Purpose: Estimating the behavior of a system using differential equations with specified initial conditions.

Questions:

In biological systems, enzymes play a crucial role in catalyzing reactions that transform one compound or substrate into another. For instance, in winemaking, the presence of alcohol $($ethanol$)$ is essential. However, undesirable enzymatic reactions, such as the conversion of ethanol into acetic acid by certain microorganisms, can lead to the souring of wine. This conversion process is controlled and facilitated by specific enzymes.

The enzyme kinetics in this scenario are modeled using the Michaelis$-$Menten kinetics model $($See Ref. $[1]),$ which provides predictions for the concentration of the substrate $($S$)$ over time. In our case, the substrate $S$ represents ethanol. The model assumes that the concentration of ethanol in wine over time follows the differential equation:

$\frac{dS}{dt}=\frac{-2S}{0\mathrm{.}5+S}$

Where time $t$ is measured in minutes, and the initial concentration of ethanol at $0$ minute is $1$ mole per liter, i$.$e$.,S(0)=1mo\frac{l}{L}.$

$($a$)$ Using the Euler method, manually calculate the ethanol concentration at ${0\mathrm{.}5}_{?}$ and $2$ minutes, considering calculations at $0\mathrm{.}5-$minute time intervals. Calculations can be performed using a calculator. Complete the following table.

$\backslash$table$[[t[$minutes$],f(t,S),\backslash$table$[[$Ethanol Concentration$],[S[mo\frac{l}{L}]$