#### Question

Derive the relation between half life and rate constant for a first order reaction

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#### Solution

The integrated rate law for the first order reaction is given by the equation

`k= 2.303/t log_10"[A]_o/[A]_t`

Where [A]_{0} = initial concentration of the reactant at t = 0

The concentration falls to [A]_{t} at time t from the start of the reaction.

The concentration of the reactant falls to [A]_{0}/2 at time t_{1/2}.

Therefore, t = t_{1/2}

[A]_{t} = [A]_{0}/2

So, the equation can be written as

`k= 2.303/t_(1/2) log_10"[A]_o /([A]_o/2)`

`=2.303/(t_1/2)log_10 2`

`=2.303/(t_1/2)*0.301=0.693/(t_1/2)`

`t_(1/2)=0.693/k`

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#### Reference Material

Solution for question: Derive the Relation Between Half Life and Rate Constant for a First Order Reaction concept: null - Half-life of a Reaction. For the courses HSC Science (General) , HSC Science (Electronics), HSC Science (Computer Science)