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Question: For a first order reaction, \((A) \to {\text{Product}}\), the concentration of A changes from 0.1M t...

For a first order reaction, (A)→Product(A) \to {\text{Product}}, the concentration of A changes from 0.1M to 0.025M in 40 minutes. The rate of reaction when the concentration of A is 0.01M, is:
(A) 1.73×10−5Mmin⁡−11.73 \times {10^{ - 5}}M{\min ^{ - 1}}
(B) 3.47×10−4Mmin⁡−13.47 \times {10^{ - 4}}M{\min ^{ - 1}}
(C) 3.47×10−5Mmin⁡−13.47 \times {10^{ - 5}}M{\min ^{ - 1}}
(D) 1.73×10−4Mmin⁡−11.73 \times {10^{ - 4}}M{\min ^{ - 1}}

Explanation

Solution

The formula to find the rate constant of the reaction is
k=2.303tlog⁡R0Rk = \dfrac{{2.303}}{t}\log \dfrac{{{R_0}}}{R}
The formula for rate of first order reaction is
Rate=k[A]Rate = k[A]

Complete step by step solution:
We know that the rate of the reaction depends upon the value of rate constant of the reaction. Rate constant of the reaction is not given here, So, we will need to find the value of the rate constant here.
- First order reaction is a reaction in which the rate of the reaction depends upon the concentration power 1 of the reactant.
- We can write the formula of the rate constant of the first order reactions as
k=2.303tlog⁡R0Rk = \dfrac{{2.303}}{t}\log \dfrac{{{R_0}}}{R}
Here, k is the rate constant, t is the time, R0{R_0} is the initial concentration of the reactant and R is the final concentration of the reactant.
So, we can put the available values into this equation as
k=2.30340log⁡0.10.025k = \dfrac{{2.303}}{{40}}\log \dfrac{{0.1}}{{0.025}}
So, we obtain
k=2.30340log⁡4k = \dfrac{{2.303}}{{40}}\log 4
Thus,
k=2.30320×log⁡2k = \dfrac{{2.303}}{{20}} \times \log 2
So, we can write that k=2.30340×0.301=0.0347min⁡−1k = \dfrac{{2.303}}{{40}} \times 0.301 = 0.0347{\min ^{ - 1}}
Now, we can write the equation of rate of first order reaction as
Rate=k[A]Rate = k[A]
We want to find the rate at 0.01M concentration. We obtained that k=0.0347min⁡−1k = 0.0347{\min ^{ - 1}}. So, we can write the above equation as
Rate=0.0347×0.01=3.47×10−4Mmin⁡−1Rate = 0.0347 \times 0.01 = 3.47 \times {10^{ - 4}}M{\min ^{ - 1}}
Thus, we can conclude that the rate of the reaction when the concentration of A is 0.01M is 3.47×10−4Mmin⁡−13.47 \times {10^{ - 4}}M{\min ^{ - 1}} .

So, the correct answer is (B).

Note: Note that we should not consider the concentration at which we want to find the rate of the reaction as the final concentration in the formula of rate constant. For all the first order reactions, the unit for the rate constant is time−1tim{e^{ - 1}}. So, here we can see that the unit is taken in min⁡−1{\min ^{ - 1}} unit.