Mathematical representation for t_{1/4} life | vedclass

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(C) For a first-order reaction,the rate constant $k$ is given by $k = \frac{2.303}{t} \log \left( \frac{[A]_0}{[A]_t} \right)$.At $t_{1/4}$,the amount

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$t_{1/4} = \frac{2.303}{k} \log \left( \frac{4}{3} \right)$

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(C) For a first-order reaction,the rate constant $k$ is given by $k = \frac{2.303}{t} \log \left( \frac{[A]_0}{[A]_t} \right)$.At $t_{1/4}$,the amount of reactant consumed is $\frac{1}{4}$ of the initial concentration $[A]_0$.Therefore,the remaining concentration $[A]_t = [A]_0 - \frac{1}{4}[A]_0 = \frac{3}{4}[A]_0$.Substituting these values into the rate equation:$k = \frac{2.303}{t_{1/4}} \log \left( \frac{[A]_0}{\frac{3}{4}[A]_0} \right)$$k = \frac{2.303}{t_{1/4}} \log \left( \frac{4}{3} \right)$Rearranging for $t_{1/4}$ gives $t_{1/4} = \frac{2.303}{k} \log \left( \frac{4}{3} \right)$.

Mathematical representation for $t_{1/4}$ life for a first-order reaction is given by:

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