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प्रश्न
Answer the following in brief.
Derive the integrated rate law for the first-order reaction.
Starting with the differential rate law equation, derive the integrated rate equation for a first order reaction.
Derive the integrated rate equation for a first order reaction.
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उत्तर
Consider the first-order reaction,
\[\ce{A -> product}\]
The differential rate law is given by
\[\ce{rate = - \frac{d[A]}{dt} = k[A]}\] ...(1)
Where [A] is the concentration of reactant at time t. Rearranging Eq. (1)
\[\ce{\frac{d[A]}{[A]} = -kdt}\] ...(2)
Let [A]0 be the initial concentration of the reactant A at time t = 0.
Suppose [A]t is the concentration of A at time = t
The equation (2) is integrated between limits [A] = [A]0 at t = 0 and [A] = [A]t at t = t
On integration,
\[\ce{ln [A]{^{[A]_t}_{[A]_0}} = -k t^t_0}\]
Substitution of limits gives
ln[A]t − ln[A]0 = −kt
or \[\ce{ln \frac{[A]_t}{[A]_0} = -kt}\] ...(3)
or \[\ce{k = \frac{1}{t} ln \frac{[A]_0}{[A]_t}}\]
Converting ln to log10, we write
\[\ce{k = \frac{2.303}{t} log_10 \frac{[A]_0}{[A]_t}}\] ....(4)
Eq. (4) gives the integrated rate law for the first-order reactions.
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