In an autocatalytic chemical reaction, the product formed is a catalyst for the raaction, If is the amount of the original
substance and is the amount of catalyst formed, the rate of chemical reaction is

Ask by Maxwell Watson. in the United States

Mar 24,2025

Question

In an autocatalytic chemical reaction, the product formed is a catalyst for the raaction, If is the amount of the original
substance and is the amount of catalyst formed, the rate of chemical reaction is

Ask by Maxwell Watson. in the United States

Mar 24,2025

UpStudy AI · Accepted Answer

We start with the differential equation
$$
\frac{dQ}{dx}=k\,x\,(Q_{0}-x).
$$

Our goal is to find $$ Q $$ as a function of $$ x $$.

1. Write the differential equation in separable form:
   $$
   dQ=k\,x\,(Q_{0}-x)dx.
   $$

2. Integrate both sides with respect to $$ x $$:
   $$
   \int dQ=\int k\,x\,(Q_{0}-x)dx.
   $$

3. The left-hand side integrates immediately to
   $$
   Q.
   $$

4. Expand the integrand on the right-hand side:
   $$
   k\,x\,(Q_{0}-x)=k\,(Q_{0}x-x^2).
   $$
   Thus,
   $$
   Q=k\int \left(Q_{0}x-x^2\right)dx.
   $$

5. Integrate term by term:
   $$
   \int Q_{0}x\,dx=Q_{0}\int x\,dx=Q_{0}\,\frac{x^2}{2},
   $$
   and
   $$
   \int x^2\,dx=\frac{x^3}{3}.
   $$
   Therefore,
   $$
   Q=k\left(\frac{Q_{0}x^2}{2}-\frac{x^3}{3}\right)+C,
   $$
   where $$ C $$ is the constant of integration.

6. If we assume that no reaction has occurred when no catalyst is formed (i.e. when $$ x=0 $$, $$ Q=0 $$), then substituting $$ x=0 $$ yields
   $$
   0=k\left(\frac{Q_{0}\cdot0}{2}-\frac{0}{3}\right)+C \quad\Longrightarrow\quad C=0.
   $$

Thus, the solution is
$$
Q=k\left(\frac{Q_{0}x^2}{2}-\frac{x^3}{3}\right).
$$
The amount of the original substance $$ Q $$ as a function of the catalyst $$ x $$ is given by: $$ Q = k \left( \frac{Q_{0}x^{2}}{2} - \frac{x^{3}}{3} \right) $$

In an autocatalytic chemical reaction, the product formed is a catalyst for the raaction, If is the amount of the original
substance and is the amount of catalyst formed, the rate of chemical reaction is

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In an autocatalytic chemical reaction, the product formed is a catalyst for the raaction, If is the amount of the original substance and is the amount of catalyst formed, the rate of chemical reaction is

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In an autocatalytic chemical reaction, the product formed is a catalyst for the raaction, If is the amount of the original
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