Question

The integrated rate equation for first order reaction, $A \to$ product, is

• A. $k=\frac{1}{t} ln \frac{\left[A\right]_{t}}{\left[A\right]_{0}}$
• B. $k=\frac{2.303}{t}+log_{10} \frac{\left[A\right]_{0}}{\left[A\right]_{t}}$
• C. $k= - \frac{1}{t}ln \frac{\left[A\right]_{t}}{\left[A\right]_{0}}$
• D. $k=2.303\,t\,log_{10} \frac{\left[A\right]_{0}}{\left[A\right]_{t}}$

Answer 1

Answer: (C)

The integrated rate equation for first order reaction,

$A \longrightarrow $ product is $k=-\frac{1}{t} \ln \frac{[A]_{t}}{[A]_{0}}$

For 1 st order reactions, $R \longrightarrow P$

Rate $=\frac{-d[R]}{d t}=k[R]$

On integrating this equation, we get

$\ln [R]=-k t+ I\dots$(i)

At $t=0, R=[R]_{0}, \ln [R]_{0}=-k \times 0+ I$

$\ln [R]_{0}=I$...(i)

Substituting the value of I in (i) and on rearrangement, we get

$\ln [R]=-k t+\ln R_{0}$

and, $k=\frac{1}{t} \ln \frac{\left[R_{0}\right]}{[R]}$