Problem 50 - Chemistry The Central Science In Si Units

Question

The density of toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right)\) is \(0.867 \mathrm{~g} / \mathrm{mL}\), and the density of thiophene \(\left(\mathrm{C}_{4} \mathrm{H}_{4} \mathrm{~S}\right)\) is \(1.065 \mathrm{~g} / \mathrm{mL}\). A solution is made by dissolving \(8.10 \mathrm{~g}\) of thiophene in \(250.0 \mathrm{~mL}\) of toluene. (a) Calculate the mole fraction of thiophene in the solution. (b) Calculate the molality of thiophene in the solution. (c) Assuming that the volumes of the solute and solvent are additive, what is the molarity of thiophene in the solution?

Step-by-Step Solution

Answer

(a) Mole fraction of thiophene = 0.0399. (b) Molality = 0.444 mol/kg. (c) Molarity = 0.374 mol/L.

Step 1: Calculate Moles of Thiophene

First, we need to determine the number of moles of thiophene in the solution. The molar mass of thiophene \( \left(\mathrm{C}_{4}\mathrm{H}_{4}\mathrm{S}\right) \) is calculated as:\[4(12.01) + 4(1.008) + 32.07 = 84.14 \text{ g/mol}\]Then, calculate moles using the mass given:\[\text{Moles of thiophene} = \frac{8.10 \text{ g}}{84.14 \text{ g/mol}} = 0.0963 \text{ mol}\]

Step 2: Calculate Moles of Toluene

The molar mass of toluene \( \left(\mathrm{C}_{7}\mathrm{H}_{8}\right) \) is:\[7(12.01) + 8(1.008) = 92.14 \text{ g/mol}\]Calculate the mass of toluene using its density and volume:\[\text{Mass of toluene} = 250.0 \text{ mL} \times 0.867 \text{ g/mL} = 216.75 \text{ g}\]Now, calculate the moles of toluene:\[\text{Moles of toluene} = \frac{216.75 \text{ g}}{92.14 \text{ g/mol}} = 2.353 \text{ mol}\]

Step 3: Calculate Mole Fraction of Thiophene

The mole fraction of thiophene is the ratio of the moles of thiophene to the total moles in the solution:\[\text{Mole fraction of thiophene} = \frac{0.0963 \text{ mol}}{0.0963 \text{ mol} + 2.353 \text{ mol}} = 0.0399\]

Step 4: Calculate Molality of Thiophene

Molality is defined as moles of solute per kilogram of solvent. Here, the solvent is toluene.\[\text{Molality} = \frac{0.0963 \text{ mol}}{0.21675 \text{ kg}} = 0.444 \text{ mol/kg}\]

Step 5: Calculate Molarity of Thiophene

First, calculate the total volume of the solution assuming additive volumes:\[\text{Total volume} = 250.0 \text{ mL} + \frac{8.10 \text{ g}}{1.065 \text{ g/mL}} = 257.60 \text{ mL}\]Convert this volume to liters:\[0.2576 \text{ L}\]Then calculate the molarity:\[\text{Molarity} = \frac{0.0963 \text{ mol}}{0.2576 \text{ L}} = 0.374 \text{ mol/L}\]

Key Concepts

Mole Fraction Calculations

When tackling mole fraction calculations, it's important to know this term refers to the ratio of moles of a particular solute to the total moles in the solution. It gives us an idea of the concentration without needing to depend on the mass or density. Calculating mole fraction involves these steps:
1. **Determine Moles of Each Component**: Start by converting the mass of each component into moles using their molar masses. For instance, given 8.10 g of thiophene, the number of moles is found using its molar mass (84.14 g/mol).
2. **Total Moles**: Once you have the moles of thiophene and toluene, sum them up to find the total number of moles in the solution.
3. **Calculate Mole Fraction**: Finally, divide the moles of the solute of interest (thiophene, in this example) by the total moles. As seen in the step-by-step example, the mole fraction of thiophene turned out to be 0.0399. This means there’s a small proportion of thiophene relative to toluene.

Molality Determination

Molality, unlike other concentration measures, is based on mass rather than volume, making it ideal for situations where temperature fluctuates, because mass doesn’t change with temperature. Here's how to determine molality:
1. **Moles of Solute**: First, calculate the moles of the solute. This was done for thiophene using its mass and molar mass.
2. **Mass of Solvent in Kilograms**: Convert the mass of the solvent (toluene) into kilograms. This is crucial as molality uses kilograms, unlike some other concentration measures that use liters.
3. **Calculate Molality**: Divide the moles of solute by the mass (in kg) of solvent. For our given example, \[ \text{Molality} = \frac{0.0963 \text{ mol}}{0.21675 \text{ kg}} = 0.444 \text{ mol/kg} \] This implies that there are 0.444 moles of thiophene for every kilogram of toluene, offering a mass-based measure of concentration.

Molarity Determination

Molarity provides the number of moles of solute per liter of solution. It's one of the most common ways of expressing concentration, especially in lab settings. Follow these steps to determine molarity:
1. **Calculate Total Solution Volume**: You have to assume the volumes are additive. That means adding the volume of toluene and the volume derived from the amount of thiophene, calculated using thiophene's density. This results in a total volume of 257.60 mL.
2. **Convert Volume to Liters**: Since molarity needs volume in liters, convert the calculated mL into L. For our solution, 257.60 mL is converted to 0.2576 L.
3. **Determine Moles of Solute**: Again, you'll need the moles of thiophene as calculated earlier.
4. **Calculate Molarity**: Finally, use the formula: \[ \text{Molarity} = \frac{0.0963 \text{ mol}}{0.2576 \text{ L}} = 0.374 \text{ mol/L} \] This result indicates that in one liter of the solution, there are 0.374 moles of thiophene. Molarity is especially helpful when preparing solutions of a specific concentration in the lab.