Question

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 aa is the amount of substance AA and bb is the amount of substance BB at time t=0t=0, and if xx is the amount of product at time tt, then the rate of formation of xx may be given by the differential equation

dxdt=k(ax)(bx),\frac{d x}{d t}=k(a-x)(b-x),

or

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

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

Solution

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In order to determine the equation that connects the amount of product xx and the time of the chemical reaction tt, we will use the given differential equation

dxdt=k(ax)(bx),\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 AA and BB is equal, i.e. the following equality applies:

a=b.a=b.

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