Question

The half – life of a first – order reaction is ${{900}}$ minutes at ${{820 K}}$. Estimate its half – life at ${{720 K}}$ if the energy of activation of the reaction is ${{250 kJ mo}}{{{l}}^{{{ - 1}}}}$ ?

Answer 1

Hint: First we will know about the first order reaction and its expression. Then we will relate the half – life with first order reaction and finally with energy of activation given for the reaction. A reaction is said to be first order if the reaction depends on the concentration of only one reactant.

Complete step by step answer:
The reaction is called a first order reaction when the rate of the reaction is dependent only on the concentration of one reactant. The rate does not depend on any other reactant used in the reaction. The power raised to the concentration of that reactant is one. First – order reaction is also called unimolecular reactions.
The rate law expression for first – order reaction is given below:
${{r = - }}\dfrac{{{{d[A]}}}}{{{{dt}}}}{{ = k [A]}}$
The half life for first order reaction is
${{{t}}_{{{1/2}}}}{{ = }}\dfrac{{{{0}}{{.693}}}}{{{{{k}}_{{1}}}}}$
It is independent of reactant concentration.
The relation of half – life with activation energy is
$\dfrac{{{{lo}}{{{g}}_{{{10}}}}{{ }}{{{t}}_{{{1/2}}}}}}{{{{lo}}{{{g}}_{{{10}}}}{{ t}}_{{{1/2}}}^{{,}}}}{{ = }}\dfrac{{{{{E}}_{{a}}}}}{{{{2}}{{.303 \times R}}}}{{ [}}\dfrac{{{{{T}}^{{,}}}{{ - T}}}}{{{{{T}}^{{,}}}{{T}}}}{{]}}$
So on putting the values
${{lo}}{{{g}}_{{{10}}}}\dfrac{{{{900 min}}}}{{{{t}}_{{{1/2}}}^{{,}}}}{{ = }}\dfrac{{{{250 kJ mo}}{{{l}}^{{{ - 1}}}}{{ \times 1000 J mo}}{{{l}}^{{{ - 1}}}}}}{{{{2}}{{.303 \times 8}}{{.314 J mo}}{{{l}}^{{{ - 1}}}}{{ }}{{{K}}^{{{ - 1}}}}}}{{[}}\dfrac{{{{720 K - 820 K}}}}{{{{720 K \times 820 K}}}}{{]}}$
${{lo}}{{{g}}_{{{10}}}}\dfrac{{{{900 min}}}}{{{{t}}_{{{1/2}}}^{{,}}}}{{ = 13057 \times [ - 1}}{{.694 \times 1}}{{{0}}^{{{ - 4}}}}{{]}}$
${{lo}}{{{g}}_{{{10}}}}\dfrac{{{{900 min}}}}{{{{t}}_{{{1/2}}}^{{,}}}}{{ = - 2}}{{.212}}$
$\dfrac{{{{900 min}}}}{{{{t}}_{{{1/2}}}^{{,}}}}{{ = 0}}{{.006145}}$
So ${{t}}_{{{1/2}}}^{{,}}{{ = 146473 min}}$

Additional information:
The first – order reaction is said to be unimolecular reaction as the rate depends only on one reactant and the second – order reaction is sometimes called bimolecular reaction as the rate depends on two reactants but in some cases the reactant concentration for a single reactant is raised to the power of two means reaction is not bimolecular.

Note: The activation energy is the necessary energy barrier for the conversion of reactant into product. It is called a necessary energy barrier as only after passing this level of energy the formation of product is possible otherwise no product will be formed.