Problem 49 - Chemistry: The Central Science

Question

The density of acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) is \(0.786 \mathrm{~g} / \mathrm{mL}\) and the density of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is \(0.791 \mathrm{~g} / \mathrm{mL}\). A solution is made by dissolving \(22.5 \mathrm{~mL}\) of \(\mathrm{CH}_{3} \mathrm{OH}\) in \(98.7 \mathrm{~mL}\) of \(\mathrm{CH}_{3} \mathrm{CN}\). (a) What is the mole fraction of methanol in the solution? (b) What is the molality of the solution? (c) Assuming that the volumes are additive, what is the molarity of \(\mathrm{CH}_{3} \mathrm{OH}\) in the solution?

Step-by-Step Solution

Answer

The mole fraction of methanol in the solution is 0.2272, the molality of the solution is 7.17 mol/kg, and the molarity of methanol in the solution is approximately 4.583 M.

Step 1: 1. Calculate the mass of each substance

To determine the mass of each substance in the solution, multiply the volume by the corresponding density: (a) Mass of acetonitrile (\(CH_3CN\)): \(Mass_{CH_3CN} = Volume_{CH_3CN} \times Density_{CH_3CN}\) \(Mass_{CH_3CN} = 98.7\:mL \times 0.786\:g/mL\) \(Mass_{CH_3CN} = 77.53\:g\) (b) Mass of methanol (\(CH_3OH\)): \(Mass_{CH_3OH} = Volume_{CH_3OH} \times Density_{CH_3OH}\) \(Mass_{CH_3OH} = 22.5\:mL \times 0.791\:g/mL\) \(Mass_{CH_3OH} = 17.795\:g\)

Step 2: 2. Calculate the moles of each substance

Next, convert the mass of each substance to moles using the molar mass: Molar mass of acetonitrile: \(41.05\:g/mol\) Molar mass of methanol: \(32.04\:g/mol\) Moles of acetonitrile: \(\text{Moles}_{CH_3CN} = \frac{Mass_{CH_3CN}}{Molar \:mass_{CH_3CN}}\) \(\text{Moles}_{CH_3CN} = \frac{77.53\:g}{41.05\:g/mol} = 1.887\:mol\) Moles of methanol: \(\text{Moles}_{CH_3OH} = \frac{Mass_{CH_3OH}}{Molar \:mass_{CH_3OH}}\) \(\text{Moles}_{CH_3OH} = \frac{17.795\:g}{32.04\:g/mol} = 0.5558\:mol\)

Step 3: 3. Calculate the mole fraction of methanol

To determine the mole fraction of methanol, divide the moles of methanol by the total moles of the solution: Mole fraction of methanol (\(X_{CH_3OH}\)): \(X_{CH_3OH} = \frac{Moles_{CH_3OH}}{Moles_{CH_3OH} + Moles_{CH_3CN}}\) \(X_{CH_3OH} = \frac{0.5558\:mol}{0.5558\:mol + 1.887\:mol} = 0.2272\) So, the mole fraction of methanol is 0.2272.

Step 4: 4. Calculate the molality of the solution

Now, calculate the molality of methanol in the solution by dividing the moles of methanol by the mass (in kg) of the solvent (acetonitrile): Molality of methanol (\(b_{CH_3OH}\)): \(b_{CH_3OH} = \frac{Moles_{CH_3OH}}{Mass_{CH_3CN} \:(kg)}\) \(b_{CH_3OH} = \frac{0.5558\:mol}{0.07753\:kg} \approx 7.17\:m\) The molality of the solution is 7.17 mol/kg.

Step 5: 5. Calculate the molarity of methanol

Finally, calculate the molarity of methanol by dividing the moles of methanol by the total volume of the solution (in liters). We're given that the volumes are additive, so the total volume is \(98.7\:mL + 22.5\:mL = 121.2\:mL = 0.1212\:L\). Molarity of methanol (\(M_{CH_3OH}\)): \(M_{CH_3OH} = \frac{Moles_{CH_3OH}}{Volume_{solution} \:(L)}\) \(M_{CH_3OH} = \frac{0.5558\:mol}{0.1212\:L} \approx 4.583\:M\) The molarity of methanol in the solution is approximately 4.583 M.

Key Concepts

Mole Fraction

To understand the mole fraction, think of it as the fraction of moles of a component compared to the total moles in the solution. It's a way to express the concentration of each component in a solution without considering the volume or mass. The mole fraction is dimensionless, which means it has no units.

• First, calculate the moles of each substance. This is done by dividing the mass of each substance by its molar mass.
• Then, find the total moles by summing the moles of all components present in the solution.
• Finally, compute the mole fraction by dividing the moles of the component by the total moles.

Calculating mole fraction is useful in mixtures where you need a ratio that does not rely on volumes or masses. For our methanol solution, the mole fraction of methanol helps in understanding its proportionate presence in the mixture without being influenced by the entire solution volume or density.

Molality

Molality is another way to express the concentration of a solution. Unlike molarity, molality is concerned with the mass of the solvent and is expressed as moles of solute per kilogram of solvent. This measure is particularly useful when dealing with temperature and pressure changes since it does not vary with temperature.

• First, determine the moles of the solute you are interested in analyzing - in this case, methanol.
• Next, measure the mass of the solvent (acetonitrile in this case) in kilograms.
• Finally, calculate the molality by dividing the moles of solute by the mass of the solvent (in kg).

Molality provides a consistent measure of concentration, especially useful in scenarios where temperature can affect volume but not mass. By knowing the molality of the methanol, you can predict how the solution will behave in different temperatures.

Molarity

Molarity refers to the number of moles of solute present in one liter of solution. It's a very common concentration measure used in chemistry. Unlike molality, molarity depends on the total volume of the solution, which includes both solute and solvent.

• First, calculate the moles of the solute, using the same method as before - dividing the mass of the solute by its molar mass.
• Next, find the total volume of the solution in liters. This volume includes both solute and solvent.
• Finally, determine the molarity by dividing the moles of solute by the total solution volume.

Molarity is helpful in reactions where volumes are additive, which simplifies the understanding of the solution's concentration. It also aligns well with volume-based laboratory measurements. For methanol, its molarity in the solution illustrates how much of it is contained in a given volume of the overall solution.