Question

A first order reaction is $75\% $ completed in $100$ minutes. How long time will it take for it’s $87.5\% $ completion.
A. $125\min $
B. $150\min $
C. $175\min $
D. $200\min $

Answer 1

Hint: The power dependence of the rate on the concentration of each reactant is known as order of the reaction. The order of the overall reaction depends upon the sum up of the powers present on each reactant species.

Complete step by step answer:
The change in concentration with respect to time in a chemical reaction is known as rate of reaction. Order of the reaction is also the rate of reaction but represented in a different way. Basically it is the power dependence of the rate on the concentration of each reactant. This order of reaction includes the concentration of all the species raised to some power present in the reaction.
As we all know that generally the rate equation is represented as:
\[{\text{rate = k}}{\left[ {\text{A}} \right]^a}{\left[ {\text{B}} \right]^b}\]
Here ${\text{A and B}}$ are the concentration of species and $a{\text{ and }}b$ are the individual orders of the two reactant species. The overall order of the reaction is the sum up of the individual orders of the species i.e. $a + b$ .
And if the value of sum ($a + b$) becomes one it is known as first order reaction. Hence one of the individual orders will be zero, so in first order reaction the rate of the reaction is dependent linearly on only one reactant species.
The time taken by the reaction to complete half of the reaction is known as its half life time, for first order reaction it is independent of the concentration of reactants.
Hence we can say that
$\underbrace {100\% }_{remaining}\xrightarrow{{halftime}}\underbrace {50\% }_{remaining}\xrightarrow{{halftime}}\underbrace {25\% }_{remaining}\xrightarrow{{halflife}}\underbrace {12.5\% }_{remaining}$
Or we can say that
$\underbrace {0\% }_{consumed}\xrightarrow{{halftime}}\underbrace {50\% }_{consumed}\xrightarrow{{halftime}}\underbrace {75\% }_{consumed}\xrightarrow{{halflife}}\underbrace {87.5\% }_{consumed}$
And we know that $75\% $ (i.e. two half life) is consumed in $100\min $.
$\begin{gathered}
   \Rightarrow 2{t_{\dfrac{1}{2}}} = 100\min \\
   \Rightarrow {t_{\dfrac{1}{2}}} = 50\min \\
\end{gathered} $
And from the above diagrammatic representations we can see that $87.5\% $ completion occurs after three half lives.
So, time taken for $87.5\% $ completion = $3 \times {t_{\dfrac{1}{2}}} = 3 \times 50 = 150\min $
Hence $150\min $ are consumed for the $87.5\% $ completion of the first order reaction.

So, the correct answer is Option B.

Note:
Half life is nothing but the probability of the time taken to decay half of the material. The half life depends upon the order of the reaction, if the reaction is first order reaction then the half is independent of the concentration of the reactant species.