Animated Solution for Chemistry - Basic Concepts in Chemistry: The ratio of masses of oxygen and nitrogen in a particular gaseous mixture is $1:4$. The ratio of number of their molecule is

The mole concept is one of the most elegant and powerful tools in chemistry. It acts as a universal bridge, allowing us to translate the macroscopic world of grams and kilograms into the microscopic world of atoms and molecules. In this problem, we are tasked with finding the ratio of the number of molecules of oxygen and nitrogen in a gaseous mixture, given only their mass ratio. Let's embark on this journey and see how the mole concept effortlessly solves this puzzle.

Imagine a sealed glass container filled with a mixture of two gases: Oxygen ($O_2$) and Nitrogen ($N_2$). We are given a crucial piece of information: the ratio of their masses is $1:4$.

Mathematically, we can express this as:

This means that for every $1\text{ gram}$ of Oxygen present in the container, there are exactly $4\text{ grams}$ of Nitrogen swirling around alongside it. However, mass alone doesn't tell us how many individual molecules are present, because different molecules have different weights. A heavier molecule will contribute more to the total mass even if there are fewer of them.

The question asks for the ratio of the number of their molecules. Counting individual molecules is practically impossible, but Avogadro's principle provides the perfect bridge. The number of molecules ($N$) is directly proportional to the number of moles ($n$), related by Avogadro's number ($N_A$):

Since $N_A$ is a universal constant ($6.022\times 10^{23}$), when we take the ratio of the number of molecules of two different substances, Avogadro's number cancels out completely. Therefore, the ratio of the number of molecules is exactly equal to the ratio of their moles:

To find the moles, we use the fundamental formula:

Here lies a classic trap where many students make a silly mistake. Oxygen and Nitrogen gases do not exist as single, isolated atoms in nature; they exist as diatomic molecules, $O_2$ and $N_2$.

Therefore, we must use their molecular masses, not their atomic masses. The atomic mass of oxygen is $16$, so the molar mass of $O_2$ is $32\text{ g/mol}$. Similarly, the atomic mass of nitrogen is $14$, making the molar mass of $N_2$ equal to $28\text{ g/mol}$.

Now, let's set up the ratio of their moles by substituting the mole formula for both gases:

By rearranging the terms, we can elegantly group the mass ratio and the molar mass ratio together:

We already know the mass ratio is $\frac{1}{4}$, and we know the molar masses. Let's substitute these values into our rearranged equation:

Now, it's just a matter of simple arithmetic. We can simplify the fraction by dividing $28$ by $4$, which gives us $7$ in the numerator. The denominator remains $32$:

This is our final answer. The ratio of the number of oxygen molecules to nitrogen molecules in the mixture is $7:32$.

Before we conclude, let's think a bit deeper. What if the examiner twisted the question and asked for the ratio of the total number of atoms instead of molecules?

Since both Oxygen and Nitrogen are diatomic gases, every single molecule of either gas contains exactly two atoms. To find the ratio of atoms, we would multiply the number of molecules of both gases by $2$. Because we are multiplying both the numerator and the denominator by the same factor, the ratio would remain completely unchanged at $7:32$!

However, if the mixture contained a monatomic gas like Helium alongside a diatomic gas, the atom ratio would be entirely different from the molecule ratio. Always read the question carefully to see whether it asks for atoms or molecules!