Animated Solution for Chemistry - Basic Concepts in Chemistry: 100 mL of ${\text{Na}}_3{\text{PO}}_4$ solution contains 3.45 g of sodium. The molarity of the solution is ...... $\times 10^{-2}{\text{ mol L}}^{-1}$. (Nearest integer) [Atomic masses - Na = 23.0 u, O = 16.0 u, P = 31.0 u]

Have you ever wondered how chemists figure out the concentration of an entire compound when they only know the mass of one specific element inside it? It’s like trying to guess the number of whole cars in a parking lot when you only know the total number of tires!

In this problem, we are given a $100\text{ mL}$ solution of sodium phosphate (${\text{Na}}_3{\text{PO}}_4$) and told that it contains exactly $3.45\text{ g}$ of sodium ($\text{Na}$). Our mission is to find the molarity of the entire ${\text{Na}}_3{\text{PO}}_4$ solution. Let's break this down step-by-step.

Before we can talk about the whole molecule, we need to understand what we have. We are given the mass of sodium. In chemistry, mass is just a stepping stone to the real currency: moles.

To find the number of moles of sodium, we use the classic formula:

We plug in our given values. The mass of sodium is $3.45\text{ g}$, and its atomic mass is $23\text{ g/mol}$.

Calculating this gives us:

So, we have $0.15\text{ moles}$ of sodium atoms swimming around in our beaker.

Now comes the crucial part. We don't want the concentration of just sodium; we want the concentration of the entire sodium phosphate molecule, ${\text{Na}}_3{\text{PO}}_4$.

Look closely at the chemical formula. The subscript "3" next to Na tells us a very important story: For every one molecule of ${\text{Na}}_3{\text{PO}}_4$, there are exactly three atoms of Na.

This means the number of moles of the complete molecule will be one-third the number of moles of sodium. We can write this relationship as:

Substituting the moles of sodium we just found:

Which gives us:

We now know exactly how much of the whole compound is in the solution!

The final step is to find the molarity. Molarity is defined as the number of moles of solute per liter of solution.

Our volume is given as $100\text{ mL}$. To use the molarity formula, we must convert this to liters by dividing by $1000$, or simply multiplying the whole expression by $1000$.

Let's plug in our moles and volume:

The math here is beautifully simple. The zeros cancel out, leaving us with:

But wait, the question has a specific format requirement! It asks for the answer in the form of $x\times 10^{-2}$.

We can easily rewrite $0.50$ as:

Therefore, our integer $x$ is $50$.

And just like that, by following the trail from mass to moles, and from atoms to molecules, we've cracked the code!