Animated Solution for Chemistry - Chemical Kinetics: For the elementary reaction $\text{M}\to \text{N}$, the rate of disappearance of $\text{M}$ increases by a factor of $8$ upon doubling the concentration of $\text{M}$. The order of the reaction with respect to $\text{M}$ is

The study of chemical kinetics is fundamentally about understanding how fast reactions occur and what factors influence that speed. In this problem, we are presented with a classic scenario: observing how a change in reactant concentration affects the overall rate of the reaction.

This relationship is mathematically captured by the order of the reaction, which is exactly what we need to find.

Every kinetic analysis begins with the rate law. For our reaction where reactant $M$ transforms into product $N$, we can express the rate of disappearance of $M$, denoted as $r$, using a general rate equation.

Here, $k$ represents the rate constant, a value that remains unchanged as long as the temperature is constant. The exponent $n$ is the order of the reaction with respect to $M$. This exponent dictates how sensitive the rate is to changes in concentration.

The problem provides us with two distinct experimental states. Let's define our initial state. We have an initial concentration $[M{]}_1$ which corresponds to an initial rate $r_1$.

Next, we are told that the concentration of $M$ is doubled. So, our new concentration is $2[M{]}_1$. Under this new condition, the rate of disappearance increases by a massive factor of $8$. Therefore, our new rate is $8r_1$. Let's write the rate law for this second state.

We now have a system of two equations. The most elegant and efficient way to solve this system and eliminate the unknown rate constant $k$ is to take the ratio of the two equations. We divide the second state by the first state.

Notice how beautifully the terms cancel out. The rate constant $k$ disappears, and the initial rate $r_1$ on the left side also cancels out. We can also factor out the concentration term on the right side.

The concentration terms $[M{]}_1$ cancel out completely, leaving us with a very simple exponential equation.

To solve for $n$, we just need to express $8$ as a power of $2$. We know that $2\times 2\times 2=8$, which means $8=2^3$.

By comparing the exponents, it becomes immediately clear that $n=3$.

Before we conclude, we must address a brilliant trap hidden in the wording of the question. The problem states this is an elementary reaction $M\to N$.

Normally, for an elementary step, the order of the reaction is exactly equal to its stoichiometric coefficient. Looking at $M\to N$, you might be tempted to say the order is $1$.

However, our rigorous mathematical calculation using the experimental data proves the order is $3$. What does this mean? It means that the equation $M\to N$ provided in the text is merely a skeletal representation.

The true, balanced elementary step must involve three molecules of $M$ colliding simultaneously, which would be written as $3M\to \text{Products}$. In kinetics, always trust the experimental data over a skeletal chemical equation!