Question

The half-life period and the initial concentration for a reaction are as follows. What is the order of reaction?

Answer 1

Hint: Half-life is the time required to consume the half amount of a substance in a reaction. The order of a reaction is an experimental value. This is basically the total number of atoms or molecules that take part in a reaction at the rate determining step. For elementary reaction the order is equal to the molecularity of the reaction.

Complete step by step answer:
Now to calculate the order of a reaction from the half-life, there is a general equation of relation between the half-life and the initial concentration of the reactant. Which is shown below.
\[{{\text{t}}_{\dfrac{{\text{1}}}{{\text{2}}}}}\infty {\left( {\text{a}} \right)^{\left( {{\text{1 - n}}} \right)}}\], where a is the initial concentration of reactant and n is the order of the reaction.
Now from the given values,
For initial concentration 350 the value of half-life is 425,
\[{\text{425}}\infty {\left( {350} \right)^{\left( {1 - n} \right)}}\]
 For initial concentration 540 the value of half-life is 275,
\[{\text{275}}\infty {\left( {540} \right)^{\left( {1 - n} \right)}}\]
Now take a ration of both reaction as follows,
\[\dfrac{{{\text{425}}}}{{275}} = {\left( {\dfrac{{350}}{{540}}} \right)^{\left( {1 - n} \right)}}\]
Now calculate this equation and find out the value of n as follows,
\[
  \dfrac{{{\text{425}}}}{{{\text{275}}}}{\text{ = }}{\left( {\dfrac{{{\text{350}}}}{{{\text{540}}}}} \right)^{\left( {{\text{1 - n}}} \right)}} \\
  {\text{log}}\left( {\dfrac{{{\text{425}}}}{{{\text{275}}}}} \right){\text{ = }}\left( {{\text{1 - n}}} \right){\text{log}}\left( {\dfrac{{{\text{350}}}}{{{\text{540}}}}} \right) \\
  {\text{log1}}{\text{.54 = }}\left( {{\text{1 - n}}} \right){\text{log0}}{\text{.6481}} \\
  {\text{0}}{\text{.189 = - }}\left( {{\text{1 - n}}} \right){\text{0}}{\text{.188}} \\
  \dfrac{{{\text{0}}{\text{.189}}}}{{{\text{0}}{\text{.188}}}}{\text{ = n - 1}} \\
  {\text{1 + 1 = n}} \\
  {\text{n = 2}} \\
  \]

Therefore, the order of the reaction is 2.

Note:
The definition of rate of a reaction is the speed of a reaction by which the concentration of reactants decreases and the concentrations of products increases per unit time. The reaction A \[ \to \] products. is a first order reaction. The rate equation of first order reaction is \[{\text{r = k}}\left[ {\text{A}} \right]\]. Where, rate is r, rate constant is k and \[\left[ {\text{A}} \right]\] is concentration of reactant A at a time t. The unit of rate depends upon the concentration of reactant and rate constant.