Question

A solution contains 0.100 mol of \(\mathrm{NaCl}\) dissolved in \(8.60 \mathrm{~mol}\) of water. (a) What is the mole fraction of \(\mathrm{NaCl} ?\) (b) The mass percent? (c) The molality?

Short answer

The mole fraction of NaCl is 0.0115, the mass percent is 3.63%, and the molality is 0.645 mol/kg.

Step-by-step solution

Calculate the mole fraction of NaCl

The mole fraction (\text{X}) of NaCl is given by \[ X_{\text{NaCl}} = \frac{\text{moles of NaCl}}{\text{moles of NaCl} + \text{moles of water}} \] \[ X_{\text{NaCl}} = \frac{0.100}{0.100 + 8.60} = \frac{0.100}{8.70} \] \[ X_{\text{NaCl}} = 0.0115 \]

Calculate the mass percent

Mass percent is calculated using the formula \[ \text{Mass percent} = \frac{\text{mass of solute}}{\text{mass of solution}} \times 100 \] First, calculate the mass of each component. Moles of NaCl = 0.100 mol, and the molar mass of NaCl is 58.44 g/mol, so mass of NaCl = 0.100 \times 58.44 = 5.844 g. \ Similarly, moles of water = 8.60 mol, and the molar mass of water is 18.015 g/mol, so mass of water = 8.60 \times 18.015 = 154.929 g. \ Thus, the total mass of the solution = 5.844 g + 154.929 g = 160.773 g. Mass percent of NaCl = \[ \text{Mass percent} = \frac{5.844}{160.773} \times 100 = 3.63\text{\text{\text{\text{\text{} \(\backslash\)%}}}} \]

Calculate the molality

Molality (\text{m}) is the number of moles of solute per kilogram of solvent. \ Moles of NaCl = 0.100 mol. \ Mass of water = 154.929 g = 0.154929 kg. \ Thus, \[ \text{molality} = \frac{0.100}{0.154929} = 0.645 \text{ mol/kg} \]

Key concepts

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It represents the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture. In this exercise, we are calculating the mole fraction of NaCl in a solution. The formula is:
\[ X_{\text{NaCl}} = \frac{\text{moles of NaCl}}{\text{moles of NaCl} + \text{moles of water}} \]
Simply put, you divide the moles of NaCl by the sum of the moles of NaCl and water.
Using the given values:
\[ X_{\text{NaCl}} = \frac{0.100}{0.100 + 8.60} = \frac{0.100}{8.70} \]
We get:
\[ X_{\text{NaCl}} = 0.0115 \]
This means that NaCl makes up 1.15% of the total moles in the solution.

Mass Percent

Mass percent is another way of expressing concentration, specifically, the mass of the solute divided by the total mass of the solution, multiplied by 100 to get a percentage. The formula is:
\[ \text{Mass percent} = \frac{\text{mass of solute}}{\text{mass of solution}} \times 100 \]
First, we calculate the mass of each component in the solution. For NaCl, we know the moles and can use its molar mass to find the mass:
\[ 0.100 \text{ mol} \times 58.44 \text{ g/mol} = 5.844 \text{ g} \]
For water, we do the same:
\[ 8.60 \text{ mol} \times 18.015 \text{ g/mol} = 154.929 \text{ g} \]
Adding these gives the total mass of the solution:
\[ 5.844 \text{ g} + 154.929 \text{ g} = 160.773 \text{ g} \]
Now, we can calculate the mass percent of NaCl:
\[ \text{Mass percent} = \frac{5.844}{160.773} \times 100 = 3.63\text{\text{\text{\text{\text{} \text{ percent}}}}} \]
So, NaCl constitutes 3.63% of the total mass of the solution.

Molality

Molality is a measure of the concentration of a solute in a solution, defined as the number of moles of solute per kilogram of solvent. The formula is:
\[ \text{molality} = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]
In this exercise, NaCl is the solute and water is the solvent. We already have the moles of NaCl (0.100 mol). Next, we need to convert the mass of water to kilograms:
\[ 154.929 \text{ g} = 0.154929 \text{ kg} \]
Now, we can calculate the molality:
\[ \text{molality} = \frac{0.100}{0.154929} = 0.645 \text{ mol/kg} \]
Thus, the molality of the NaCl solution is 0.645 mol/kg, meaning there are 0.645 moles of NaCl for every kilogram of water in the solution.