Animated Solution for Chemistry - Chemical Kinetics: The reaction, $2X\to B$ is a zeroth order reaction. If the initial concentration of $X$ is $0.2\text{ M}$, the half-life is $6\text{ h}$. When the initial concentration of $X$ is $0.5\text{ M}$, the time required to reach its final concentration of $0.2\text{ M}$ will be

The study of chemical kinetics is like watching a beautifully choreographed dance of molecules. Some dances are complex, changing tempo as the music plays. But others, like the zero-order reaction, are steadfast and unyielding. They march to the beat of their own drum, completely ignoring how many molecules are left on the dance floor.

In this problem, we are introduced to a reaction: $2X\to B$, and we are explicitly told it follows zero-order kinetics. This is our golden ticket. It tells us that the rate at which $X$ disappears is absolutely constant.

Imagine a factory assembly line. No matter how huge the pile of raw materials is, the workers can only produce a certain number of items per hour. That's zero-order for you. Mathematically, this means the concentration of our reactant $X$ decreases linearly with time. If we were to plot the concentration of $X$ against time, we would see a perfectly straight line sloping downwards.

The master equation governing this linear descent is the integrated rate law:

Here, $[X{]}_0$ is our starting concentration, $[X{]}_t$ is the concentration at any given time $t$, and $k$ is our steadfast rate constant.

Before we can predict the future of this reaction, we need to know its speed limit—the rate constant $k$. The problem gives us a crucial clue: when the initial concentration is $0.2\text{ M}$, the half-life ($t_{1/2}$) is $6\text{ h}$.

For a zero-order reaction, the half-life is directly proportional to the initial concentration. It makes sense, right? If the factory works at a constant speed, a larger pile of raw materials will naturally take longer to cut in half. The formula is:

Let's substitute our known values into this equation:

Now, it's just a matter of simple algebra to isolate $k$:

We've found our rate constant! This $k$ is the fundamental speed of our reaction, and it will remain constant as long as the temperature doesn't change.

Now we are ready to tackle the main question. We are given a new scenario: a fresh batch starting with an initial concentration of $0.5\text{ M}$. We want to find out exactly how long it will take for this concentration to drop to $0.2\text{ M}$.

We bring back our master equation, the integrated rate law, and plug in our new initial concentration, our target final concentration, and our newly discovered rate constant:

Subtracting the concentrations gives us the total amount of $X$ that needs to react:

To find the time $t$, we simply multiply $0.3$ by $60$:

And there we have it! It will take exactly $18.0\text{ h}$ for the concentration to reach $0.2\text{ M}$. This perfectly matches option (b). The beauty of zero-order kinetics lies in this predictable, linear elegance. Once you know the speed, you can easily predict the journey!