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Initial concentration of reactant for \[{n^{th}}\]order reaction is '\[a\]'. Which of the following relations is correct about \[{t_{1/2}}\]​ of the reaction?
(A)\[\;ln\;{t_{1/2}} = ln\left( {constant} \right) - \left( {n - 1} \right)lo{g_{e}}a\]
(B)\[\;ln\;{t_{1/2}} = ln\;n + ln\left( {constant} \right) - ln\;a\]
(C)\[{t_{1/2}}ln\;n = ln\left( {constant} \right) + ln\;{a_0}\] ​
(D)\[ln\;\;{t_{1/2}} = n\;\;ln\;\;{a_0}\]

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Last updated date: 16th Jun 2024
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Answer
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Hint: The convergences of reactants and the pace of a reaction with these fixations, we will view from the outset and second request reactions just as half-life. For a Pseudo-\[{n^{th}}\]-Order Reaction, the reaction rate consistent \[k\] is supplanted by the apparent reaction rate steady \[k'\].

Complete step by step answer:
If the reaction isn't worked out explicitly to show an estimation of \[{\nu _A}\], the worth is thought to be \[1\] and doesn't appear in these conditions.
One approach to determine the request \[n\] and to get an estimated incentive for \[k\] or \[k'\], is with the strategy for half-lives. The half-life \[{t_{1/2}}\]is characterized as the time needed for the initial fixation to be divided: \[{\left[ A \right]_{{t_{1/2}}}}{\text{ }} = {\text{ }}{\left[ A \right]_t} = 0/2\]
This shows that there is an overall relationship for all estimations of \[n\] (counting \[n{\text{ }} = {\text{ }}1\]) between the initial focus and the half-life for a reaction concentrated with various initial fixations at a similar temperature: \[\dfrac{{{t_{1/2}}}}{{{{\left[ A \right]}_t}}} = {0^{n{\text{ }} - {\text{ }}1}}{\text{ }} = {\text{ }}constant\].
On the off chance that the reaction is First-Order, the half-life won't change with fixation.
On the off chance that the request is more noteworthy than one, the half-life will diminish as the initial focus is expanded.
In the event that the request is short of what one, the half-life will increment as the initial fixation is expanded.
The request for the reaction (\[n\]) might be found by assurance of the half-lives for a reaction learned at two initial focuses. On the off chance that the subsequent focus is equivalent to multiple times the first: \[\dfrac{{{{\left( {{t_{1/2}}} \right)}_1}}}{{{{\left( {{t_{1/2}}} \right)}_2}}} = {\text{ }}{10^{n{\text{ }} - {\text{ }}1}}\].
Whenever n has been determined, \[k\] or \[k'\] can be determined from the relationship above,
\[\dfrac{{\left( {{2^{n{\text{ }} - {\text{ }}1}}{\text{ }} - {\text{ }}1} \right)}}{{{{\left[ A \right]}_t}}} = {0^{n{\text{ }} - {\text{ }}1}}{\text{ }} = {\text{ }}{\nu _A}^{\left( {n{\text{ }} - {\text{ }}1} \right)}{\text{ }}k{t_{1/2}}\].
So, the required answer is as follows…
\[{t_{1/2}} \propto \dfrac{{1}}{{{a^{n - 1}}}}\]
\[{t_{1/2}} = k\dfrac{{1}}{{{a^{n - 1}}}}\]
\[ln\;\;{t_{1/2}} = ln\;\;k - (n - 1)lo{g_e}a\]
Hence, the correct option is (A).

Note:
The best estimations of the reaction rate consistent (\[k\]) can be acquired with information taken in the centre third of the reaction (from\[{\left[ A \right]_t}{\text{ }} = {\text{ }}\left( {\dfrac{2}{3}} \right){\text{ }}{\left[ A \right]_t} = 0{\text{ }}to{\text{ }}{\left[ A \right]_t}{\text{ }} = {\text{ }}\left( {\dfrac{1}{3}} \right){\text{ }}{\left[ A \right]_t} = 0\]). Straight Least Squares relapse with \[Y{\text{ }} = {\text{ }}1/{\left[ A \right]_t}^{n{\text{ }} - {\text{ }}1}\]and \[X{\text{ }} = {\text{ }}t\] gives \[{\nu _A}^{\left( {n{\text{ }} - {\text{ }}1} \right)k}\] or \[{\nu _A}^{\left( {n{\text{ }} - {\text{ }}1} \right)k'}\]as the slant.