# Decomposition of X exhibits a rate constant of 0.05µg/year. How many years are required for the decomposition of 5µg of X into 2.5µg?

A. 50

B. 25

C. 20

D. 40

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**Hint:**To check the order of reaction see the units of rate constant. In this question according to the units of rate constant it is a zero order reaction. Here we have to find the half life of a reaction and to find the half life time period of Zero order reaction we have a formula that is ${{t}_{\dfrac{1}{2}}}=\dfrac{{{a}_{o}}}{2K}$ where K is the rate constant and ${{a}_{o}}$ is the initial concentration value.

**Complete step by step solution:**

From your chemistry lessons you have learned about the rate constant, Zero order reaction.

The rate constant is known as the proportionality constant which explains the relationship between the rate of a chemical reaction and the molar concentration of the reactants. It is also known as reaction rate constant and is denoted by 'K' and its units are determined by the order of reaction. Rate constant is expressed as $K=\dfrac{Rate}{{{[Conc.]}^{n}}}$

Where K= rate constant.

Conc.= molar concentration of reactants

n = order of reaction

Here we are going to deal with Zero order reaction.

Zero order reaction is a type of reaction in which the rate of reaction does not depend upon the increase or decrease in the molar concentration. Thus the rate of these reactions are always equal to the rate constant.

For the zero order reaction the rate of reaction is written as, $rate=k{{[A]}^{0}}$

Here n= 0 (zero order reaction)

-The unit of rate constant for Zero order reaction is equal to the unit of rate of reaction because the unit of concentration of reactant will not be taken in account and thus the unit is $mol\,{{L}^{-1}}{{s}^{-1}}$

-So in the question the rate constant which is given is of zero order reaction according to its unit.

-From the question the value of rate constant (K)= 0.05 µg/year.

-In the question we are asked to find the time period for the decomposition of 5µg ( initial concentration) of X into 2.5µg (final concentration), here the final concentration is the half of the initial concentration. So, we will find the half life of the reaction and the formula to find it is,

\[{{t}_{\dfrac{1}{2}}}=\dfrac{{{a}_{o}}}{2K}\]

where K is the rate constant and ${{a}_{o}}$ is the initial concentration value.

\[\therefore {{t}_{\dfrac{1}{2}}}=\dfrac{5}{2\times 0.05}=50years\]

**Thus the correct option will be (A).**

**Note:**Order of reaction is defined as the relationship that relates the concentration of reacting species with the rate of a chemical reaction. It is determined as the sum of the power of the rate of the concentration of each of the reactants. Order of reaction can be positive, negative, integer of Zero. The trick to find the unit of rate constant directly is $K={{\left( \dfrac{mol}{L} \right)}^{1-n}}\times {{\sec }^{-1}}$ where n is the order of reaction.