15} \right]}^y}}}\]
\[x = 2\]
So, the order of the reaction with respect to the ${\text{NO}}$ is 2 and with respect to the ${\text{Cl}}$ is 1.
So, the rate law is as follows:
Substitute 2 for x and 1 for y in rate law\[\left( 1 \right)\].
\[{\text{rate = K}}{\left[ {{\text{NO}}} \right]^2}{\left[ {{\text{C}}{{\text{l}}_2}} \right]^1}\]
(ii) The value and unit of rate constant is determined as follows:
Rearrange the rate law for rate constant.
\[{\text{K}}\,{\text{ = }}\dfrac{{{\text{rate}}}}{{{{\left[ {{\text{NO}}} \right]}^2}{{\left[ {{\text{C}}{{\text{l}}_2}} \right]}^1}}}\]
Substitute 0.60 for rate, \[{\text{0}}{\text{.15}}\]for${\text{NO}}$ concentration and \[{\text{0}}{\text{.15}}\]for${\text{Cl}}$ concentration from Experiment -$1$.
\[{\text{K}}\,{\text{ = }}\dfrac{{0.60\,{\text{M/min}}{\text{.}}}}{{{{\left[ {0.15\,{\text{M}}} \right]}^2}{{\left[ {0.15{\text{ M}}} \right]}^1}}}\]
\[{\text{K}}\,{\text{ = 177}}{\text{.77}}\,{{\text{M}}^{ - 1}}{\text{.mi}}{{\text{n}}^{ - 1}}\]
Therefore, the value of rate constant is \[{\text{177}}{\text{.77}}\] and unit of rate constant is \[{{\text{M}}^{ - 1}}{\text{.mi}}{{\text{n}}^{ - 1}}\].
(iii) Determine the rate of disappearance of chlorine in experiment-4 is as follows:
Substitute \[{\text{177}}{\text{.77}}\] for rate constant in experiment-4.
Experiment -4 \[{\text{rate = 177}}{\text{.77}}\,{\left[ {{\text{0}}{\text{.25}}} \right]^2}{\left[ {0.25} \right]^1}\]
\[{\text{rate = }}2.77\]
Therefore, the rate of experiment-4 is \[{\text{2}}{\text{.77}}\,{\text{M/min}}\].

Note: The order of the reaction does not depend upon the change in concentration. In each experiment, the concentration of reactants changes but the order is the same. Order is an experimental value that can't be derived theoretically.