Question

A and B are two different chemical species undergoing first order decomposition with rate constants ${K_A}$ and ${K_B}$ which are in the ratio $3:2$, respectively. If the initial concentration of A and B are in the ratio $[{A_ \circ }]:[{B_ \circ }] = 3:2$, what would be the ratio $[A]:[B]$ after three half lives of A?

Answer 1

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 ${t_{\dfrac{1}{2}}}$.
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.

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. 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 ${s^{ - 1}}$. The unit of second order of reaction is $1/Ms$.
Rate constant: It is defined as a constant in the equation of rate of reaction. As we know that rate of reaction is directly proportional to the product of concentration of the reactants raised to the power of their order. The proportional constant in this is known as rate constant.
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 ${t_{\dfrac{1}{2}}}$.
The relation between the half-time and concentration of reactant is as follows:
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.
After one half-life the concentration of reactant reduces to half and after two half lives the concentration of reactant reduces to one-fourth of the initial concentration and similarly after three half lives the concentration of reactant reduces to one-eighth of the initial concentration.
Here in the question we are given with the ratio of initial concentrations and the ratio of their rate constants. And we know that half life of a reaction is inversely proportional to its rate constant. So if the ratio of rate constant is $3:2$ then the ratio of their half-lives will be $2:3$. Now let the half life of A be $x$ then half-life of B will be $\dfrac{3}{2}x$. Now when A completes its three half lives then B will do its two half lives because the ratio is $2:3$. Hence after three half live of A the concentration of A will be $\dfrac{{[{A_ \circ }]}}{8}$ and after two half live of B the concentration of B will be $\dfrac{{[{B_ \circ }]}}{4}$. Now the ratio of these two will be as $\dfrac{{[{A_ \circ }]}}{8}:\dfrac{{[{B_ \circ }]}}{4}$ and we already know the ratio of $[{A_ \circ }]:[{B_ \circ }] = 3:2$ after putting this ratio we get the final ratio as $\dfrac{3}{4}$.

Note:
Activation energy: It is defined as the least required energy for a chemical reaction to happen. It is represented by ${E_a}$.
Frequency factor is defined as the rate of molecular collisions that occur during the chemical reaction. It is also known as pre-exponential factor and is represented as A.