Animated Solution for Chemistry - Basic Concepts in Chemistry: 4g equimolar mixture of NaOH and ${\text{Na}}_2{\text{CO}}_3$ contains $x$ g of NaOH and $y$ g of ${\text{Na}}_2{\text{CO}}_3$. The value of $x$ is …… g. (Nearest integer)

The Magic of Equimolar Mixtures

Imagine you are standing in a chemistry lab. You have a beaker resting on a highly precise weighing scale, and the digital display reads exactly $4.00\text{ g}$. Inside this beaker is a mixture of two distinct compounds: sodium hydroxide ($\text{NaOH}$) and sodium carbonate (${\text{Na}}_2{\text{CO}}_3$).

The problem tells us that the mass of $\text{NaOH}$ is $x\text{ g}$, and the mass of ${\text{Na}}_2{\text{CO}}_3$ is $y\text{ g}$. Naturally, the sum of the masses of the individual components must equal the total mass of the mixture. This gives us our very first, intuitive mathematical relationship:

Now, we must look closely at the terminology used in the question. There is a magic word here: equimolar. What does equimolar actually mean in the physical world? It means that despite having different masses, the number of molecules (or moles) of both substances in the mixture is exactly the same. This is the master key to unlocking the entire problem.

We know from the fundamental principles of stoichiometry that the number of moles ($n$) is calculated by dividing the given mass by the molar mass of the substance. Therefore, we can equate the moles of sodium hydroxide to the moles of sodium carbonate:

Before we can substitute our unknown masses into this mole equation, we need to determine the molar masses of both compounds. Let's calculate them quickly. For sodium hydroxide, we add the atomic masses of sodium, oxygen, and hydrogen: $23+16+1=40\text{ g/mol}$. For sodium carbonate, the calculation is slightly longer: $2(23)+12+3(16)=106\text{ g/mol}$.

Substituting these values into our equimolar condition, we get a beautiful ratio:

We now have a system of two linear equations with two variables. The most elegant way to solve this is by the method of substitution. Let's express $y$ in terms of $x$ using our mole equation. By rearranging the terms, we find:

Simplifying that fraction, we get:

This equation tells us exactly how the mass of sodium carbonate scales with the mass of sodium hydroxide in this specific mixture. Now, let's bring back our very first mass equation, $x+y=4$, and replace $y$ with the expression we just derived.

Adding the like terms on the left side, we successfully reduce our problem to a simple linear equation with just one variable:

Finally, let's solve for $x$. Dividing both sides by $3.65$, we get:

When you calculate this division, it comes out to approximately $\mathrm{1.0958...}\text{ g}$. But wait, we must always respect the constraints of the question! The examiner specifically asks for the answer rounded to the nearest integer.

Rounding $1.096$ to the nearest whole number gives us exactly $1$. Therefore, the mass of sodium hydroxide in the mixture is:

Before we wrap up, let's think about a fascinating variation of this problem. What if the examiner had used the word equimass instead of equimolar? In an equimass mixture, the masses would be identical, meaning $x$ would simply equal $y$, and both would be exactly $2\text{ g}$. It is a tiny change in wording, but it completely alters the mathematical reality of the mixture. Always read these prefix terms very carefully!