Derive the expression for integrated rate law (equation) for the … - Vidyadip

Question

Derive the expression for integrated rate law (equation) for the first-order reaction.

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Answer

Consider the following first-order reaction, $A \rightarrow B$ The rate of the chemical reaction is given by the rate law expression as, Rate, $R=k[A]$ where $[A]$ is the concentration of the reactant $A$ and $k$ is the velocity constant or specific rate of the reaction.
The instantaneous rate is given by,
$\begin{aligned}
& R=k[ A ]=\frac{-d[ A ]}{d t} \\
& \therefore \frac{-d[ A ]}{[ A ]}=k \cdot d t
\end{aligned}$

If $\left[A_0\right]$ is the initial concentration of the reactant and $[A]_t$ at time $t$, then by integrating the above equation,
$\begin{aligned}
& -\int_{\left[ A _0\right]}^{[ A ]_t} \frac{d[ A ]}{[ A ]}=k \int_0^t d t \\
& \therefore-\log _{ e } \frac{[ A ]_t}{\left[ A _0\right]}=k t \\
& \therefore \log _{ e } \frac{\left[ A _0\right]}{[ A ]_t}=k t \\
& \therefore k=\frac{1}{t} \log _{ e } \frac{\left[ A _0\right]}{[ A ]_t} \quad \\
& \therefore k=\frac{2.303}{t} \log _{10} \frac{\left[ A _0\right]}{[ A ]_t}
\end{aligned}$

This is the integrated rate equation for the first order reaction. This is also called integrated rate law.

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