Calculus / Calculus: Early Transcendentals 9 / Chapter 9.3 / Problem 42

Calculus: Early Transcendentals | 9th Edition | ISBN: 9781337613927 | Authors: Daniel K. Clegg, Saleem Watson, James Stewart

?In an elementary chemical reaction, single molecules of two reactants and form a

Chapter 8, Problem 42

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In an elementary chemical reaction, single molecules of two reactants $A$ and $B$ form a molecule of the product $C$ :
$A+B→C$ . The law of mass action states that the rate of reaction is proportional to the product of the concentrations of $A$ and $B$ :

$\frac{d[C]}{dt}=k[\ A][B]$

(See Example 3.7.4.) Thus, if the initial concentrations are $[A]=a$ moles $/L$ and $[B]=b$ moles $/L$ and we write $x=[C]$ , then we have

$\frac{dx}{dt}=k(a-x)(b-x)$

Assuming that $a\ne{}b$ , find $x$ as a function of $t$ . Use the fact that the initial concentration of $C$ is $0.$ (b) Find $x(t)$ assuming that $a=b$ . How does this expression for $x(t)$ simplify if it is known that $[C]=\frac{1}{2}a$ after 20 seconds?

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