Question

A solution at \(50^{\circ} \mathrm{C}\) containing 2.0 \(\mathrm{mol}\) of liquid \(\mathrm{A}\) and 3.0 \(\mathrm{mol}\) of liquid \(\mathrm{B}\) has a total vapor pressure of \(240 .\) torr. If pure A has a vapor pressure of 150 . torr at \(50^{\circ} \mathrm{C},\) what is the vapor pres- sure of pure \(\mathrm{B}\) at \(50^{\circ} \mathrm{C} ?\)

Short answer

The vapor pressure of pure B at \(50^{\circ}\mathrm{C}\) is 300 torr.

Step-by-step solution

Determine the mole fractions of A and B

The first step is to calculate the mole fractions of A and B in the solution. Mole fraction is the ratio of the moles of a substance to the total moles of the solution. Mole fraction of A = moles of A / (moles of A + moles of B) Mole fraction of B = moles of B / (moles of A + moles of B)

Calculate the mole fractions of A and B

Using the given values, let's calculate the mole fractions of A and B. Mole fraction of A = 2.0 mol / (2.0 mol + 3.0 mol) = 2.0 / 5.0 = 0.4 Mole fraction of B = 3.0 mol / (2.0 mol + 3.0 mol) = 3.0 / 5.0 = 0.6

Apply Raoult's Law

Raoult's Law can be applied to calculate the vapor pressures of A and B. P_solution = P_A * mole_fraction_A + P_B * mole_fraction_B Given, P_solution = 240 torr and P_A = 150 torr, we can solve for P_B.

Solve for P_B

Let's plug in the values into the equation from step 3 and solve for P_B. 240 = (150 * 0.4) + P_B * 0.6 240 = 60 + 0.6 * P_B 240 - 60 = 0.6 * P_B 180 = 0.6 * P_B Now, divide by 0.6 to find the vapor pressure of pure B: P_B = 180 / 0.6 = 300 torr

State the final answer

The vapor pressure of pure B at \(50^{\circ}\mathrm{C}\) is 300 torr.

Key concepts

Vapor Pressure

Vapor pressure is a key concept in understanding how mixtures like solutions behave. Simply put, vapor pressure is the pressure exerted by the vapor of a liquid when it's in equilibrium with its liquid phase in a closed system. When the air above a liquid contains the maximum amount of vapor from that liquid, the liquid is said to be in equilibrium with its vapor. This means the rate of evaporation equals the rate of condensation.

Several factors can influence vapor pressure:

• Temperature: Increasing the temperature increases the kinetic energy of the molecules, which increases vapor pressure.
• Nature of the liquid: Different liquids have different inherent vapor pressures based on the strength of intermolecular forces.

When discussing solutions, Raoult's Law is often applied. According to Raoult's Law, the vapor pressure of a solution is the sum of the vapor pressures of each component, each multiplied by its respective mole fraction. In simpler terms, the total vapor pressure depends on both the amount of each substance and their respective abilities to generate vapor at a given temperature.

Mole Fraction

Mole fraction is a measure of concentration, which tells us how many moles of a particular component are present compared to the total number of moles in the solution. It's a concise way to express the proportion of a component in a mixture.

Calculating mole fraction can be done using the formula:

• For a component A: \(X_A = \frac{n_A}{n_{total}}\)
• Where \(n_A\) is the number of moles of component A, and \(n_{total}\) is the total number of moles in the solution.

In our problem, we used this concept to determine that the mole fraction for liquid A and B are 0.4 and 0.6, respectively. Understanding mole fraction is crucial because it directly affects vapor pressure calculations through Raoult's Law. This ratio helps in predicting how much each substance's vapor contributes to the total vapor pressure in a solution.

Solutions Chemistry

Solutions chemistry involves studying homogeneous mixtures of two or more substances. A key topic within this realm is understanding how different substances interact in a solution and how they affect properties like boiling point, freezing point, and vapor pressure. In our exercise, we are dealing with a liquid-liquid solution, where both components contribute to the overall properties of the mixture.

Some important points to remember about solutions:

• Components: Solutions generally consist of solute(s) and solvent(s). In our case, liquids A and B are both solutes and solvents simultaneously.
• Interactions: Interactions between molecules can affect how a solution behaves. This is why considering mole fractions and vapor pressures is essential in prediction and calculation.
• Raoult's Law: This law helps understand how mixing affects vapor pressure. It is especially useful in determining specific properties when the solution contains two or more volatile components.

By being aware of these principles, one can gain a better grip on predicting the behavior of solutions and using formulas like Raoult's Law to find properties such as vapor pressure.