Question

the first order reaction is found to have a rate constant k=2 min / sec . find the half life period of the reaction ?
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Answer 1

For first order reaction, we know that :-

$\boxed{\boxed{\rm \large k = \dfrac{ ln(2) }{t_ \frac{1}{2} } } } \\ \\ \sf where : \rm \large k = rate \: constant \\ \large\rm t_ \frac{1}{2} = half \: life$

It is given that :-

Rate constant k =2

Now , we have to find out half life period of the reaction.

Let half life period be x.

According to the question :-

$\rm \longrightarrow 2 = \dfrac{ ln(2) }{x } \\ \\ \\ \rm \longrightarrow 2 x= { ln(2) } \\ \\ \\ \\ \rm \longrightarrow \: x= \dfrac{ ln(2) }{2 }\\ \\ \\ \\ \rm \longrightarrow \: x= \dfrac{ 0.693}{2 }\\ \\ \\ \\ \rm \longrightarrow \: x= \dfrac{ 693}{2000}\\ \\ \\ \\ \rm \longrightarrow \: x=0.3465$

Answer :

The half life period of the reaction = 0.3465 seconds

Additional Information :

For first order reaction :-

$\boxed{ \rm k \times t = log( \dfrac{a}{b} ) } \\ \\ \rm where : a = initial \: concentration \\ \\ \rm b= concentration \: after \: time \: t$

Answer 2

${\large{\bold{\sf{\underline{Understanding \; the \; question}}}}}$

${\bullet}$ This question says that the first order reaction is found to have a rate constant k=2 min / sec . We have to find the half life period of that reaction ?

${\large{\bold{\sf{\underline{Given \; that}}}}}$

${\bullet}$ The first order reaction is found to have a rate constant k = 2 min/sec.

${\large{\bold{\sf{\underline{To \; find}}}}}$

${\bullet}$ The half life period of that reaction.

${\large{\bold{\sf{\underline{Solution}}}}}$

${\bullet}$ The half life period of that reaction = ${\bold{\sf{\dfrac{In(2)}{2}}}}$

${\large{\bold{\sf{\underline{Full \; Solution}}}}}$

~ As we already know that in the first order reaction

$\; \; \;{\large{\sf{\bold{k = \dfrac{IN(2)}{t_{ \dfrac{1}{2}}}}}}}$

Where,

➙ k means constant rate

➙ ${\bold{\sf{t _{\dfrac{1}{2}}}}}$ means

Here,

➙ k is 2 min/sec

➙ Let half period as a

~ Putting the values

➨ 2 = ${\bold{\sf{\dfrac{In(2)}{a}}}}$

• Let's cross multiply

➨ 2a = 1n (2)

• ( ÷ = × ) ; ( × = ÷ )

➨ a = ${\bold{\sf{\dfrac{In(2)}{2}}}}$

• Henceforth, ${\bold{\sf{\dfrac{In(2)}{2}}}}$ is the half life period of that reaction