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?

Short answer

(a) The mole fraction of thiophene in the solution is 0.0438. (b) The molality of thiophene in the solution is 0.4975 mol/kg. (c) The molarity of thiophene in the solution is 0.4170 mol/L.

Step-by-step solution

Calculate the molecular weights of thiophene and toluene

To determine the molecular weights of thiophene and toluene, first find the molecular weights of each element present in each compound: Thiophene: \(\mathrm{C}_4 \mathrm{H}_4 \mathrm{S}\) Toluene: \(\mathrm{C}_7 \mathrm{H}_8\) Molecular weight of carbon (C) = 12.01 g/mol Molecular weight of hydrogen (H) = 1.008 g/mol Molecular weight of sulfur (S) = 32.07 g/mol Using these values, we can calculate the molecular weights for each compound: Molecular weight of thiophene (C4H4S) = \((4 × 12.01) + (4 × 1.008) + 32.07 = 84.14 g/mol\) Molecular weight of toluene (C7H8) = \((7 × 12.01) + (8 × 1.008) = 92.14 g/mol\)

Calculate the number of moles for each component in the solution

Now we can find the moles of each component in the solution. Moles of thiophene = \(\frac{mass~of~thiophene}{molecular~weight~of~thiophene} = \frac{9.08~g}{84.14~g/mol} = 0.1078~mol\) To calculate the moles of toluene, we first need to determine the mass of toluene in grams. Mass of toluene = Volume of toluene × Density of toluene \(= 250.0 ~mL × 0.867 \frac{g}{mL} = 216.75 g\) Moles of toluene = \(\frac{mass~of~toluene}{molecular~weight~of~toluene} = \frac{216.75~g}{92.14~g/mol} = 2.352~mol\)

Calculate the mole fraction of thiophene

The mole fraction of a component in a solution is defined as the ratio of the number of moles of the component to the total number of moles of all components in the solution. Mole fraction of thiophene = \(\frac{moles~of~thiophene}{moles~of~thiophene + moles~of~toluene} = \frac{0.1078}{0.1078 + 2.352} = 0.0438\) The mole fraction of thiophene in the solution is 0.0438.

Calculate the molality of thiophene

Molality is defined as the number of moles of solute per kilogram of solvent. Molality of thiophene = \(\frac{moles~of~thiophene}{mass~of~toluene × (1kg/1000g)} = \frac{0.1078~mol}{216.75~g × \frac{1~kg}{1000~g}} = 0.4975~mol/kg\) The molality of thiophene in the solution is 0.4975 mol/kg.

Calculate the molarity of thiophene

Assuming that the volumes of the solute and solvent are additive, we can calculate the total volume of the solution. Total volume of the solution = Volume of thiophene + Volume of toluene Using the density of thiophene, we can calculate the volume of thiophene that corresponds to the given mass: Volume of thiophene = \(\frac{mass~of~thiophene}{density~of~thiophene} = \frac{9.08~g}{1.065~\frac{g}{mL}} = 8.53~mL\) Total volume of the solution = 8.53 mL + 250.0 mL = 258.53 mL Molarity is defined as the number of moles of solute per liter of solution. Molarity of thiophene = \(\frac{moles~of~thiophene}{total~volume~of~solution × (1L/1000mL)} = \frac{0.1078~mol}{258.53~mL × \frac{1~L}{1000~mL}} = 0.4170~\frac{mol}{L}\) The molarity of thiophene in the solution is 0.4170 mol/L.