Answer
(a) $\frac{d[C]}{dt} = \frac{a^2k}{(akt+1)^2}$
(b) $\frac{dx}{dt}=k(a-x)^2$
(c) As $t \to \infty$ the concentration of C approaches $a$
(d) As $t \to \infty$ the rate of reaction approaches $0$
(e) As the reaction continues, all of reactant A and all of reactant B combine to form the product C.
Since all of reactant A and all of reactant B have been changed into product C, there are no reactants left to cause a reaction so the reaction is complete.
Work Step by Step
(a) $[C] = \frac{a^2kt}{akt+1}$
We can find the rate of reaction at time $t$:
$\frac{d[C]}{dt} = \frac{a^2k(akt+1)-(a^2kt)(ak)}{(akt+1)^2}$
$\frac{d[C]}{dt} = \frac{a^2k}{(akt+1)^2}$
(b) $k(a-x)^2$
$= k(a-\frac{a^2kt}{akt+1})^2$
$= k(\frac{(a)(akt+1)}{akt+1}-\frac{a^2kt}{akt+1})^2$
$= k(\frac{a}{akt+1})^2$
$= \frac{a^2k}{(akt+1)^2}$
$= \frac{d[C]}{dt}$
$= \frac{dx}{dt}$
(c) $\lim\limits_{t \to \infty}[C] = \lim\limits_{t \to \infty}\frac{a^2kt}{akt+1}$
$=\lim\limits_{t \to \infty} (\frac{a^2kt/t}{akt/t+1/t})$
$=\lim\limits_{t \to \infty} (\frac{a^2k}{ak+1/t})$
$=\frac{a^2k}{ak+0}$
$= a$
(d) $\lim\limits_{t \to \infty} \frac{a^2k}{(akt+1)^2} = 0$
(e) As the reaction continues, all of reactant A and all of reactant B combine to form the product C.
Since all of reactant A and all of reactant B have been changed into product C, there are no reactants left to cause a reaction so the reaction is complete.
