Tamilnadu BoardEnglish MediumSTD 12ChemistryChemical Kinetics5 Marks
Question
Derive the integrated rate law for a first order reaction?
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Answer
A reaction whose rate depends on the reactant concentration raised to the first power is called a first order reaction. First order reaction is $A \rightarrow$ product. Rate law can be expressed as, Rate $=$ $k [ A ]^1$. Where, $k$ is the first order rate constant
$\frac{-d[A]}{d t}= k [ A ]^1$
$\frac{-d[A]}{[A]}= k \cdot dt .$
Integrate the above equation (I) between the limits of time $t=0$ and time equal to $t$, while the concentration varies from initial concentration $\left[A_0\right]$ to $[A]$ at the later time.
$\int_{ A _0}^{ A } \frac{-d[ A ]}{[ A ]}=k \int_{\text { }}^t d t$
$-\ln [ A ]_{ A _0}^{ A }=k(t)_0^t$
$-\ln [ A ]-\left(-\operatorname{In}\left[ A _0\right]\right)= k ( t -0)$
$-\ln [ A ]+\operatorname{In}\left[ A _0\right]= kt$
$\ln \left(\frac{\left[ A _0\right]}{[ A ]}\right)=k t$
This equation (2) is in natural logarithm. To convert it into usual logarithm with base 10, we have to multiply the term by 2.303
$2.303 \log \left(\frac{\left[ A _0\right]}{[ A ]}\right)= kt$
$k =\frac{2.303}{ t } \log \left(\frac{\left[ A _0\right]}{[ A ]}\right)$
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