For a reaction A → 2 B + C the half lives are 100   s and 50   s when the concentration of…

The relation between half-life and initial concentration of ${\mathrm{n}}^{\mathrm{th}}$ order reaction can be written as follows,

${\mathrm{t}}_{\frac{1}{2}}\propto \frac{1}{{\left[{\mathrm{A}}_0\right]}^{\mathrm{n}-1}}$

$\left[100\right]\propto \frac{1}{{\left(0.5\right)}^{\mathrm{n}-1}}$

$\left(50\right)\propto \frac{1}{{\left(1\right)}^{\mathrm{n}-1}}$

${\left[2\right]}^1={\left[\frac{1}{0.5}\right]}^{\mathrm{n}-1}$

${\left[2\right]}^1={\left[2\right]}^{\mathrm{n}-1}$

$\mathrm{n}-1=1$

$\mathrm{n}=2$

order $=2$

For the second order reactions half-life is inversely proportional to initial concentration.