Question
Show that $t_{1 / 2}$ is given by eqn $17 \mathrm{B} .6$ for a reaction that is $n$ th order in A.(b) Derive an expression for the time it takes for the concentration of a substance to fall to one-third the initial value in an $n$ th-order reaction.
Step 1
Step 1: The rate of reaction is given by $r=k[A]^n$, where $k$ is the rate constant, $[A]$ is the concentration of A, and $n$ is the order of the reaction. Show more…
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Show that the ratio $t_{1 / 2} / t_{3 / 4},$ where $t_{1 / 2}$ is the half-life and $t_{34}$ is the time for the concentration of $\mathrm{A}$ to decrease to $\frac{3}{4}$ of its initial value (implying that $\left.t_{3 / 4}<t_{1 / 2}\right),$ can be written as a function of $n$ alone, and can therefore be used as a rapid assessment of the order of a reaction.
Chemical kinetics
Integrated rate laws
The following expression shows the dependence of the half-life of a reaction $\left(t_{2}\right)$ on the initial reactant concentration $[\mathrm{A}]_{0}:$ $$ t_{\frac{1}{2}} \propto \frac{1}{[\mathrm{~A}]_{0}^{n-1}} $$ where $n$ is the order of the reaction. Verify this dependence for zero-, first-, and second-order reactions.
The following expression shows the dependence of the half-life of a reaction $\left(t_{1 / 2}\right)$ on the initial reactant concentration $[\mathrm{A}]_{0}$ : $$t_{1 / 2} \propto \frac{1}{[\mathrm{A}]_{0}^{n-1}}$$ where $n$ is the order of the reaction. Verify this dependence for zeroth-, first-, and second-order reactions.
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