Problem 50 - Chemistry: The Central Science in SI Units, Global Edition

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

\(Answer: (a)~0.048;~(b)~0.549~mol/kg;~(c)~0.462~mol/L\)

Step 1: Calculate the number of moles of thiophene

To calculate the number of moles of thiophene, we will use the mass of thiophene and its molar mass, given by: Molar mass of thiophene (\(C_4H_4S\)) = (\(4 * 12.01\) g/mol (for Carbon) + \(4 * 1.01\) g/mol (for Hydrogen) + \(32.07\) g/mol (for Sulfur)) g/mol Molar mass of thiophene = \(68.13\) g/mol Now, we can calculate the moles of thiophene as follows: Number of moles of thiophene = \(\frac{Mass~of~thiophene}{Molar~mass~of~thiophene}\) Number of moles of thiophene = \(\frac{8.10~g}{68.13~g/mol}\) Number of moles of thiophene = \(0.119~mol\)

Step 2: Calculate the number of moles of toluene

To calculate the number of moles of toluene, we will first calculate the mass of toluene using its density and the given volume. Then we will use the molar mass of toluene to find the number of moles. Mass of toluene = Density of toluene * Volume of toluene Mass of toluene = (\(0.867~g/mL\)) * (\(250.0~mL\)) Mass of toluene = \(216.75~g\) Now, let's find the molar mass of toluene (\(C_7H_8\)): Molar mass of toluene = (\(7 * 12.01\) g/mol (for Carbon) + \(8 * 1.01\) g/mol (for Hydrogen)) g/mol Molar mass of toluene = \(92.14\) g/mol Now, we can calculate the moles of toluene as follows: Number of moles of toluene = \(\frac{Mass~of~toluene}{Molar~mass~of~toluene}\) Number of moles of toluene = \(\frac{216.75~g}{92.14~g/mol}\) Number of moles of toluene = \(2.352~mol\)

Step 3: Calculate the mole fraction of thiophene

Mole fraction of thiophene is given by the ratio of the moles of thiophene to the total moles of both substances. Mole fraction of thiophene = \(\frac{Moles~of~thiophene}{Moles~of~thiophene + Moles~of~toluene}\) Mole fraction of thiophene = \(\frac{0.119~mol}{0.119~mol + 2.352~mol}\) Mole fraction of thiophene = 0.048

Step 4: Calculate the molality of thiophene

Molality (m) is the number of moles of solute per kilogram of solvent. We already have the number of moles of thiophene from Step 1, and the mass of the solvent (toluene) in grams. We can now calculate the molality as follows: Molality of thiophene = \(\frac{Moles~of~thiophene}{Mass~of~toluene~in~kg}\) Molality of thiophene = \(\frac{0.119~mol}{0.21675~kg}\) Molality of thiophene = \(0.549~mol/kg\)

Step 5: Calculate the molarity of thiophene, assuming the volumes are additive

To calculate the molarity, we will need to find the total volume of the solution, and then divide the number of moles of thiophene by this volume in liters. Assuming the volumes are additive: Total volume of solution = Volume of toluene + Volume of thiophene The volume of thiophene can be calculated using its mass and density: Volume of thiophene = \(\frac{Mass~of~thiophene}{Density~of~thiophene}\) Volume of thiophene = \(\frac{8.10~g}{1.065~g/mL}\) Volume of thiophene = \(7.606~mL\) Total volume of solution = \(250.0~mL + 7.606~mL = 257.606~mL\) Now, we will convert this volume to liters and then calculate the molarity: Total volume of solution in liters = \(0.257606~L\) Molarity of thiophene = \(\frac{Moles~of~thiophene}{Total~volume~of~solution~in~L}\) Molarity of thiophene = \(\frac{0.119~mol}{0.257606~L}\) Molarity of thiophene = \(0.462~mol/L\) In conclusion: (a) The mole fraction of thiophene in the solution is 0.048. (b) The molality of thiophene in the solution is \(0.549~mol/kg\). (c) Given the volume of the solute and solvent are additive, the molarity of thiophene in the solution is \(0.462~mol/L\).

Key Concepts

Molality

Molality is a concentration term often used when dealing with solutions, especially in chemistry. It describes the number of moles of solute present in 1 kilogram of solvent. This concept is different from molarity, which depends on the total volume of the solution. Unlike molarity, molality is not affected by temperature changes. This makes it a useful measure when working with systems experiencing temperature variations. To calculate molality, you divide the moles of the solute by the mass of the solvent (not the solution) in kilograms. For example, in the exercise provided, the molality of thiophene in toluene is calculated as follows: We have 0.119 moles of thiophene and a mass of toluene equal to 216.75 grams, which is 0.21675 kilograms. Hence, the molality would be \(0.549 \, \text{mol/kg}\). Understanding this concept helps predict how the solute will behave under different conditions without having to worry about changes in volume due to temperature alterations.

Molarity

Molarity, another important concentration unit, measures the number of moles of a solute per liter of solution. This value is significant in quantifying how much of a substance is dissolved in a given volume of solution. Molarity is temperature dependent as it accounts for the volume of the solution, which can change with temperature. In our example, we first calculate the total volume of the solution by adding the volume of toluene and thiophene, given their densities and masses. Then, knowing the moles of thiophene (0.119 mol), and the total solution volume in liters (0.257606 L), we find the molarity by dividing the moles by the volume, resulting in \(0.462 \, \text{mol/L}\). Molarity is especially important when performing chemical reactions needing precise mixture and concentration.

Solution Density

Solution density refers to the mass of the solution per unit of volume. This property is crucial when dealing with solutions where you need to convert between mass and volume, as density links them. For instance, in the exercise, knowing the density of each component allows us to calculate the respective volumes and masses involved, which is critical for finding concentrations like molarity or molality. The density of toluene (0.867 g/mL) and thiophene (1.065 g/mL) allows us to compute the mass of toluene from its volume and determine how much thiophene contributes to the total volume, hence impacting the concentration ratios. Understanding solution density is key to making precise calculations in solution chemistry, especially when converting between different concentration units.

Molar Mass Calculations

Molar mass, often expressed in g/mol, is the mass of a given substance (chemical element or chemical compound) divided by the amount of substance in moles. It serves as a bridge between the macroscopic and microscopic worlds in chemistry, allowing us to translate mole-based values to practical, weight-based amounts, and vice versa. In our exercise, molar mass calculations are pivotal in determining the amount of thiophene dissolved. The molar mass of thiophene (68.13 g/mol) is calculated by summing the atomic masses of its constituent elements: carbon, hydrogen, and sulfur. This fundamental computation allows us to convert from mass in grams to moles, using the formula \(\frac{\text{mass}}{\text{molar mass}}\). Once we have the moles, we can easily transition to other concentration metrics like molality and molarity. Mastering molar mass calculations is a fundamental skill enabling careful and accurate chemical analysis and solutions preparation.