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.

Short answer

The mass percent of toluene is 28.42%, mole fraction is 0.2517, molality is 4.303 mol/kg, and the molarity is 2.687 mol/L.

Step-by-step solution

Find the mass of toluene and benzene

First, find the mass for toluene and benzene using their volumes and densities. Mass = Volume x Density For toluene: Mass = \(50.0 \mathrm{mL} \times 0.867 \mathrm{g/mL}\) Mass = 43.35 g For benzene: Mass = \(125.0 \mathrm{mL} \times 0.874 \mathrm{g/mL}\) Mass = 109.25 g

Calculate the number of moles of toluene and benzene

Next, find the number of moles using the molar mass of toluene (C7H8) and benzene (C6H6). For toluene (molar mass: 92.14 g/mol): Moles = \(43.35 \mathrm{g} \div 92.14 \mathrm{g/mol}\) Moles = 0.4703 mol For benzene (molar mass: 78.11 g/mol): Moles = \(109.25 \mathrm{g} \div 78.11 \mathrm{g/mol}\) Moles = 1.3987 mol

Calculate the mass percent

Mass percent = \(\frac{\text{mass of solute}}{\text{total mass of solution}} \times 100\) For toluene: Mass percent = \(\frac{43.35 \mathrm{g}}{43.35 \mathrm{g} + 109.25 \mathrm{g}} \times 100\) Mass percent = 28.42%

Calculate the mole fraction

Mole fraction = \(\frac{\text{moles of solute}}{\text{total moles}}\) For toluene: Mole fraction = \(\frac{0.4703 \text{mol}}{0.4703 \text{mol} + 1.3987 \text{mol}}\) Mole fraction = 0.2517

Calculate the molality

Molality = \(\frac{\text{moles of solute}}{\text{mass of solvent (in kg)}}\) For toluene: Molality = \(\frac{0.4703 \mathrm{mol}}{0.10925 \mathrm{kg}}\) Molality = 4.303 mol/kg

Calculate the molarity

Molarity = \(\frac{\text{moles of solute}}{\text{total volume of solution (in L)}}\) For toluene: Total volume of solution = 50.0 mL + 125 mL = 175 mL = 0.175 L Molarity = \(\frac{0.4703 \mathrm{mol}}{0.175 \mathrm{L}}\) Molarity = 2.687 mol/L The mass percent of toluene is 28.42%, mole fraction is 0.2517, molality is 4.303 mol/kg, and the molarity is 2.687 mol/L.

Key concepts

Mass Percent

Understanding the mass percent is pivotal in the study of chemistry, especially in the domain of solutions. It is a way to express the concentration of a component in a mixture or solution. Fundamentally, the mass percent can be described as the mass of the solute divided by the total mass of the solution, then multiplied by 100 to convert it into a percentage. This gives us an easily interpretable figure indicating how much of the total solution is composed of the specific component.

For instance, when we say the mass percent of toluene in a solution is 28.42%, this indicates that for every 100 grams of the solution, 28.42 grams are toluene. This concept is particularly useful for practical applications such as preparing solutions in a lab, or calculating reagents for a reaction.

Mole Fraction

Diving into mole fraction, we enter the realm of quantities based on the number of molecules rather than their mass. The mole fraction is defined as the ratio of the number of moles of a component to the total number of moles of all components in the mixture. It’s a dimensionless quantity since it's a ratio of similar units, and it helps us understand the proportion of the molecules of one substance relative to the total in the mixture.

In our exercise, the mole fraction of toluene equals 0.2517, which means that toluene constitutes approximately 25.17% of the total moles in the solution. Remember that mole fractions always add up to one when you consider all the components of a mixture. It's often used alongside mass percent to provide a comprehensive understanding of a solution's makeup.

Molality

Moving on to molality, another concentration term frequently used in chemistry, which is especially useful when dealing with temperature changes. Molality is defined as the number of moles of solute per kilogram of solvent. Unlike molarity (which we'll discuss next), molality is not impacted by changes in temperature since mass remains constant regardless of temperature fluctuations.

To calculate it, you take the number of moles of your solute and divide it by the mass of the solvent in kilograms, leading to units of moles per kilogram (mol/kg). For instance, a molality of 4.303 mol/kg for toluene suggests a relatively concentrated solution, and this measure is crucial when dealing with boiling point elevation and freezing point depression.

Molarity

Finally, we explore molarity, one of the most common units of concentration in chemistry. Molarity is denoted as the number of moles of a solute per liter of solution. This concentration metric is volume-based, making it susceptible to changes in temperature, as liquids expand or contract with temperature changes.

The formula involves dividing the number of moles of solute by the total volume of the solution in liters. For example, with a molarity of 2.687 mol/L for toluene, one can infer that each liter of the total solution contains 2.687 moles of toluene. Molarity is highly valuable for stoichiometric calculations in chemical reactions because it directly relates to the volume of solution used.