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Question: A first order reaction has a specific rate of \({10^{ - 2}}{\text{ se}}{{\text{c}}^{ - 1}}\). How mu...

A first order reaction has a specific rate of 10−2 sec−1{10^{ - 2}}{\text{ se}}{{\text{c}}^{ - 1}}. How much time will it take for 20 g of the reactant to reduce to 5 g?
A) 693.0 sec693.0{\text{ sec}}
B) 238.6 sec238.6{\text{ sec}}
C) 138.6 sec138.6{\text{ sec}}
D) 346.5 sec346.5{\text{ sec}}

Explanation

Solution

To solve this question we must know the equation for the rate constant of first order reaction. The initial and final concentrations of the reactants are given. Thus, we can calculate the time required for the reactant concentration to reduce from the initial concentration to the final concentration.

Formula Used: k=2.303tlog⁡[a]0[a]k = \dfrac{{2.303}}{t}\log \dfrac{{{{\left[ a \right]}^0}}}{{\left[ a \right]}}

Complete step by step answer:
We know the equation for the rate constant of a first order reaction is,
k=2.303tlog⁡[a]0[a]k = \dfrac{{2.303}}{t}\log \dfrac{{{{\left[ a \right]}^0}}}{{\left[ a \right]}}
Where k is the rate constant of a first order reaction,
t is time,
[a]0{\left[ a \right]^0} is the initial concentration of the reactant,
[a]\left[ a \right] is the final concentration of the reactant.
Rearrange the equation for time as follows:
t=2.303klog⁡[a]0[a]t = \dfrac{{2.303}}{k}\log \dfrac{{{{\left[ a \right]}^0}}}{{\left[ a \right]}}
Substitute 10−2 sec−1{10^{ - 2}}{\text{ se}}{{\text{c}}^{ - 1}} for the rate constant of the first order reaction, 20 g20{\text{ g}} for the initial concentration of the reactant, 5 g5{\text{ g}} for the final concentration of the reactant and solve for the time required for the 20 g20{\text{ g}} of the reactant to reduce to 5 g5{\text{ g}}. Thus,
t=2.30310−2 sec−1log⁡20 g5 gt = \dfrac{{2.303}}{{{{10}^{ - 2}}{\text{ se}}{{\text{c}}^{ - 1}}}}\log \dfrac{{20{\text{ g}}}}{{5{\text{ g}}}}
t=138.6 sect = 138.6{\text{ sec}}
Thus, the time required for the 20 g20{\text{ g}} of the reactant to reduce to 5 g5{\text{ g}} is 138.6 sec138.6{\text{ sec}}.

Thus, the correct option is (C) 138.6 sec138.6{\text{ sec}}.

Note: The sum of the powers of concentration of the reactants in the rate equation of the chemical equation is known as the order of the reaction. The reaction in which the rate of the reaction is directly proportional to the concentration of the reactant species is known as the first order reaction. The unit of rate constant for first order reaction is sec−1{\text{se}}{{\text{c}}^{ - 1}}. The units do not contain concentration terms. Thus, we can say that the rate constant of a first order reaction is independent of the concentration of the reactant.