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Q. Half life of first orders reaction and half life of $2^{\text {nd }}$ order reaction are equal. Hence the ratio of their rates at the start of the reaction is/are-

Chemical Kinetics

Solution:

Half life of first order rxn , $t _{1 / 2}=\frac{0.693}{ k }$

Half life of $2^{ nd }$ order reaction, $t _{1 / 2}=\frac{1}{( A )_{0} k }$

Since, half lives are equal, $\therefore =[ A ]_{0}=\frac{1}{0.693}$

Rate of first order reaction $= k \frac{1}{(0.693)^{1}} ...(i)$

Rate of 2 nd order reaction $= k \frac{1}{(0.693)^{2}}....(2)$

-Ratio of rates $=\frac{\frac{1}{0.693}}{\left(\frac{1}{0.693}\right)^{2}}$

$=\frac{1}{0.693} \times 0.693 \times 0.693=0.693$