Question

A first order reaction is $10\% $ complete in $20\min $. The time taken for $19\% $ completion is:
A. $30\min $
B. $40\min $
C. $50\min $
D. $38\min $

Answer 1

Hint: A first order reaction is a type of chemical reaction in which the rate of reaction is directly proportional to the concentration of the reactants. The sum of the powers of the concentration of reactants i.e. order of the reaction of the first order is equal to one.

Complete step by step answer:
As per the question, time taken ($t$ ) by $10\% $ of the reactant to get converted into the product = $20\min $.
Let the dissociated moles of the reactants be $ = x$
According to the mathematical representation of the first order reaction, we have:
$kt = 2.303\log \dfrac{a}{{a - x}}$
Where, $k = $ rate constant
$a = $ Initial concentration of the reactants
$a - x = $ Concentration of the reactants after time $t$= Final concentration of the reactants
Let the initial concentration of the reactants be $100$ .
Substituting these values in the above given equation, we have:
$ 20 = \dfrac{{2.303}}{k}\log \left( {\dfrac{{100}}{{100 - 90}}} \right)$
$\implies k = \dfrac{{2.303}}{{20}}\log \left( {\dfrac{{100}}{{90}}} \right) = 0.1151 \times 0.0453$
Thus, the rate constant is equal to = $k = 0.00521{\min ^{ - 1}}$
Now, it is given that the reaction now gets completed to an extent when $19\% $ of the reactants get converted into corresponding products.
Let the time taken by the reaction for the conversion of $19\% $ of the reactants into product be = $T$
Again substituting the values of $k$ and $x$ in the equation, we have:
$T = \dfrac{{2.303}}{{0.00521}}\log \left( {\dfrac{{100}}{{100 - 19}}} \right)$
Thus, the time required for the completion of $19\% $ of the reaction = $T = 440.03 \times 0.0913 \approx 40\min $.
The correct option is: B. $40\min $.

So, the correct answer is “Option B”.

Note:
In the above reaction, we can clearly notice that it takes almost double the amount of time for the completion of double the reaction. In a first order reaction, the rate of the reaction is directly proportional to the initial concentration of the reactants and the slope of the equation is the rate constant.