## Related questions with answers

Many chemical reactions are the result of the interaction of two molecules that undergo a change to produce a new product. The rate of the reaction typically depends on the concentrations of the two kinds of molecules. If $a$ is the amount of substance $A$ and $b$ is the amount of substance $B$ at time $t=0$, and if $x$ is the amount of product at time $t$, then the rate of formation of $x$ may be given by the differential equation

$\frac{d x}{d t}=k(a-x)(b-x),$

or

$\frac{1}{(a-x)(b-x)} \frac{d x}{d t}=k,$

where $k$ is a constant for the reaction. Integrate both sides of this equation to obtain a relation between $x$ and $t$ if $a=b$. Assume that $x=0$ when $t=0$.

Solution

VerifiedIn order to determine the equation that connects the amount of product $x$ and the time of the chemical reaction $t$, we will use the given differential equation

$\dfrac{dx}{dt}=k\left(a-x\right)\left(b-x\right),$

and then, using the initial condition that at the beginning amount of product is zero, determine the constant that will appear in the resulting expression.

Please note that in this case, the amount of substances $A$ and $B$ is equal, i.e. the following equality applies:

$a=b.$

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