Animated Solution for Chemistry - Chemical Kinetics: The rate law for a reaction between the substances $A$ and $B$ is given by $\text{rate}=k[A{]}^n[B{]}^m$. On doubling the concentration of $A$ and halving the concentration of $B$, the ratio of the new rate to the earlier rate of the reaction will be as

Unraveling the Rate Law

Imagine you are a master chef, but instead of baking cakes, you are orchestrating chemical reactions. You have two magical ingredients, $A$ and $B$. You know that mixing them produces a spectacular product, but you want to control the speed of this reaction. How do you do it? This is the fundamental question of chemical kinetics.

In chemistry, the speed at which reactants turn into products is governed by a mathematical equation known as the Rate Law. For our reaction between substances $A$ and $B$, the rate law is given as:

Let's break down this recipe for speed. The term $r$ is the rate of the reaction. The square brackets $[A]$ and $[B]$ represent the concentrations of our ingredients. The exponents $n$ and $m$ are the orders of the reaction with respect to $A$ and $B$. They tell us exactly how sensitive the reaction speed is to changes in each ingredient. Finally, $k$ is the rate constant—a unique fingerprint for this specific reaction at a given temperature.

Tweaking the Recipe

Now, let's perform a thought experiment. What happens if we mess with the recipe? The problem states that we are going to double the concentration of $A$ and halve the concentration of $B$.

Let's define our new, modified state: - The new concentration of $A$ is $[A{]}^{\prime }=2[A]$. - The new concentration of $B$ is $[B{]}^{\prime }=\frac{[B]}{2}$.

With these new ingredients, our reaction will proceed at a new rate, which we will call $r^{\prime }$. By substituting our new concentrations into the original rate law, we get the equation for the new rate:

Setting up the Ratio

To find out exactly how much the rate has changed, we need to compare the new rate to the old rate. We do this by setting up a mathematical ratio, dividing $r^{\prime }$ by $r$:

At first glance, this fraction might look a bit intimidating, but it is actually perfectly set up for a beautiful algebraic simplification.

The Beauty of Algebra

This is where the magic happens. Let's expand the terms in the numerator using the standard rules of exponents. The term $[2A{]}^n$ becomes $2^n\cdot [A{]}^n$, and the term ${\left[\frac{B}{2}\right]}^m$ becomes ${\left(\frac{1}{2}\right)}^m\cdot [B{]}^m$.

Substituting these expanded forms back into our ratio gives:

Now, watch the symphony of cancellation! The rate constant $k$ appears in both the numerator and the denominator, so it cancels out. The concentration terms $[A{]}^n$ and $[B{]}^m$ also appear in both places, so they vanish as well. We are left with only the numerical factors:

The Power of Exponents

We are almost at the finish line. We know from the laws of exponents that $\frac{1}{2}$ is the same as $2^{-1}$. Therefore, ${\left(\frac{1}{2}\right)}^m$ can be rewritten as $2^{-m}$.

Multiplying $2^n$ by $2^{-m}$ allows us to simply add the exponents together:

Our final result, $2^{n-m}$, is a powerful testament to the role of exponents in chemistry. It elegantly captures how opposing changes in concentration—doubling one and halving another—compete to determine the final speed of the reaction based entirely on their respective reaction orders.