A constant volume batch reactor is sterilized by inactivating a vitamin using heating. The reaction kinetics of the inactivation is expressed by the following differential equation,

$e^{\frac{y}{dt}+kC}=0$

where $C$ is the concentration of the vitamin at time $t.k$ is the rate constant which is expressed with Arrhenius equation as$,$

$k=k_o$exp$(-\frac{E_a}{RT}),C$

Here, $k_o$ is the frequency factor $1{10}^{10}mi{n}^{-1},E_a$ is the activation energy $90k\frac{J}{g}-$mol,$R$ is the gas constant $8\mathrm{.}314\frac{J}{K}.g-$mol, and $T$ is the absolute temperature in $K.$ The temperature changes with respect to time $t$ as$,$

$T(K)=428\mathrm{.}9-0\mathrm{.}88t+0\mathrm{.}12t^2$

The initial concentration of vitamin is $1m\frac{g}{L},$ and the reactor volume is $5L.$

Calculate the time where the concentration of the vitamin is half of its initial value using Euler method by choosing a suitable step size $h.$