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. $k=\frac{2.303}{t}+lo{g}_{10}\frac{{\left[A\right]}_0}{{\left[A\right]}_t}$
• C. $k=-\frac{1}{t}ln\frac{{\left[A\right]}_t}{{\left[A\right]}_0}$
• D. $k=2.303tlo{g}_{10}\frac{{\left[A\right]}_0}{{\left[A\right]}_t}$

Answer 1

Answer: (C)

The integrated rate equation for first order reaction,

$A⟶$ product is $k=-\frac{1}{t}\ln \frac{[A{]}_t}{[A{]}_0}$

For 1 st order reactions, $R⟶P$

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

On integrating this equation, we get

$\ln [R]=-kt+I\dots$(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

$\ln [R]=-kt+\ln R_0$

and, $k=\frac{1}{t}\ln \frac{\left[R_0\right]}{[R]}$