Problem 60 - Chemistry: The Central Science

Question

(a) What is an ideal solution? (b) The vapor pressure of pure water at \(60^{\circ} \mathrm{C}\) is 149 torr. The vapor pressure of water over a solution at \(60^{\circ} \mathrm{C}\) containing equal numbers of moles of water and ethylene glycol (a nonvolatile solute) is 67 torr. Is the solution ideal according to Raoult's law? Explain.

Step-by-Step Solution

Answer

An ideal solution is a mixture of substances that obeys Raoult's law, which states that the vapor pressure of a component is proportional to its mole fraction and the vapor pressure of the pure component. The given solution has equal moles of water and ethylene glycol, resulting in a mole fraction of 0.5 for water. The expected partial pressure of water according to Raoult's law is 74.5 torr, but the experimental value is 67 torr. Since these values differ, the solution does not obey Raoult's law and is not an ideal solution.

Step 1: Define an ideal solution

An ideal solution is a mixture of two or more substances that obeys Raoult's law, which states that the vapor pressure of a component in an ideal solution is proportional to its mole fraction in the solution and to the vapor pressure of the pure component.

Step 2: Write the equation for Raoult's law

Raoult's law is given by the equation: \[P_A = X_A \times P_A^*\] Where: - \(P_A\) is the partial pressure of component A in the vapor phase - \(X_A\) is the mole fraction of component A in the solution - \(P_A^*\) is the vapor pressure of pure component A

Step 3: Calculate the mole fraction of water in the solution

The solution contains equal numbers of moles of water (W) and ethylene glycol (EG). The mole fraction of water is therefore: \[X_W = \frac{\text{moles of W}}{\text{moles of W} + \text{moles of EG}} = \frac{1}{1 + 1} = 0.5\]

Step 4: Calculate the expected partial pressure of water according to Raoult's law

Using Raoult's law and the mole fraction calculated in step 3, we can determine the expected partial pressure of water in the vapor phase: \[P_W = X_W \times P_W^*\] Substituting the vapor pressure of pure water (\(P_W^* = 149 \; \text{torr}\)) and the mole fraction of water (\(X_W = 0.5\)): \[P_W = 0.5 \times 149 \; \text{torr} = 74.5 \; \text{torr}\]

Step 5: Compare the expected and experimental vapor pressures

The expected partial pressure of water according to Raoult's law is 74.5 torr; however, the experimental vapor pressure of water over the solution is given to be 67 torr. These values are different, indicating that the solution does not perfectly obey Raoult's law and is therefore not an ideal solution.