Question

What is the half life period of a reaction? Calculate the half life period of a first order reaction?

Answer 1

Half life period of a reaction is the time of a reaction at which the concentration of the reactants becomes half.

Complete step by step answer: The half life of a reaction is the time required for the concentration of a reactant molecule to reduce to half of the concentration present at the beginning. This time period is called the half-life of reaction and this governs the reaction rate of a reaction. It is represented or denoted as ${t_{1/2}}$ .
A first order reaction is a chemical reaction the rate of which is solely dependent on the concentration of only one reactant species. The differential form of first order reaction is written as
$rate = - \dfrac{{d[A]}}{{dt}} = k[A]$
Where $\left[ A \right]$ is the concentration of the reactant $A$ , and $k$ is the reaction coefficient.
Let the initial concentration of the first order reaction is ${\left[ A \right]^0}$ and the concentration at half time ${t_{1/2}}$ = $\dfrac{{{{\left[ A \right]}^0}}}{2}$.
Thus the integrated rate law of the first order reaction is
$\ln \dfrac{{{{[A]}^0}}}{{[A]}} = kt$
At half time the $t$ becomes ${t_{1/2}}$ and the concentration becomes $\dfrac{{{{\left[ A \right]}^0}}}{2}$. So inserting the values in the rate equation,
$\ln \dfrac{{{{[A]}^0}}}{{\dfrac{{{{[A]}^0}}}{2}}} = k{t_{1/2}}$
$\ln 2 = k{t_{1/2}}$
${t_{1/2}} = \dfrac{{0.693}}{k}$.
Thus the half-life of a first-order reaction is independent of the concentration of reactant. Actually half life is a constant. This is different for zeroth- and second-order reactions.

Note:
The half-life of a reaction is referred as the time required for the reactant concentration to change from ${\left[ A \right]^0}$ to $\dfrac{{{{\left[ A \right]}^0}}}{2}$. In a situation when two reactions have the same order, the faster reaction has a shorter half-life and the slower reaction has a longer half-life.