Question

In an autocatalytic chemical reaction a substance $A$ is converted into a substance $B$ in such a manner that $$ \frac{d x}{d t}=k x(a-x) $$ where $x$ is the concentration of substance $B$ at time $t, a$ is the initial concentration of substance $A$, and $k$ is a positive constant. Determine the value of $x$ at which the rate $d x / d t$ of the reaction is maximum.

          In an autocatalytic chemical reaction a substance $A$ is converted into a substance $B$ in such a manner that
$$
\frac{d x}{d t}=k x(a-x)
$$
where $x$ is the concentration of substance $B$ at time $t, a$ is the initial concentration of substance $A$, and $k$ is a positive constant. Determine the value of $x$ at which the rate $d x / d t$ of the reaction is maximum.

Added by Jackie P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/12/2022

Step 1

Step 1: Given the differential equation for the rate of the reaction: \[ \frac{dx}{dt} = kx(a - x) \] we need to find the value of $ x $ at which this rate is maximum. Show more…

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In an autocatalytic chemical reaction a substance $A$ is converted into a substance $B$ in such a manner that $$ \frac{d x}{d t}=k x(a-x) $$ where $x$ is the concentration of substance $B$ at time $t, a$ is the initial concentration of substance $A$, and $k$ is a positive constant. Determine the value of $x$ at which the rate $d x / d t$ of the reaction is maximum.