Question

The $ {{t}_{1/2}} $ of reaction becomes $ 1/4th $ as the initial concentration of the reactant is doubled. What is the order of reaction?

Answer 1

Hint :We know that for an nth order reaction the rate is proportional to nth power of the reactant concentration. Half-life is the time in which the concentration of the reactant gets reduced to half. For an nth order reaction the half-life.

Complete Step By Step Answer:
Let us first talk about the rate of reaction and order of reaction. Rate of reaction: The rate of a reaction is the speed at which a chemical reaction happens. Order of the reaction: It is defined as the power dependence of the rate of reaction on the concentration of the reactants. Rate of reaction: The rate of a reaction is the speed at which a chemical reaction happens. Order of the reaction: It is defined as the power dependence of the rate of reaction on the concentration of the reactants. For example: if order of reaction is one then rate of reaction depends linearly on the concentration of one reactant. The unit of first order of reaction is $ {{t}_{1/2}}. $ The unit of second order of reaction is $ {{t}_{1/2}}. $ Half-time: It is defined as the time duration in which the concentration of a reactant drops to one-half of its initial concentration. It is represented by $ {{t}_{1/2}}. $
Here, $ {{t}_{1/2}}\alpha \dfrac{1}{{{a}^{n-1}}} $ here n is order and a is the initial condition.
Also as reaction becomes one fourth of initial concentration is doubled;
 $ \dfrac{{{t}_{1/2}}}{2}=\dfrac{1}{{{\left( 2a \right)}^{n-1}}} $ ……………….. $ \left( 1 \right) $
On further solving we get;
 $ {{t}_{1/2}}=\dfrac{k}{{{a}^{n-1}}} $ ……………….. $ \left( 2 \right) $
From equation $ \left( 1 \right) $ & $ \left( 2 \right) $ : $ \left[ \dfrac{1}{2}=\dfrac{{{a}^{n-1}}}{{{\left( 2a \right)}^{n-1}}} \right]\Rightarrow \left[ \dfrac{1}{2}={{\left( \dfrac{1}{2} \right)}^{n-1}} \right] $
Thus here we get; $ n-1=1 $
 $ \therefore n=2 $ .

Additional Information:
The half-life definition, chemistry is the time it takes for half an initial amount to disintegrate. The time that is required for half of a reactant to be converted into products. The time it takes for half of a given sample to undergo radioactive decay. The half-life definition is given by the time that it takes for one-half of the atoms of a nuclide or of an unstable element to decay the radioactively into another nuclide or element. For a given half-life reaction, the $ {{t}_{1/2}} $ is the time required for its concentration to reach a value, the arithmetic mean of its initial and final or equilibrium) value. For an entirely consumed reactant, it is the time taken for the reactant concentration to fall to one half of its initial value.

Note :
Remember that the concepts of half-life play a vital role in the administration of drugs into the target, especially in the elimination phase, where half-life is used to discover how quickly a drug decreases in the target once it has been absorbed in a period of time relation between the half-time and concentration of reactant. Half-time of a reaction is inversely proportional to the concentration of the reactant raised to the power of its order of reaction minus one. Here in the question we are given with the initial concentration and half-time for the reaction two times.