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 \(\mathrm{g} / \mathrm{mL} . \mathrm{A}\) 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?

Short answer

To find the mole fraction, molality, and molarity of methanol (CH₃OH) in the given solution, follow these steps: 1. Calculate the mass of each component using their densities and volumes. 2. Calculate the moles of each component using their molar masses. 3. Calculate the mole fraction of methanol by dividing its moles by the total moles of the solution. 4. Calculate the molality of methanol by dividing its moles by the mass of the solvent (acetonitrile, CH₃CN) in kilograms. 5. Calculate the molarity of methanol by dividing its moles by the total volume of the solution in liters.

Step-by-step solution

Calculate the mass of each component

Since we know the density and volume of both acetonitrile (CH₃CN) and methanol (CH₃OH), we can determine the mass of each component by using the formula: mass = density * volume Mass of acetonitrile (CH₃CN): \( mass_{CH_3 CN} = density_{CH_3 CN} * volume_{CH_3 CN} \) Mass of methanol (CH₃OH): \( mass_{CH_3 OH} = density_{CH_3 OH} * volume_{CH_3 OH} \)

Calculate the moles of each component

To find the moles, we can use the molar mass of each substance: Moles of acetonitrile (CH₃CN): \( moles_{CH_3 CN} = \frac{mass_{CH_3 CN}}{molar\, mass_{CH_3 CN}} \) Moles of methanol (CH₃OH): \( moles_{CH_3 OH} = \frac{mass_{CH_3 OH}}{molar\, mass_{CH_3 OH}} \)

Calculate the mole fraction of methanol (CH₃OH)

The mole fraction of methanol can be found by dividing the moles of methanol by the total moles of the solution: Mole fraction of methanol (CH₃OH): \( X_{CH_3 OH} = \frac{moles_{CH_3 OH}}{moles_{CH_3 OH} + moles_{CH_3 CN}} \)

Calculate the molality of the methanol (CH₃OH) solution

Molality is the ratio of moles of solute (CH₃OH) to the mass of the solvent (CH₃CN) in kilograms. Molality of methanol (CH₃OH) in the solution: \( molality_{CH_3 OH} = \frac{moles_{CH_3 OH}}{mass_{CH_3 CN}(kg)} \)

Calculate the molarity of methanol (CH₃OH) in the solution

Molarity is the ratio of moles of solute (CH₃OH) to the total volume of the solution in liters. Since the volumes of CH₃OH and CH₃CN are given, we can find the total volume and convert it to liters. Total volume of the solution in liters: \( V_{total} = \frac{volume_{CH_3 OH} + volume_{CH_3 CN}}{1000} \) Molarity of methanol (CH₃OH) in the solution: \( molarity_{CH_3 OH} = \frac{moles_{CH_3 OH}}{V_{total}} \)

Key concepts

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It tells us how many moles of one component are present compared to the total number of moles in the mixture.
To find the mole fraction, you first need to calculate the number of moles of each substance. You can do this by dividing the mass of the substance by its molar mass. For example, if you have methanol \((CH_3OH)\) and acetonitrile \((CH_3CN)\), you'll calculate the moles for each separately using their respective molar masses.
Once you have the moles for both substances, the mole fraction of methanol is calculated using the formula:

• \( X_{CH_3OH} = \frac{moles_{CH_3OH}}{moles_{CH_3OH} + moles_{CH_3CN}} \)

This gives you a value between 0 and 1, representing the proportion of methanol's moles in the total mixture.

Molality

Molality is another way to indicate the concentration of a solution, focusing on mass instead of volume. It is defined as the number of moles of solute divided by the mass of the solvent in kilograms.
In the given problem, the molality of methanol \(CH_3OH\) can be found by first determining the moles of methanol and the mass of acetonitrile \(CH_3CN\), which acts as the solvent.
The formula for molality is displayed as:

• \( molality_{CH_3OH} = \frac{moles_{CH_3OH}}{mass_{CH_3CN}(kg)} \)

This approach is particularly useful because it doesn't change with temperature, as it relies solely on mass, unlike volume-based measures.

Molarity

Molarity is a common method for measuring the concentration of a solution, expressed in moles per liter of solution. It represents how many moles of solute are present in one liter of the solution.
This is particularly useful in chemical reactions and laboratory settings. To calculate molarity, you divide the number of moles of solute by the total volume of the solution in liters. In the problem, both the volumes of methanol and acetonitrile are given, and assuming these volumes are additive helps us correct the total solution volume.

• The formula used is: \( molarity_{CH_3OH} = \frac{moles_{CH_3OH}}{V_{total}} \)

Where \(V_{total}\) is the sum of the volumes of methanol and acetonitrile in liters. This method might differ with temperature changes, as volume expands or contracts with temperature.

Density Calculation

Density is a vital concept for understanding solution concentration. It's defined as the mass per unit volume of a substance. In the context of solutions, density helps in understanding how much substance there is per given space.
To compute the mass of a component, when you know its density and volume, you use the relationship:

• \( mass = density \times volume \)

In the given example, densities for both methanol \(CH_3OH\) and acetonitrile \(CH_3CN\) are used to calculate their masses, which are further utilized in mole fraction, molality, and molarity calculations.
Density is beneficial for converting between mass and volume, providing a bridge between topics like molarity and molality, which are often dependent on these values.