Question

A solution is prepared by mixing \(25 \mathrm{~mL}\) 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 properties of the solution prepared by mixing 25 mL of pentane and 45 mL of hexane are as follows: Mass percent of pentane is approximately \(34.67\%\), mole fraction of pentane is approximately \(0.387\), molality of pentane is approximately \(7.34 \: \text{mol/kg}\), and molarity of pentane is approximately \(3.27 \: \text{mol/L}\).

Step-by-step solution

Calculate the mass of pentane and hexane

First, we need to find the mass of both pentane and hexane in the solution. We have the volume and density of both substances. Use the formula mass = volume × density for each: For pentane, mass = 25 mL × 0.63 g/mL = 15.75 g For hexane, mass = 45 mL × 0.66 g/mL = 29.7 g

Calculate the moles of pentane and hexane

Now we need to find the number of moles for pentane and hexane. Divide the mass (m) by the molar mass (M) for each: For pentane (C5H12), M = (5 × 12.01) + (12 × 1.008) = 72.15 g/mol moles of pentane = 15.75 g / 72.15 g/mol ≈ 0.218 mol For hexane (C6H14), M = (6 × 12.01) + (14 × 1.008) = 86.18 g/mol moles of hexane = 29.7 g / 86.18 g/mol ≈ 0.345 mol

Calculate the mass percent of pentane

To find the mass percent of pentane in the solution, divide the mass of pentane by the total mass (mass of pentane + mass of hexane) and multiply by 100: Mass percent of pentane = (15.75 g / (15.75 g + 29.7 g)) × 100 ≈ 34.67%

Calculate the mole fraction of pentane

To find the mole fraction of pentane, divide the moles of pentane by the total moles (moles of pentane + moles of hexane): Mole fraction of pentane = 0.218 mol / (0.218 mol + 0.345 mol) ≈ 0.387

Calculate the molality of pentane

To find the molality of pentane, divide the moles of pentane by the mass of hexane in kilograms: Molality of pentane = 0.218 mol / (29.7 g × 1 kg/1000 g) ≈ 7.34 mol/kg

Calculate the molarity of pentane

To find the molarity of pentane, divide the moles of pentane by the total volume of the solution (volume of pentane + volume of hexane) in liters: Molarity of pentane = 0.218 mol / ((25 mL + 45 mL) × 1 L/1000 mL) ≈ 3.27 mol/L Now, we have found all the required properties of the solution: Mass percent of pentane ≈ 34.67% Mole fraction of pentane ≈ 0.387 Molality of pentane ≈ 7.34 mol/kg Molarity of pentane ≈ 3.27 mol/L

Key concepts

Mass Percent

Understanding the mass percent concept is essential when studying solutions in chemistry. Mass percent, also known as mass concentration or weight percent, is a way of expressing the concentration of a component in a mixture or solution. It represents the mass of the solute (in this case, pentane) divided by the total mass of the solution and then multiplied by 100 to get a percentage. In simple terms, it answers the question: 'What percentage of the total weight of this solution is made up of the solute?'

To calculate mass percent, use the formula:
\[\text{Mass percent} = \left(\frac{\text{mass of solute}}{\text{mass of solute} + \text{mass of solvent}}\right) \times 100\%\]

• First, evaluate the mass of the solute (the substance being dissolved) and the solvent (substance in which the solute is being dissolved).
• Next, add the masses of solute and solvent to get the total mass of the solution.
• Then, divide the mass of the solute by the total mass and multiply by 100 to obtain the percentage. This value shows how concentrated the solute is in the solution.

Using the mass percent calculation helps chemists to quickly understand the ratio of substances in a solution which is fundamental in fields such as pharmacology, where proper dosages are crucial.

Mole Fraction

The mole fraction is another critical concept used to measure the concentration of a solution. Unlike mass percent, which is based on the weight, mole fraction relies on the number of moles, which correlates to the quantity of molecules or atoms. The mole fraction of a component in a solution is the ratio of the number of moles of that component to the total number of moles of all components in the solution.

To calculate mole fraction, the formula is:
\[\text{Mole fraction} = \frac{\text{moles of component}}{\text{total moles of all components}}\]

• Determine the moles of each substance present in the solution.
• Add up the moles of all substances to find the total moles in the solution.
• Divide the moles of the component of interest by the total moles to obtain its mole fraction.

The mole fraction is unitless and always less than or equal to one. It is particularly useful in gas mixtures and when dealing with reactions at the molecular level because it directly relates to the number of particles involved.

Molality

Molality is a measure of solution concentration expressed in moles of solute per kilogram of solvent. This concentration unit is temperature-independent, unlike some other measures, because mass does not change with temperature. This is particularly advantageous when working with reactions or processes that involve significant temperature changes.

The formula for molality is:
\[\text{Molality (m)} = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\]

• It's important to note that the weight here is of the solvent only, not the total solution.
• The mass of the solvent must be converted to kilograms to match the units of molality (mole/kg).

Molality is especially relevant in colligative properties like boiling point elevation and freezing point depression, where the effect on the solution is directly proportional to the number of solute particles rather than their mass.

Molarity

Molarity is perhaps the most commonly used unit of concentration in chemistry. It is defined as the number of moles of solute per liter of solution. This measure is dependent on the volume of the solution, which can change with temperature due to the expansion and contraction of liquids. Thus, molarity is generally used at a specified temperature.

The formula for molarity is:
\[\text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}}\]

• First, calculate the number of moles of the solute present in the solution.
• Next, determine the total volume of the solution in liters.
• Divide the moles of solute by the volume of the solution to obtain its molarity.

Because molarity directly relates the volume of the solution to the amount of solute, it is incredibly useful for stoichiometric calculations in chemical reactions that take place in solution. Molarity is also essential when preparing solutions for titrations, a common method for determining the concentration of an unknown solution.