Question

The first-order reaction is 20 $\% $ complete in 20 minutes. Calculate the time it will take the reaction to complete 80 $\% $.

Answer 1

Hint: A first-order reaction is a reaction whose rate is determined by the change in concentration one concentration term only. Or in other words, a reaction whose rate varies as the first power of the concentration of a reactant. Let us consider the following reaction:
${{A}} \to {{B}}$, where A is the reactant and B is the product.
For the above reaction to being a first-order reaction,
Rate $ = $ ${{ - }}\dfrac{{{{d[A]}}}}{{{{dt}}}}{{ = k[A]}}$, where k is the rate constant.

Complete step by step answer:
Firstly we will try and find out the rate constant of the reaction:
The rate constant is given by: ${{k = }}\dfrac{{{1}}}{{{t}}}{{ln}}\left( {\dfrac{{{{{{[A]}}}_{{i}}}}}{{{{{{[A]}}}_{{t}}}}}} \right)$, where t is time, ${{{[A]}}_{{i}}}$ is the initial concentration of the reactant, and is the concentration of reaction at time t.
The reaction is 20 $\% $ complete in 20 minutes. So if we have started with 100 g we are now left with 80 g of the reactant. Putting all the values in the expression for k, we get:
${{k = }}\dfrac{{{1}}}{{{{20}}}}{{ln}}\left( {\dfrac{{{{100}}}}{{{{80}}}}} \right)$, which gives k $ = $ 0.0111571 ${{minut}}{{{e}}^{{{ - 1}}}}$
Using this value of k we will calculate the time required for the reaction to complete 80 $\% $. Let us assume that reaction is 80% complete in time t, so at time t we are left with 20 $\% $ of reactant, putting all these values together with the value of rate constant we get:
${{0}}{{.0111571 = }}\dfrac{{{1}}}{{{t}}}{{ln}}\left( {\dfrac{{{{100}}}}{{{{20}}}}} \right)$
This gives us t $ = $ 144.25 minutes

Thus, the reaction is 80 $\% $ complete in 144.25 minutes.

Note:
The order of a reaction is defined as the sum of powers of the reactants of a chemical reaction in its rate law equation. While the rate of a reaction is defined as the speed at which the reaction proceeds it is rate constant multiplied by the concentration of each reactant each raised to some power which may or may not be equal to the stoichiometric coefficients of the reactants.
${{aA + bB}} \to c{{C + dD}}$
The rate of a reaction is:
Rate ${{ = k[A}}{{{]}}^{{x}}}{{{[B]}}^{{y}}}$
And therefore the order the reaction is \[x + y\]