Question

A solution is prepared by mixing 25 mL pentane \(\left(\mathrm{C}_{5} \mathrm{H}_{12}, d=\right.\) \(\left.0.63 \mathrm{g} / \mathrm{cm}^{3}\right)\) with \(45 \mathrm{mL}\) hexane \(\left(\mathrm{C}_{6} \mathrm{H}_{14}, d=0.66 \mathrm{g} / \mathrm{cm}^{3}\right)\) Assuming that the volumes add on mixing, calculate the mass percent, mole fraction, molality, and molarity of the pentane.

Short answer

The mass percent of pentane in the solution is approximately 34.65%, the mole fraction of pentane is approximately 0.387, the molality of pentane is approximately 7.34 mol/kg, and the molarity of pentane is approximately 3.11 mol/L.

Step-by-step solution

Calculate the mass of pentane and hexane

To calculate the mass of pentane and hexane, we use the given densities. Mass = Density × Volume For pentane: Mass of pentane (C5H12) = (0.63 g/cm^3) × (25 mL) = 15.75 g For hexane: Mass of hexane (C6H14) = (0.66 g/cm^3) × (45 mL) = 29.7 g

Calculate the moles of pentane and hexane

To calculate the number of moles, we use the molecular formula and the molar mass of pentane and hexane. Moles = Mass/Molar Mass For pentane (C5H12): Molar mass of pentane = (5 × 12.01) + (12 × 1.01) = 72.15 g/mol Moles of pentane = 15.75 g / 72.15 g/mol ≈ 0.218 mol For hexane (C6H14): Molar mass of hexane = (6 × 12.01) + (14 × 1.01) = 86.17 g/mol Moles of hexane = 29.7 g / 86.17 g/mol ≈ 0.345 mol

Calculate the mass percent of pentane

Mass percent is calculated with this equation: Mass percent = (Mass of component/Total mass) × 100 Mass percent of pentane = (15.75 g / (15.75 g + 29.7 g)) × 100 ≈ 34.65%

Calculate the mole fraction of pentane

Mole fraction is calculated with this equation: Mole fraction = Moles of component/Total moles Mole fraction of pentane = 0.218 mol / (0.218 mol + 0.345 mol) ≈ 0.387

Calculate the molality of pentane

Molality is calculated with this equation: Molality = Moles of solute/Mass of solvent (in kg) Molality of pentane = 0.218 mol / (29.7 g / 1000) ≈ 7.34 mol/kg

Calculate the molarity of pentane

Molarity is calculated with this equation: Molarity = Moles of solute/Total volume of solution (in L) Total volume of the solution = 25 mL + 45 mL = 70 mL = 0.07 L Molarity of pentane = 0.218 mol / 0.07 L ≈ 3.11 mol/L The mass percent of pentane is approximately 34.65%, the mole fraction of pentane is approximately 0.387, the molality of pentane is approximately 7.34 mol/kg, and the molarity of pentane is approximately 3.11 mol/L.

Key concepts

Mass Percent

When dealing with solutions in chemistry, understanding the composition is crucial. One such measure is the mass percent, which indicates the mass of a solute in relation to the total mass of the solution. It is expressed as a percentage, showing how much of the solution’s weight is due to the solute.

To calculate the mass percent, you simply divide the mass of the solute by the total mass of the solution and then multiply by 100. For example, if you dissolve 5 grams of salt in 95 grams of water, the mass percent of salt would be \( \frac{5g}{5g + 95g} \times 100 = 5\% \). This measure is particularly helpful in scenarios where temperature does not affect the solution's volume, since the mass does not change with temperature like volume might.

Mole Fraction

Another way to express the concentration of a solution is by using the mole fraction. As its name implies, mole fraction represents the ratio of moles of a particular compound to the total moles of all substances in the solution. A mole fraction is a unitless number, which showcases the proportion of the solute in the mixture at the molecular level.

To find the mole fraction, you would take the amount (in moles) of the compound of interest and divide it by the sum of moles of all the components present. For instance, if a solution is composed of 1 mole of ethanol mixed with 3 moles of water, the mole fraction of ethanol is \( \frac{1}{1+3} = 0.25 \), meaning ethanol comprises 25% of the total moles in the solution. This method is particularly useful when dealing with gas mixtures and colligative properties.

Molality

Molality measures the concentration of a solution by comparing the number of moles of solute to the mass of the solvent in kilograms. Importantly, molality is not affected by changes in temperature, making it a valuable measure for experiments that involve temperature variations.

To calculate molality, one would divide the moles of solute by the mass of the solvent in kilograms. For example, if you have 2 moles of sugar dissolved in 1 kilogram of water, the solution's molality is \( \frac{2 \, \text{moles}}{1 \, \text{kg}} = 2 \, \text{mol/kg} \). Because it is temperature-independent, molality is often used in boiling point elevation and freezing point depression calculations.

Molarity

Lastly, molarity is one of the most common ways to express the concentration of a solution in chemistry. Molarity is defined by the number of moles of solute divided by the total volume of the solution in liters. It emphasizes the volume aspect of concentration, which is particularly useful for reactions occurring in liquid phase.

The formula for molarity is \( \frac{\text{moles of solute}}{\text{liters of solution}} \). For example, if a solution contains 3 moles of sodium chloride in 2 liters of water, the molarity would be \( \frac{3 \, \text{moles}}{2 \, \text{L}} = 1.5 \, \text{M} \), where 'M' stands for molar. Note that molarity can be influenced by temperature since it depends on the volume of the solution, which can expand or contract with temperature changes.