What is the half-life time of a reaction? Derive the equation of half-life time $(t_{1/2})$ for a zero-order reaction.
(N/A) Half-life time: The half-life of a reaction is the time in which the concentration of a reactant is reduced to one-half of its initial concentration. It is represented as $(t_{1/2})$.
Derivation for zero-order reaction:
For a zero-order reaction,the integrated rate equation is given by:
$k = \frac{[R]_{0} - [R]}{t}$
At half-life:
$t = t_{1/2}$
$[R] = \frac{1}{2} [R]_{0}$
Substituting these values into the rate equation:
$k = \frac{[R]_{0} - \frac{1}{2} [R]_{0}}{t_{1/2}}$
$k = \frac{\frac{1}{2} [R]_{0}}{t_{1/2}}$
Rearranging for $t_{1/2}$:
$t_{1/2} = \frac{[R]_{0}}{2k}$
Thus,for a zero-order reaction,the half-life is directly proportional to the initial concentration of the reactant.
Derivation for zero-order reaction:
For a zero-order reaction,the integrated rate equation is given by:
$k = \frac{[R]_{0} - [R]}{t}$
At half-life:
$t = t_{1/2}$
$[R] = \frac{1}{2} [R]_{0}$
Substituting these values into the rate equation:
$k = \frac{[R]_{0} - \frac{1}{2} [R]_{0}}{t_{1/2}}$
$k = \frac{\frac{1}{2} [R]_{0}}{t_{1/2}}$
Rearranging for $t_{1/2}$:
$t_{1/2} = \frac{[R]_{0}}{2k}$
Thus,for a zero-order reaction,the half-life is directly proportional to the initial concentration of the reactant.
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