Problem 45 - 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 \(22.5 \mathrm{~mL} \mathrm{CH}_{3} \mathrm{OH}\) in \(98.7 \mathrm{~mL}\) \(\mathrm{CH}_{3} \mathrm{CN}\). (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

(a) The mole fraction of methanol (\(\mathrm{CH}_{3} \mathrm{OH}\)) in the solution is 0.227. (b) The molality of the solution is 7.15 mol/kg. (c) The molarity of methanol (\(\mathrm{CH}_{3} \mathrm{OH}\)) in the solution is 4.58 mol/L, assuming the volumes are additive.

Step 1: Finding molar mass of CH3OH and CH3CN

First, we need to find the molar mass of methanol (CH3OH) and acetonitrile (CH3CN). We can find the molar mass by adding the molar masses of each element present in the molecules. For CH3OH: Molar mass = Carbon(C) + 4*Hydrogen(H) + Oxygen(O) For CH3CN: Molar mass = Carbon(C) + 3*Hydrogen(H) + 1*Nitrogen(N) Molar mass of C = 12.01 g/mol Molar mass of H = 1.01 g/mol Molar mass of O = 16.00 g/mol Molar mass of N = 14.01 g/mol Molar mass of CH3OH = 12.01 + 4*1.01 + 16.00 = 32.05 g/mol Molar mass of CH3CN = 12.01 + 3*1.01 + 14.01 = 41.06 g/mol

Step 2: Finding the mass of CH3OH and CH3CN using their densities and volumes

Now, we need to find the mass of methanol (CH3OH) and acetonitrile (CH3CN) used in making the solution. Mass = Density × Volume Mass of CH3OH = Density of CH3OH × Volume of CH3OH Mass of CH3OH = 0.791 g/mL × 22.5 mL = 17.798 g Mass of CH3CN = Density of CH3CN × Volume of CH3CN Mass of CH3CN = 0.786 g/mL × 98.7 mL = 77.573 g

Step 3: Finding the moles of CH3OH and CH3CN

Now, we need to find the number of moles of methanol and acetonitrile in the solution by using their masses and molar masses: Moles = Mass / Molar mass Moles of CH3OH = Mass of CH3OH / Molar mass of CH3OH Moles of CH3OH = 17.798 g / 32.05 g/mol = 0.555 moles Moles of CH3CN = Mass of CH3CN / Molar mass of CH3CN Moles of CH3CN = 77.573 g / 41.06 g/mol = 1.888 moles

Step 4: Finding mole fraction of methanol (a)

Mole fraction is defined as the ratio of the moles of one component to the total moles of the solution. To find the mole fraction of methanol, we can use the following formula: Mole fraction of CH3OH = Moles of CH3OH / (Moles of CH3OH + Moles of CH3CN) Mole fraction of CH3OH = 0.555 / (0.555 + 1.888) = 0.555 / 2.443 = 0.227 The mole fraction of methanol in the solution is 0.227.

Step 5: Finding the molality of the solution (b)

Molality is defined as the moles of solute (CH3OH in this case) per kilogram of solvent (CH3CN in this case). To find the molality, we can use the following formula: Molality = Moles of CH3OH / Mass of CH3CN (in kg) Let's convert the mass of CH3CN to kg: Mass of CH3CN (in kg) = 77.573 g / 1000 = 0.0776 kg Molality = 0.555 moles / 0.0776 kg = 7.15 mol/kg The molality of the solution is 7.15 mol/kg.

Step 6: Finding the molarity of methanol (c)

Molarity is defined as the moles of solute (CH3OH in this case) per liter of solution. Assuming volumes are additive, we can find the molarity by using the following formula: Molarity = Moles of CH3OH / Volume of the solution (in L) Let's find the volume of the solution: Volume of the solution = Volume of CH3OH + Volume of CH3CN Volume of the solution = 22.5 mL + 98.7 mL = 121.2 mL Let's convert the volume of the solution to liters: Volume of the solution (in L) = 121.2 mL / 1000 = 0.1212 L Molarity = 0.555 moles / 0.1212 L = 4.58 mol/L The molarity of methanol in the solution is 4.58 mol/L.

Key Concepts

Mole Fraction

Mole fraction is a way to express the concentration of a component in a mixture. It is defined as the ratio of the moles of one component to the total number of moles in the solution. This method does not depend on the temperature or pressure, making it very useful for understanding the composition of a solution across various conditions.

To calculate the mole fraction of methanol, we first determine the moles of each component using their respective masses and molar masses:
- Moles of methanol - Moles of acetonitrile

Then, using the formula:

• Mole fraction of methanol = \( \frac{{\text{{moles of methanol}}}}{{\text{{total moles in solution}}}} \)

By plugging in the moles of methanol and the total moles from both methanol and acetonitrile, you'll obtain a clear idea of methanol's presence in the solution's composition. The mole fraction is unitless and provides a relative measure compared to other solutes or solvents in the mixture. This can be particularly insightful when comparing similar compounds or assessing vapor pressures in solutions.

Molality

Molality is another way to measure concentration, specifically in terms of moles of solute relative to the mass of the solvent in kilograms. This is different from molarity because it's based on the mass of the solvent rather than the volume of the solution. Molality remains constant with temperature changes, which makes it ideal for situations involving temperature variations.

To determine molality, follow these steps:

• Convert the mass of the solvent (here, acetonitrile) to kilograms.
• Use the formula: Molality = \( \frac{{\text{{moles of solute}}}}{{\text{{mass of solvent in kg}}}} \)

Molality gives insights for experiments or reactions occurring at different temperatures, especially when physical properties like boiling points or freezing points must be accurately understood. It also allows for straightforward calculations when dealing with colligative properties of solutions.

Molarity

Molarity is one of the most common methods for expressing concentration. It is defined as moles of solute per liter of solution. This unit of measure depends on the volume of the solution, which can change with temperature. Therefore, it's essential to consider possible changes during experiments.

To find the molarity of methanol in the solution:

• Add the volumes of methanol and acetonitrile to get the total solution volume.
• Convert this volume from milliliters to liters.
• Use the formula: Molarity = \( \frac{{\text{{moles of solute}}}}{{\text{{volume of solution in liters}}}} \)

Molarity is extremely useful for stoichiometric calculations in chemical reactions and allows chemists to prepare solutions with precise concentrations for various applications. It offers easy-to-use conversion from lab reagents and is broadly applied in chemical industry and research.