91Ó°ÊÓ

The rate law for a chemical reaction is an expression that provides a relationship between the rate of the reaction and the concentration of the reactants participating in it.

General rate law

\({\bf{Rate = k(A}}{{\bf{)}}^{\bf{m}}}\)

From the table we can make the following equation

\(\begin{align}4.17 \times {10^{ - 4}}mol{L^{ - 1}}{h^{ - 1}} &= k{(0.230)^m}\,\,\,\,\,\,......(1)\\9.99 \times {10^{ - 4}}mol{L^{ - 1}}{h^{ - 1}} &= k{(0.356)^m}\,\,\,\,\,\,\,......(2)\\2.44 \times {10^{ - 3}}mol{L^{ - 1}}{h^{ - 1}} &= k{(0.557)^m}\,\,\,\,\,\,\,......(3)\end{align}\)

The order of reaction refers to the power dependence of the rate on the concentration of each reactant.

\(\begin{align}\frac{{Rate(R{}_1)}}{{Rate(R{}_2)}} &= \frac{{4.17 \times {{10}^{ - 4}}\,mol{L^{ - 1}}{s^{ - 1}}}}{{9.99 \times {{10}^{ - 4}}mol{L^{ - 1}}{s^{ - 1}}}}\\ &= \frac{{k{{(.230)}^m}}}{{k{{(0.356)}^m}}}\\0.4174 &= \frac{{k{{(.230)}^m}}}{{k{{(0.356)}^m}}}\end{align}\)

Taking natural log both side

\(\begin{align}\ln \,0.4174 &= m\,(\,\ln \,0.230) - m\,(\,\ln \,0.356)\\ - 0.8737 &= - 1.4697m + 1.0328m\\m &= \frac{{ - 0.8737}}{{ - 0.4369}}\\ &= 2\end{align}\)

Hence, the reaction is of second order.

The rate constant is the proportionality constant in a rate equation.

It can be calculated as:

\(\begin{align}4.17 \times {10^{ - 4}}mol{L^{ - 1}}{s^{ - 1}} &= K{(2.30M)^2}\\k &= \frac{{4.17 \times {{10}^{ - 4}}mol{L^{ - 1}}{s^{ - 1}}}}{{{{(0.059M)}^2}}}\\ &= 7.88 \times {10^{ - 3}}{L^1}Mo{l^{ - 1}}{s^{ - 1}}\end{align}\)