Question

The first order reaction is completed 90% in 40 minutes. Calculate its half-life period. (log 10 =1)

Answer 1

Hint:In first order reaction the reaction rate is dependent upon only one reactant. The formula used for calculating the rate constant is $$K={\displaystyle \frac{2.303}{t}}\log {\displaystyle \frac{a}{a-x}}$$, where K is rate constant, t is time, a is initial concentration.

Complete step by step answer:
Given,
90% reaction is completed in 40 minutes.
log 10 = 1
The reaction is said as a first order reaction when the reaction rate depends on the concentration of only one reactant. The unit referring the first order reaction is $s^{-1}$.
The rate constant for the first order reaction is given by the formula as shown below.
$$K={\displaystyle \frac{2.303}{t}}\log {\displaystyle \frac{a}{a-x}}$$
Where,
K is the rate constant.
t is the time
$a$ is the initial concentration.
$a-x$ is the left out concentration.
Let, the initial concentration be 100.
Substitute the values in above equation
$$\Rightarrow K={\displaystyle \frac{2.303}{40}}\log {\displaystyle \frac{100}{100-90}}$$
$$\Rightarrow K={\displaystyle \frac{2.303}{40}}\log {\displaystyle \frac{100}{10}}$$
And hence on solving we have,
$$\Rightarrow K={\displaystyle \frac{2.303}{40}}\log 10$$
$$\Rightarrow K={\displaystyle \frac{2.303}{40}}\times 1$$
Now on doing the simplification, we have
$$\Rightarrow K=5.757\times {10}^{-2}\stackrel{-1}{min}$$
Thus, the rate constant value of first order reaction is $$5.757\times {10}^{-2}\stackrel{-1}{min}$$.
The half-life of the reaction is defined as the time required to decrease the amount of reactant consumed by one half.
The half-life of first order reaction is given as shown below.
$$t_{1/2}={\displaystyle \frac{0.693}{k}}$$
Where,
K is the rate constant.
To calculate the half-life of first order reaction, substitute the value of K in the equation.
$$\Rightarrow t_{1/2}={\displaystyle \frac{0.693}{5.757\times {10}^{-2}}}$$
$$\Rightarrow t_{1/2}=10.3min$$
Thus, the half-life of first order reaction is 10.3 min.

Note:
As the half-life of first order reaction is inversely proportional to the rate constant. So, for a shorter half-life, the reaction will be fast and the rate constant value will be larger. For, longer value of half-life the reaction will be slow and the rate constant value will be low.