Question

Chemical Kinetics Suppose a gas consists of molecules of type A. When the gas is heated a second substance B is formed by molecular collision. Let $\mathit{A}\left(t\right)$and $\mathit{B}\left(t\right)$denote, in turn, the number of molecules of types $\mathbf{A}$and $\mathbf{B}$present at time ${\mathit{t}}^{\mathbf{3}}\mathbf{}\mathbf{}\mathbf{}\mathbf{0}$. A mathematical model for the rate at which the number of molecules of type A decreases is $\frac{\mathbf{d}\mathbf{A}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathit{k}{\mathit{A}}^{\mathbf{2}}\mathbf{,}\mathit{k}\mathbf{>}\mathbf{0}$.

(a) Determine$\mathit{A}\left(t\right)$if $\mathit{A}\left(0\right)\mathbf{=}{\mathit{A}}_{\mathbf{0}}$.

(b) Determine the number of molecules of substance B present at timelocalid="1668438283398" $\mathit{t}$if it is assumed that$\mathit{A}\left(t\right)\mathbf{+}\mathbf{}\mathit{B}\left(t\right)\mathbf{=}{\mathit{A}}_{\mathbf{0}}$.

Short answer

$$\begin{gathered} \left(\mathrm{a}\right)A=\frac{A_0}{A_{0}kt+1} \\ \left(\mathrm{b}\right)B\left(t\right)=A_0-\frac{A_0}{A_{0}kt+1} \end{gathered}$$

Step-by-step solution

Definition

Mathematical modeling refers to the process of creating a mathematical representation of a real-world scenario to make a prediction or provide insight.

Find

(a)

A mathematical model for the rate at which the number of molecules of type $A$decreases is $\frac{dA}{dt}=-k{A}^2,k>0$.

Integrate $\frac{dA}{dt}=-k{A}^2$

$$\begin{gathered} \tilde{O}{A}^{-2}dA=\tilde{O}-kdt \\ -A^{-1}=-kt+c \\ A=\frac{1}{kt-c} \\ A\left(0\right)=-\frac{1}{c} \\ c=-\frac{1}{A_0} \\ \mathrm{So},A=\frac{A_0}{A_{0}kt+1} \end{gathered}$$

Find number of molecule

(b)

It is assumed that $A\left(t\right)+B\left(t\right)=A_0$.

Determine the number of molecules of $B$present at time $t$.

$$\begin{gathered} B\left(t\right)=A_0-A\left(t\right) \\ B\left(t\right)=A_0-\frac{A_0}{A_{0}kt+1} \end{gathered}$$