Animated Solution for Chemistry - Basic Concepts in Chemistry: How many moles of magnesium phosphate, ${\text{Mg}}_3({\text{PO}}_4{)}_2$ will contain 0.25 mole of oxygen atoms?

Welcome to one of the most fundamental and fascinating concepts in all of chemistry: the mole concept!

When we look at a chemical reaction or a chemical formula, we are essentially looking at a recipe. But atoms and molecules are so incredibly tiny that counting them one by one is practically impossible. Imagine trying to count the grains of sand on a beach; counting atoms in a beaker is infinitely harder!

This is where the "mole" comes to our rescue. Just like a "dozen" represents 12 items, a "score" represents 20 items, a "mole" represents $6.022\times 10^{23}$ items. This massive number, known as Avogadro's number, allows chemists to bridge the gap between the microscopic world of atoms and the macroscopic world of grams and liters that we can actually measure in a laboratory.

Why is this specific number so special? Because it perfectly links the atomic mass of an element (measured in atomic mass units, or amu) to its molar mass (measured in grams). For instance, one atom of carbon-12 weighs exactly 12 amu, and one mole of carbon-12 atoms weighs exactly 12 grams. It is a beautiful, elegant system that makes chemical calculations possible.

In this problem, we are not just dealing with a simple element; we are dealing with a complex ionic compound: magnesium phosphate.

Our mission is to find out how many moles of this entire compound are needed if we want to have exactly $0.25$ moles of oxygen atoms. To solve this, we need to become chemical detectives and decode the formula of the compound itself.

Let's take a very close look at the chemical formula provided in the question: ${\text{Mg}}_3({\text{PO}}_4{)}_2$.

This formula is packed with information. It tells us exactly what ingredients make up one single molecule (or more accurately, one formula unit, since this is an ionic compound) of magnesium phosphate.

First, we see ${\text{Mg}}_3$. This tells us that there are $3$ atoms of magnesium.

Next, we see the phosphate ion, ${\text{PO}}_4$, enclosed in parentheses with a subscript $2$ outside: $({\text{PO}}_4{)}_2$.

This is where many students make a silly mistake. The subscript outside the parentheses multiplies everything inside. It is exactly like the distributive property in algebra: $2(x+4y)=2x+8y$.

Inside the parentheses, we have $1$ phosphorus atom and $4$ oxygen atoms. Because the whole group is multiplied by $2$, we actually have $1\times 2=2$ phosphorus atoms, and most importantly for our problem, $4\times 2=8$ oxygen atoms.

So, if we scale this up from a single molecule to a whole mole, the logic remains exactly the same.

This stoichiometric ratio is the absolute key to unlocking the entire problem. Once you have this relationship, the chemistry part is over, and it simply becomes a math puzzle!

A Chemist's Best Friend

Now that we have our master relationship, we can use the unitary method.

The unitary method is a simple but incredibly powerful mathematical tool. If you know the value of a certain quantity, you can find the value of a single unit, and then multiply to find the value of any other quantity. It is the exact same logic you use when shopping: if 8 apples cost 1 dollar, how much do 0.25 apples cost? (Even though you can't buy a quarter of an apple, the math holds up!)

We know that:

But the question doesn't ask about $8$ moles of oxygen. It asks about $0.25$ moles of oxygen.

So, we ask ourselves: if $8$ moles of oxygen come from $1$ mole of the compound, where does $1$ mole of oxygen come from?

By dividing both sides by $8$, we find that:

Now, to find the amount of compound that contains $0.25$ moles of oxygen, we simply multiply this single-unit value by $0.25$:

This logical bridge is flawless. It guarantees that our proportions remain perfectly balanced.

Let's execute the final mathematical step. We need to calculate the value of $\frac{1}{8}\times 0.25$.

While you could use a calculator, it is always better to build your mental math muscles, especially for competitive exams like JEE and NEET where time is of the essence and calculators are often not allowed.

Let's convert the decimal $0.25$ into a fraction. We know that $0.25$ is exactly $\frac{1}{4}$.

Substituting this into our expression, we get:

Multiplying the denominators, $8\times 4=32$. So, our answer is $\frac{1}{32}$.

Now, we just need to convert $\frac{1}{32}$ back into a decimal. We know that $\frac{1}{2}=0.5$. Half of that is $\frac{1}{4}=0.25$. Half of that is $\frac{1}{8}=0.125$. Half of that is $\frac{1}{16}=0.0625$. And finally, half of that is $\frac{1}{32}=0.03125$.

So, the number of moles of the compound is $0.03125$.

To match the options provided in the question, we need to express this in scientific notation. Scientific notation requires a number between 1 and 10 multiplied by a power of 10.

We move the decimal point two places to the right, which means we must multiply by $10^{-2}$. This gives us:

This perfectly matches option (b).

By carefully decoding the chemical formula, establishing the correct stoichiometric ratio, and executing a clean unitary calculation, we have successfully solved the problem.

But don't stop here! Challenge yourself. What if the question had asked for the number of moles of the compound containing $0.5$ moles of phosphorus atoms? You would look at the formula, see that there are $2$ moles of phosphorus per mole of compound, and set up a new unitary ratio: $2\text{ moles P}⟶1\text{ mole compound}$.

Always remember to trust the stoichiometry—it is the map that guides you through the microscopic world! Keep practicing, keep questioning, and the beautiful logic of chemistry will always reveal itself to you.