Problem 46 - Chemistry: The Central Science

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 \(9.08 \mathrm{~g}\) 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

The mole fraction of thiophene in the solution is \(0.044\). The molality of thiophene in the solution is \(0.498 \, \mathrm{mol/kg}\). Assuming that the volumes of solute and solvent are additive, the molarity of thiophene in the solution is \(0.418\, \mathrm{mol/L}\).

Step 1: 1. Calculate the number of moles of thiophene and toluene

To calculate the number of moles, we first need the molar mass of each substance. For thiophene \(\mathrm{(C_4H_4S)}\), we have: 4 carbon atoms: \(4 \times 12.01 = \) 48.04 g/mol 4 hydrogen atoms: \(4 \times 1.008 = \) 4.032 g/mol 1 sulfur atom: \(1 \times 32.07 = \) 32.07 g/mol So, the molar mass of thiophene = 48.04 + 4.032 + 32.07 = 84.142 g/mol. We are given that there are 9.08 g of thiophene, so we can find the number of moles of thiophene by: number of moles of thiophene = \(\frac{mass}{molar~mass} = \frac{9.08 \mathrm{~g}}{84.142\mathrm{~g/mol}}\) = 0.108 moles. Now, we need to find the number of moles of toluene \(\mathrm{(C_7H_8)}\). First, let's find the molar mass of toluene: 7 carbon atoms: \(7 \times 12.01 = \) 84.07 g/mol 8 hydrogen atoms: \(8 \times 1.008 = \) 8.064 g/mol So, the molar mass of toluene = 84.07 + 8.064 = 92.134 g/mol. We are given that there are 250.0 mL of toluene with a density of 0.867 g/mL. Therefore, the mass of toluene = volume × density = 250.0 mL × 0.867 g/mL = 216.75 g. Now we can find the number of moles of toluene: number of moles of toluene = \(\frac{mass}{molar~mass} = \frac{216.75 \mathrm{~g}}{92.134 \mathrm{~g/mol}}\) = 2.352 moles.

Step 2: (a) Calculate the mole fraction of thiophene in the solution

To find the mole fraction of thiophene, we need to divide the number of moles of thiophene by the total number of moles in the solution (thiophene and toluene). Mole fraction of thiophene = \(\frac{moles~of~thiophene}{moles~of~thiophene + moles~of~toluene} = \frac{0.108}{(0.108+2.352)}\) = 0.044.

Step 3: (b) Calculate the molality of thiophene in the solution

To find the molality of thiophene, we need to divide the number of moles of thiophene by the mass of toluene (in kg). Molality of thiophene = \(\frac{moles~of~thiophene}{mass~of~toluene~(kg)} = \frac{0.108}{0.21675 \mathrm{kg}}\) = 0.498 mol/kg.

Step 4: (c) Calculate the molarity of thiophene in the solution

To find the molarity of thiophene, we need to divide the number of moles of thiophene by the total volume of the solution (in liters). First, let's find the volume of thiophene, which can be calculated from the given mass (9.08 g) and the density (1.065 g/mL). The volume of thiophene = mass / density = \( \frac{9.08?}{1.065}\) = 8.525 mL. Now we can find the total volume of the solution (assuming the volumes are additive): 8.525 mL (thiophene) + 250.0 mL (toluene) = 258.525 mL = 0.258525 L. Now, we can calculate the molarity of thiophene: Molarity of thiophene = \(\frac{moles~of~thiophene}{total~volume~(L)} = \frac{0.108}{0.258525 \,\mathrm{L}}\) = 0.418 mol/L.

Key Concepts

Mole Fraction

The 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 moles of all components in the solution. For instance, in our solution of thiophene and toluene, the mole fraction of thiophene is calculated by dividing the moles of thiophene by the total moles in the solution.

This gives us a dimensionless number, typically expressed as a fraction or a percentage. It provides insight into the proportion of a particular component in the mixture without being affected by temperature changes, making it particularly useful in chemical equilibrium calculations. In our exercise, the fraction was calculated as:

\[\text{Mole fraction of thiophene} = \frac{0.108}{0.108 + 2.352} = 0.044\]

This means that in every part of the solution, about 4.4% is thiophene by mole count.

Molality

Molality measures the concentration of a solute in a solution in terms of moles per kilogram of solvent. Unlike molarity, molality is based on the mass of the solvent rather than the volume of the solution. This makes it particularly useful in studies involving temperature changes, as it does not vary with temperature.

In our case, the molality of thiophene is determined by the number of its moles divided by the mass of the toluene solvent in kilograms:

\[\text{Molality of thiophene} = \frac{0.108}{0.21675} = 0.498 \, \text{mol/kg}\]

This value indicates how many moles of thiophene are present for each kilogram of toluene, providing a reliable measure of concentration.

Molarity

Molarity is a common unit of concentration used by chemists, defined as the moles of solute per liter of solution. With molarity, the focus is on the total volume rather than mass, meaning it's influenced by temperature changes that affect volume.

In the scenario given, the molarity of thiophene was determined by considering the additive volume of the solute and solvent:

1. The volume of thiophene was calculated using its mass and density.
2. The total solution volume was found by adding thiophene's volume to the volume of toluene.

The calculation for molarity is:

\[\text{Molarity of thiophene} = \frac{0.108}{0.258525} = 0.418 \, \text{mol/L}\]

This means in every liter of the solution, there are around 0.418 moles of thiophene. Molarity is widely used in laboratory settings because it directly relates to the volume of liquid solutions, simplifying preparation and analysis.