Problem 59 - Chemistry: The Central Science

Question

Consider two solutions, one formed by adding \(10 \mathrm{~g}\) of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) to \(1 \mathrm{~L}\) of water and the other formed by adding \(10 \mathrm{~g}\) of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) to \(1 \mathrm{~L}\) of water. Are the vapor pressures over the two solutions the same? Why or why not?

Step-by-Step Solution

Answer

The vapor pressures over the two solutions are not the same. The glucose solution has a higher molality (0.0556 mol/kg) than the sucrose solution (0.0292 mol/kg), which leads to a lower mole fraction of the solvent (water) in the glucose solution. According to Raoult's law, this results in a lower vapor pressure over the glucose solution compared to the sucrose solution.

Step 1: Compute the molality of solutions

First, compute the molality of each solution by dividing the ratio of the mass of the solute to the mass of the solvent. Remember that we will need the molecular weights of glucose (C6H12O6) and sucrose (C12H22O11) to compute the number of moles. The molecular weights are approximately 180g/mol for glucose and 342g/mol for sucrose, and we are using 1000g of water (solvent) for each case. For glucose solution: Molality = (10g) / (180g/mol) / (1000g) = 0.0556 mol/kg For sucrose solution: Molality = (10g) / (342g/mol) / (1000g) = 0.0292 mol/kg

Step 2: Compare the molalities and vapor pressure

We can see that the molality of the glucose solution (0.0556 mol/kg) is greater than the molality of the sucrose solution (0.0292 mol/kg). According to the Raoult's law, the vapor pressure of a solution is directly proportional to the mole fraction of the solvent, and as the molality of the glucose solution is higher, the mole fraction of the solvent (water) would be lower, resulting in a lower vapor pressure over the glucose solution compared to the sucrose solution. Therefore, the vapor pressures over the two solutions are not the same. The vapor pressure over glucose solution will be lower than the vapor pressure over sucrose solution due to the higher molality of glucose solution, which causes a lower mole fraction of the solvent and thus, a lower vapor pressure.