Animated Solution for Chemistry - Chemical Kinetics: In a first order reaction, the concentration of the reactant, decreases from $0.8\text{ M}$ to $0.4\text{ M}$ in $15\text{ min}$. The time taken for the concentration to change from $0.1\text{ M}$ to $0.025\text{ M}$ is

Chemical kinetics is not just about memorizing formulas; it is about recognizing the beautiful mathematical patterns that govern how fast reactions occur. In this problem, we are presented with a classic scenario involving a first-order reaction. The question asks us to find the time required for a specific concentration drop, given an initial rate of decay. While we could dive straight into the integrated rate law, there is a much more elegant and intuitive way to solve this.

Let's look closely at the first piece of information provided: the concentration of the reactant decreases from $0.8\text{ M}$ to $0.4\text{ M}$ in exactly $15\text{ min}$.

What is the relationship between $0.8$ and $0.4$? It is exactly half! This simple observation is the key to unlocking the entire problem. The time it takes for a reactant's concentration to reduce to exactly half of its initial value is known as the half-life ($t_{1/2}$) of the reaction.

Here is the most crucial property of first-order reactions: their half-life is absolutely constant. It does not matter if you start with $1000\text{ M}$ or $0.001\text{ M}$; the time it takes to halve that amount will always be the same.

Mathematically, this is because the half-life for a first-order reaction is given by the equation:

Notice that the initial concentration ($C_0$) is completely absent from this formula! Therefore, based on our initial observation, we can confidently state that the half-life of this specific reaction is $15\text{ min}$.

Now, let's tackle the second part of the question. We need to find the time it takes for the concentration to drop from $0.1\text{ M}$ to $0.025\text{ M}$.

Instead of plugging these numbers into a complex logarithmic equation, let's use our half-life logic. Imagine you start with $0.1\text{ M}$. After one half-life ($15\text{ min}$), the concentration will halve:

We are not at our target yet. Let's let another half-life pass. The concentration will halve again:

And there it is! We have reached our target concentration of $0.025\text{ M}$.

By tracing the decay, we can clearly see that it took exactly two half-lives to go from $0.1\text{ M}$ to $0.025\text{ M}$.

Since we already established that one half-life is $15\text{ min}$, the total time required is simply:

We arrived at the correct answer without touching a single logarithm. While the integrated rate law ($t=\frac{2.303}{k}\log \frac{C_0}{C_t}$) will absolutely give you the same result, recognizing half-life multiples is a powerful shortcut that saves precious time in competitive exams like JEE and NEET. Always keep an eye out for these elegant patterns!