Question

You have a solution of two volatile liquids, \(\mathrm{A}\) and \(\mathrm{B}\) (assume ideal behavior). Pure liquid A has a vapor pressure of \(350.0\) torr and pure liquid \(\mathrm{B}\) has a vapor pressure of \(100.0\) torr at the temperature of the solution. The vapor at equilibrium above the solution has double the mole fraction of substance A that the solution does. What is the mole fraction of liquid \(\mathrm{A}\) in the solution?

Short answer

The mole fraction of liquid A in the solution is \(x_A = \frac{1}{3}\) or approximately 0.333.

Step-by-step solution

Define Variables and Known Values

Let's first define some variables that we will be using to solve this problem: - \(x_A\) is the mole fraction of liquid A in the solution. - \(x_B\) is the mole fraction of liquid B in the solution. - \(P_A\) is the vapor pressure of pure liquid A at the given temperature (350.0 torr). - \(P_B\) is the vapor pressure of pure liquid B at the given temperature (100.0 torr). - \(y_A\) is the mole fraction of substance A in the vapor at equilibrium. Since \(x_A\) and \(x_B\) represent mole fractions, we know that: \(x_A + x_B = 1\) Also, it is given that the mole fraction of A in the vapor phase is double the mole fraction of A in the liquid phase. Therefore: \(y_A = 2x_A\) Now we can use Dalton's Law of Partial Pressures and Raoult's Law to express the partial pressures of A and B in terms of their mole fractions and pure vapor pressures.

Apply Dalton's Law of Partial Pressures and Raoult's Law

According to Raoult's Law, the partial pressure of A in the vapor phase is given by: \(P_A^* = x_A P_A\) Similarly, for B: \(P_B^* = x_B P_B\) Now, using Dalton's Law of Partial Pressures, we can write the total pressure as: \(P_{total} = P_A^* + P_B^*\) As we know that the mole fraction of A in the vapor phase is double that of the mole fraction of A in the liquid phase, we have: \(y_A = \frac{P_A^*}{P_{total}} = 2x_A\)

Solve for the Mole Fraction of Liquid A

Now we can substitute the expressions for \(P_A^*\), \(P_B^*\), and \(P_{total}\) into the formula for \(y_A\): \(2x_A = \frac{x_A P_A}{x_A P_A + x_B P_B}\) Let's eliminate \(x_B\) using the expression \(x_A + x_B = 1\): \(2x_A = \frac{x_A P_A}{x_A P_A + (1 - x_A) P_B}\) Now we can solve for \(x_A\): \(2x_A (x_A P_A + (1 - x_A) P_B) = x_A P_A\) \(2x_A^2 P_A + 2x_A(1 - x_A)P_B = x_A P_A\) \(x_A^2 P_A + x_A(1 - x_A)P_B = \frac{1}{2} x_A P_A\) Divide both sides by \(x_A\): \(x_A P_A + (1 - x_A)P_B = \frac{1}{2} P_A\) Now solve for \(x_A\): \(x_A = \frac{P_A - 2P_B}{P_A + P_B}\) Substitute the values of \(P_A\) and \(P_B\): \(x_A = \frac{350.0 - 2(100.0)}{350.0 + 100.0}\) \(x_A = \frac{150}{450}\) \(x_A = \frac{1}{3}\) So, the mole fraction of liquid A in the solution is \(\frac{1}{3}\) or approximately 0.333.

Key concepts

Understanding Raoult's Law

Raoult's Law is a fundamental principle that applies to the behavior of solutions, specifically dealing with the vapor pressure of a solvent or a component in an ideal solution. According to this law, the partial vapor pressure of each component of an ideal solution is directly proportional to its mole fraction in the solution. In simpler terms, if you have a higher proportion of a substance in a solution, it will contribute more to the overall vapor pressure.

When describing an ideal solution, Raoult's Law can be represented by the equation:\[ P_i^* = x_i P_i^\circ \]Where \( P_i^* \) is the partial vapor pressure of component \( i \), \( x_i \) is the mole fraction of \( i \) in the liquid phase, and \( P_i^\circ \) is the vapor pressure of the pure component \( i \). Knowing this, we can predict the behavior of a solution under certain conditions and determine various thermodynamic properties such as boiling points and freezing points.

Dalton's Law of Partial Pressures

When we move from Raoult's Law to consider the vapor phase above a solution, Dalton's Law of Partial Pressures becomes vital. This law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. For students working on mole fraction calculations, understanding Dalton's Law is essential because it helps determine the contribution of each gas to the total pressure.

Mathematically, Dalton's Law can be expressed as:\[ P_{total} = \sum P_i^* \]Where \( P_{total} \) is the total pressure of the mixture, and \( P_i^* \) is the partial pressure of each gas \( i \), which we can calculate using Raoult's Law as previously described. The problem we examined showcases how this law is used in conjunction with Raoult's Law to deduce the composition of the vapor above a solution.

Vapor Pressure in Solutions

Vapor pressure is a crucial concept linking Raoult's Law and Dalton's Law in the study of solutions. It is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. Understanding vapor pressure is not only necessary for solving problems but also for grasping the physical properties of substances.

The vapor pressure of a pure liquid is the pressure exerted when the liquid and its vapor are in equilibrium. With solutions, this concept expands as the vapor pressure over the solution is a result of the contributions from all volatile components. The problem we worked on uses the ideal behavior assumptions to simplify the calculations, but in reality, many solutions deviate from this ideal because of interactions between molecules. Thus, understanding these underlying principles is key for students to comprehend more complex systems they may encounter in the real world.