What is the half-life of a first-order reacti | vedclass

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(C) For a first-order reaction,the rate constant $k$ is given by the formula: $k = \frac{2.303}{t} \log \frac{[A]_0}{[A]_t}$.Given,$[A]_0 = 0.4 \ M$,$

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$\frac{x}{2} \ 	ext{hours}$

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(C) For a first-order reaction,the rate constant $k$ is given by the formula: $k = \frac{2.303}{t} \log \frac{[A]_0}{[A]_t}$.Given,$[A]_0 = 0.4 \ M$,$[A]_t = 0.1 \ M$,and $t = x \ 	ext{hours}$.Substituting these values: $k = \frac{2.303}{x} \log \frac{0.4}{0.1} = \frac{2.303}{x} \log 4 = \frac{2.303}{x} 	imes 2 \log 2$.Since $\log 2 \approx 0.3010$,$k = \frac{2.303 	imes 2 	imes 0.3010}{x} = \frac{1.386}{x}$.The half-life $t_{1/2}$ is given by: $t_{1/2} = \frac{0.693}{k}$.Substituting $k$: $t_{1/2} = \frac{0.693}{1.386 / x} = \frac{0.693 	imes x}{1.386} = \frac{x}{2} \ 	ext{hours}$.

What is the half-life of a first-order reaction if the time required to decrease the concentration of the reactant from $0.4 \ M$ to $0.1 \ M$ is $x \ \text{hours}$?

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