Question

The thermal decomposition of \[ [HCOOH]\] is a first order reaction with a rate constant of \[2.4 \times {10^{ - 3}}{s^{ - 1}}\] at a certain temperature. How long will it take for three fourths of the initial quantity of \[ [HCOOH]\] to decompose?
A. 578 sec
B. 225 sec
C. 436 sec
D. 57.8 sec

Answer 1

Hint: To solve this question we need to know the first order rate law equation. And then substituting the values we will get the answer.

Complete step by step answer:
The question will be solved as given below,
We are given \[k = 2.4 \times {10^{ - 3}}{s^{ - 1}}\]
Now, according to the first order rate law,
\[k = \dfrac{{2.303}}{t}\log \dfrac{{{{[HCOOH]}_0}}}{{{{[HCOOH]}_t}}}\]
Now,
\[{[HCOOH]_t} = {[HCOOH]_0} - \dfrac{3}{4}{[HCOOH]_0} = \dfrac{1}{4}{[HCOOH]_0}\]
Substituting for \[{[HCOOH]_t}\] in the rate constant equation, we obtain,
\[\Rightarrow k = \dfrac{{2.303}}{t}\log \dfrac{{{{[HCOOH]}_0}}}{{\dfrac{1}{4}{{[HCOOH]}_0}}}\]
\[\Rightarrow k = \dfrac{{2.303}}{t}[\log 1 - \log 0.25]\]
\[\Rightarrow k = \dfrac{{2.303 \times 0.6021}}{{2.4 \times {{10}^{ - 3}}}}s = 578\sec \]

Therefore, the correct answer is option A.

Additional Information:
> Rate equation or rate law for a chemical reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters. There are four types of order of reaction and they are zero order, first order, second order and Pseudo first order.
> First Order reaction depends on the concentration of only one reactant and it is also called unimolecular reaction. Other reactants can be present but each will be of zero order. The half-life is independent of the starting concentration. First order reactions often have the general form \[A \to products\].
> The differential rate for a first order reaction is as follows:
\[rate = - \dfrac{{\Delta [A]}}{{\Delta t}} = k[A]\]
> As the reaction rate is directly proportional to the concentration of one of the reactants, if the concentration of A is doubled the reaction rate doubles and if it is increased by 10 times then the reaction rate also increases by 10 times.

Note:  The equation for every order is different and therefore the use of appropriate equations is necessary as per the order of the reaction. The general equation can be obtained by solving the equation,
\[rate = - \dfrac{{\Delta [A]}}{{\Delta t}} = k[A]^{\rm{n}}\], where n is the order of the reaction.