Animated Solution for Chemistry - Chemical Kinetics: Consider the reaction, $2A+B\to \text{product}$. When concentration of $B$ alone was doubled, the half-life did not change. When the concentration of $A$ alone was doubled, the rate increased by two times. The unit of rate constant for this reaction is

Chemical kinetics is like being a detective at a molecular crime scene. You are given clues about how the speed of a reaction changes when you tweak the starting conditions, and your job is to uncover the hidden "rate law." In this problem, we are investigating the reaction $2A+B\to \text{Products}$. Our ultimate goal is to find the unit of the rate constant, $k$. But to do that, we first need to determine the overall order of the reaction.

The first major clue lies in the behavior of reactant $B$. The problem states that when the concentration of $B$ alone is doubled, the half-life ($t_{1/2}$) of the reaction does not change.

Why is this significant? The half-life of a reaction is the time it takes for the concentration of a reactant to drop to half of its initial value. For a general reaction of order $n$, the half-life is proportional to the initial concentration raised to the power of $1-n$:

If the half-life is completely independent of the initial concentration, it means the exponent must be zero. Therefore, $n-1=0$, which gives us $n=1$. This tells us that the reaction is first-order with respect to $B$.

Now let's look at the second clue involving reactant $A$. When the concentration of $A$ alone is doubled, the rate of the reaction increases by exactly two times.

This is a classic case of direct proportionality. If we write the rate dependence as $\text{Rate}\propto [A{]}^x$, doubling $[A]$ gives us a rate that is $2^x$ times faster. Since the rate doubled ($2^1$), we can immediately conclude that $x=1$. Thus, the reaction is also first-order with respect to $A$.

The overall order of a reaction is simply the sum of the individual orders with respect to each reactant.

Since the reaction is first-order with respect to $A$ and first-order with respect to $B$, the overall order $n$ is:

We are dealing with a second-order reaction overall. The complete rate law can be written as $\text{Rate}=k[A{]}^1[B{]}^1$.

Units of the Rate Constant

With the overall order in hand, finding the unit of the rate constant $k$ is a breeze. The general formula for the unit of $k$ for an $n^{\text{th}}$-order reaction is:

Substitute our overall order $n=2$ into the formula:

And there we have it! By carefully analyzing the experimental clues, we deduced the reaction orders and flawlessly calculated the unit of the rate constant.