Problem 34 - Chemistry An Atoms First Approach

Question

A solution is prepared by mixing \(50.0 \mathrm{mL}\) toluene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{3},\right.\) \(d=0.867 \mathrm{g} / \mathrm{cm}^{3}\) with \(125 \mathrm{mL}\) benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}, d=0.874 \mathrm{g} / \mathrm{cm}^{3}\right)\) Assuming that the volumes add on mixing, calculate the mass percent, mole fraction, molality, and molarity of the toluene.

Step-by-Step Solution

Answer

The properties of the toluene solution are: mass percent of toluene is 28.4%, mole fraction of toluene is 0.251, molality of toluene is 4.30 mol/kg, and molarity of toluene is 2.68 mol/L.

Step 1: Calculate the masses of toluene and benzene

Use the volumes and densities of toluene and benzene to calculate their respective masses: Mass of toluene = Volume of toluene × Density of toluene Mass of toluene = \(50.0 \mathrm{mL} \times 0.867 \frac{\mathrm{g}}{\mathrm{cm}^3} = 43.35 \mathrm{g}\) Mass of benzene = Volume of benzene × Density of benzene Mass of benzene = \(125 \mathrm{mL} \times 0.874 \frac{\mathrm{g}}{\mathrm{cm}^3}= 109.25 \mathrm{g}\)

Step 2: Calculate the moles of toluene and benzene

Using the molar masses of toluene (M_toluene = \(92.14\ \mathrm{g/mol}\)) and benzene (M_benzene = \(78.11\ \mathrm{g/mol}\)), calculate the moles of each substance: Moles of toluene = \(\frac{\mathrm{Mass}\ \mathrm{of}\ \mathrm{toluene}}{\mathrm{Molar}\ \mathrm{mass}\ \mathrm{of}\ \mathrm{toluene}} = \frac{43.35 \mathrm{g}}{92.14 \frac{\mathrm{g}}{\mathrm{mol}}}= 0.470 \mathrm{mol}\) Moles of benzene = \(\frac{\mathrm{Mass}\ \mathrm{of}\ \mathrm{benzene}}{\mathrm{Molar}\ \mathrm{mass}\ \mathrm{of}\ \mathrm{benzene}} = \frac{109.25 \mathrm{g}}{78.11 \frac{\mathrm{g}}{\mathrm{mol}}}= 1.398 \mathrm{mol}\)

Step 3: Calculate the mass percent of toluene

Mass percent of toluene = \(\frac{\mathrm{Mass}\ \mathrm{of}\ \mathrm{toluene}}{\mathrm{Total}\ \mathrm{mass}\ \mathrm{of}\ \mathrm{solution}} \times 100\% = \frac{43.35 \mathrm{g}}{(43.35 \mathrm{g} + 109.25 \mathrm{g})} \times 100\% = 28.4\%\)

Step 4: Calculate the mole fraction of toluene

Mole fraction of toluene (X_toluene) = \(\frac{\mathrm{Moles}\ \mathrm{of}\ \mathrm{toluene}}{\mathrm{Total}\ \mathrm{moles}\ \mathrm{of}\ \mathrm{solution}} = \frac{0.470 \mathrm{mol}}{(0.470 \mathrm{mol} + 1.398 \mathrm{mol})} = 0.251\)

Step 5: Calculate the molality of toluene

Molality of toluene = \(\frac{\mathrm{Moles}\ \mathrm{of}\ \mathrm{toluene}}{\mathrm{Mass}\ \mathrm{of}\ \mathrm{benzene}\ \mathrm{in}\ \mathrm{kg}} = \frac{0.470 \mathrm{mol}}{0.10925 \mathrm{kg}} = 4.30\ \frac{\mathrm{mol}}{\mathrm{kg}}\)

Step 6: Calculate the molarity of toluene

Molarity of toluene = \(\frac{\mathrm{Moles}\ \mathrm{of}\ \mathrm{toluene}}{\mathrm{Total}\ \mathrm{volume}\ \mathrm{of}\ \mathrm{solution}\ \mathrm{in}\ \mathrm{L}} = \frac{0.470 \mathrm{mol}}{\frac{175 \mathrm{mL}}{1000}} = 2.68\ \frac{\mathrm{mol}}{\mathrm{L}}\) Finally, we have the following properties of the toluene solution: 1. Mass percent of toluene: 28.4% 2. Mole fraction of toluene: 0.251 3. Molality of toluene: 4.30 mol/kg 4. Molarity of toluene: 2.68 mol/L

Key Concepts

Mass Percent

Understanding the mass percent of a substance in a solution is fundamental in chemistry. It defines the concentration of a component by the proportion of its mass to the total mass of the solution. To calculate the mass percent, use the formula:
\[ \text{Mass percent} = \left(\frac{\text{Mass of solute}}{\text{Total mass of solution}}\right) \times 100\% \]
In the given exercise, the mass percent of toluene is found by dividing the mass of toluene by the total mass of toluene and benzene together. Multiplying this ratio by 100 gives us the concentration in terms of mass percentage. This concept is particularly useful in industries like pharmacology and food science where precise composition is crucial.

Mole Fraction

The mole fraction provides a way of expressing the ratio of the number of moles of one component to the total number of moles in a mixture. The formula is:
\[ \text{Mole fraction} = \frac{\text{Moles of component}}{\text{Total moles in mixture}} \]
One can see it as a measure of the 'participation' of each component in the total amount of substance. The mole fraction is dimensionless and always less than or equal to one. It's an essential concept for understanding colligative properties and chemical equilibria as it helps to relate the amounts of various substances involved.

Molality

Molality is a measure of the concentration of a solute in a solution based on the mass of the solvent. It is calculated as:
\[ \text{Molality} = \frac{\text{Moles of solute}}{\text{Mass of solvent in kilograms}} \]
Unlike molarity, molality is not affected by temperature changes, because it is based on mass rather than volume. This makes it a valuable variable for use in situations where temperature can vary significantly, such as in chemical thermodynamics and kinetics.

Molarity

Molarity is perhaps the most commonly used unit of concentration in chemistry. The molarity of a solution is the number of moles of solute per liter of solution and is expressed as:
\[ \text{Molarity} = \frac{\text{Moles of solute}}{\text{Volume of solution in liters}} \]
It directly relates the volume of a solution to the amount of substance it contains, making it extremely useful in stoichiometry for reactions in the solution phase. Moreover, molarity changes with temperature due to the expansion or contraction of the solution's volume, so it's most accurate at a specified temperature.