Animated Solution for Chemistry - Basic Concepts in Chemistry: $8\text{ g}$ of $\text{NaOH}$ is dissolved in $18\text{ g}$ of ${\text{H}}_2\text{O}$. Mole fraction of $\text{NaOH}$ in solution and molality (in ${\text{mol kg}}^{-1}$) of the solution respectively are

Imagine you are in a chemistry lab, holding a beaker. You carefully weigh out exactly $8\text{ g}$ of solid sodium hydroxide ($\text{NaOH}$) and dissolve it into $18\text{ g}$ of pure water (${\text{H}}_2\text{O}$). Our objective is to determine two fundamental concentration terms for this newly formed solution: the mole fraction of $\text{NaOH}$ and the molality of the solution.

Before we can plug numbers into any concentration formulas, we must convert our raw masses into moles. The mole is the bridge between the macroscopic world of grams and the microscopic world of atoms and molecules.

The molar mass of $\text{NaOH}$ is $40{\text{ g mol}}^{-1}$ (since $\text{Na}=23$, $\text{O}=16$, and $\text{H}=1$). For water, the molar mass is $18{\text{ g mol}}^{-1}$. Let's calculate the moles of each component:

Mole Fraction

Mole fraction, denoted by the Greek letter $\chi$ (chi), is a beautifully simple concept. It is the ratio of the number of moles of a specific component to the total number of moles of all components in the solution. Because it is a ratio of moles to moles, it is a unitless quantity.

For our solute, sodium hydroxide, the mole fraction is given by:

Substituting the values we just calculated:

Simplifying this fraction gives us $\frac{1}{6}$, which evaluates to approximately $0.167$.

Molality

Next, we tackle molality ($m$). It is crucial not to confuse molality with molarity ($M$). While molarity depends on the volume of the solution, molality depends strictly on the mass of the solvent. This makes molality incredibly useful in thermodynamics because, unlike volume, mass does not change with temperature.

The formula for molality is:

We already know we have $0.2\text{ mol}$ of $\text{NaOH}$. Our solvent is water, and we have $18\text{ g}$ of it. To use the molality formula, we must convert this mass into kilograms by dividing by $1000$:

Now, we substitute these values into our molality equation:

We have successfully decoded the solution's concentration! The mole fraction of $\text{NaOH}$ is $0.167$, and the molality is $11.11{\text{ mol kg}}^{-1}$.