Suppose two chemicals \(A\) and \(B\) are combined to form a compound \(X\text{.}\) Initially, there are \(60\) units of \(A\) and \(40\) units of \(B\text{.}\) For every \(2\) units of \(B\text{,}\) \(3\) units of \(A\) are used to form \(X\text{.}\) After \(10\) minutes, \(5\) units of \(X\) are present. Write a boundary-value problem for \(X\text{,}\) using time in minutes.
Section 1.5 Second-order chemical reactions
The Law of Mass states that when temperature is held constant, the range of change in the amount of a compound is proportional to the product of the remaining amounts of chemicals used to form it. Suppose that we start with \(a\) units of chemical \(A\) and \(b\) units of chemical \(B\text{.}\) For every \(M\) units of chemical \(A\) used, \(N\) units of chemical \(B\) are used when forming a compound \(X\text{.}\) The rate of change in the compound \(X\) formed is given by the differential equation
\begin{equation*}
\dfrac{\mathrm{d}X}{\mathrm{d}t}=k\underbrace{\left( a - \dfrac{M}{M+N} X \right)}_{\text{remaining amount of } A}\underbrace{\left( b-\dfrac{N}{N+M}X \right)}_{\text{remaining amount of } B},
\end{equation*}
where \(k\) is the constant of proportionality.
