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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 = \(0.119~mol\)

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 = \(2.352~mol\)

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

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 = \(0.549~mol/kg\)

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 \(0.462~mol/L\).