Animated Solution for Chemistry - Basic Concepts in Chemistry: A $100\text{ mL}$ solution was made by adding $1.43\text{ g}$ of ${\text{Na}}_2{\text{CO}}_3\cdot x{\text{H}}_2\text{O}$. The normality of the solution is $0.1\text{ N}$. The value of $x$ is ......... .

Have you ever looked at a chemical formula with an unknown variable like $x$ and wondered how to crack it? In this problem, we are given a solution of a hydrated salt, ${\text{Na}}_2{\text{CO}}_3\cdot x{\text{H}}_2\text{O}$, and we need to find the exact number of water molecules of crystallization, represented by $x$.

This is a classic physical chemistry puzzle that tests our understanding of concentration terms, specifically Normality. Let's break it down step-by-step and uncover the identity of this mysterious salt.

Imagine you are in a laboratory. You carefully weigh out $1.43\text{ g}$ of our unknown hydrated salt and dissolve it in water to make exactly $100\text{ mL}$ of solution.

We are given a crucial piece of information: the final normality of this solution is $0.1\text{ N}$. To use this information, we first need to ensure our units are consistent. The volume must be in liters for the normality formula.

Molar Mass and Equivalent Weight

Before we can use the normality formula, we need to understand the chemical properties of our solute. First, let's find its molar mass in terms of $x$. The molar mass is the sum of the atomic masses of all atoms in the formula:

Now, normality deals with equivalent weight, not just molar mass. The bridge between them is the n-factor. For a salt like sodium carbonate (${\text{Na}}_2{\text{CO}}_3$), the n-factor is the total positive charge on the cations. Since it dissociates into two ${\text{Na}}^{+}$ ions, the n-factor is $2$. The equivalent weight ($E$) is the molar mass divided by the n-factor:

With the equivalent weight expressed in terms of $x$, we can now calculate the number of gram equivalents ($G_{eq}$) of the solute we added to the beaker.

Now, we bring in the master equation for normality. Normality is defined as the number of gram equivalents of solute per liter of solution: Substitute the known values into this formula:

The chemistry is done; now it's just algebra. Let's solve for $x$. First, multiply both sides by the volume ($0.1\text{ L}$) to clear the denominator:

Next, rearrange the equation to isolate the term with $x$: Dividing by $0.01$ is the same as multiplying by $100$, so:

And there we have it! The value of $x$ is exactly $10$. Our mysterious salt is sodium carbonate decahydrate, commonly known as washing soda. By carefully applying the concepts of equivalent weight and normality, we've successfully decoded the chemical formula.