Liquid A has vapor pressure \(x\) , and liquid \(\mathrm{B}\) has vapor pressure \(y .\) What is the mole fraction of the liquid mixture if the vapor above the solution is \(30 . \% \mathrm{A}\) by moles? $50 . \% \mathrm{A} ? 80 . \% \mathrm{A}\( ? (Calculate in terms of \)x\( and \)y . )$ Liquid A has vapor pressure \(x,\) liquid \(\mathrm{B}\) has vapor pressure y. What is the mole fraction of the vapor above the solution if the liquid mixture is \(30 . \% \mathrm{A}\) by moles? $50 . \% \mathrm{A} ? 80 . \% \mathrm{A}\( ? (Calculate in terms of \)x\( and \)y . )$

According to Raoult's Law, the partial pressure of a component i in a mixture is equal to its mole fraction times its vapor pressure. We write the partial pressure of A and B with respect to their mole fraction in the liquid phase and vapor pressures as: \(P_A = x_A \times x\) \(P_B = x_B \times y\)

We are given the mole fraction of A in the vapor phase for each case (30%, 50%, and 80%). We can find the mole fraction of B in the vapor phase simply by subtracting the mole fraction of A from 1: Case 1: \(30\%\) \(y_A = 0.3\) \(y_B = 1 - y_A = 0.7\) Case 2: \(50\%\) \(y_A = 0.5\) \(y_B = 1 - y_A = 0.5\) Case 3: \(80\%\) \(y_A = 0.8\) \(y_B = 1 - y_A = 0.2\)

We can use the partial pressure equations from Step 1 to find the mole fractions of A and B for each case. We will solve for \(x_A\) and \(x_B\) in terms of the given vapor pressures x and y: Case 1: \(30\% A\) \(P_A = x_A \times x \Rightarrow x_A = \frac{0.3 \times (x + y)}{x}\) \(P_B = x_B \times y \Rightarrow x_B = \frac{0.7 \times (x + y)}{y}\) Case 2: \(50\% A\) \(P_A = x_A \times x \Rightarrow x_A = \frac{0.5 \times (x + y)}{x}\) \(P_B = x_B \times y \Rightarrow x_B = \frac{0.5 \times (x + y)}{y}\) Case 3: \(80\% A\) \(P_A = x_A \times x \Rightarrow x_A = \frac{0.8 \times (x + y)}{x}\) \(P_B = x_B \times y \Rightarrow x_B = \frac{0.2 \times (x + y)}{y}\) Part 2: Finding the mole fraction of the vapor for given mole fractions of the liquid mixture In this part, we will follow the same steps as before, but we will calculate the mole fraction of the vapor for given mole fractions of the liquid mixture rather than vice versa.

We are given the mole fraction of A in the liquid phase for each case (30%, 50%, and 80%). We can find the mole fraction of B in the liquid phase simply by subtracting the mole fraction of A from 1: Case 1: \(30\%\) \(x_A = 0.3\) \(x_B = 1 - x_A = 0.7\) Case 2: \(50\%\) \(x_A = 0.5\) \(x_B = 1 - x_A = 0.5\) Case 3: \(80\%\) \(x_A = 0.8\) \(x_B = 1 - x_A = 0.2\)

We can use the partial pressure equations from Step 1 to find the mole fractions of A and B for each case. We will solve for \(y_A\) and \(y_B\) in terms of the given vapor pressures x and y: Case 1: \(30\% A\) \(P_A = x_A \times x \Rightarrow y_A = \frac{0.3 \times x}{x + y}\) \(P_B = x_B \times y \Rightarrow y_B = \frac{0.7 \times y}{x + y}\) Case 2: \(50\% A\) \(P_A = x_A \times x \Rightarrow y_A = \frac{0.5 \times x}{x + y}\) \(P_B = x_B \times y \Rightarrow y_B = \frac{0.5 \times y}{x + y}\) Case 3: \(80\% A\) \(P_A = x_A \times x \Rightarrow y_A = \frac{0.8 \times x}{x + y}\) \(P_B = x_B \times y \Rightarrow y_B = \frac{0.2 \times y}{x + y}\) In conclusion, we have calculated the mole fraction of the liquid mixture and of the vapor for the given cases in terms of the vapor pressures x and y.