Question

At \(80^{\circ} \mathrm{C}\), the vapour pressure of pure liquid 'A' is 520 \(\mathrm{mm} \mathrm{Hg}\) and that of pure liquid 'B' is \(1000 \mathrm{~mm} \mathrm{Hg}\). If a mixture solution of 'A' and 'B' boils at \(80^{\circ} \mathrm{C}\) and \(\mathrm{I}\) atm pressure, the amount of 'A' in the mixture is ( \(1 \mathrm{~atm}=\) \(760 \mathrm{~mm} \mathrm{Hg}\) ). (a) \(52 \mathrm{~mol}\) per cent (b) 34 mol per cent (c) 48 mol per cent (d) \(50 \mathrm{~mol}\) per cent

Short answer

The amount of 'A' in the mixture is 50 mol percent.

Step-by-step solution

Convert Atmospheric Pressure

First, convert the atmospheric pressure from atm to mm Hg to match the units given in the problem. Since 1 atm = 760 mm Hg, the atmospheric pressure at which the solution boils is 760 mm Hg.

Apply Raoult's Law

For a mixture boiling at its boiling point, Raoult's Law can be used. According to Raoult's Law for ideal solutions, the total vapour pressure of the solution is equal to the sum of the partial vapour pressures of the components.Therefore, \( P_{solution} = P^{0}_{A} \cdot x_{A} + P^{0}_{B} \cdot x_{B} \), where \( P^{0}_{A} \) and \( P^{0}_{B} \) are the vapour pressures of pure A and B, and \( x_{A} \) and \( x_{B} \) are the mole fractions of A and B, respectively.

Set Up the Equation for Mole Fractions

The total pressure of the mixture is 760 mm Hg. Substitute the given values into the equation:\[ 760 = 520 \cdot x_{A} + 1000 \cdot (1 - x_{A}) \]Here, we used the relationship \( x_{A} + x_{B} = 1 \).

Solve for Mole Fraction of A

Simplify and solve the equation:\[ 760 = 520x_{A} + 1000 - 1000x_{A} \]\[ 760 = 1000 - 480x_{A} \]\[ 480x_{A} = 1000 - 760 \]\[ 480x_{A} = 240 \]\[ x_{A} = \frac{240}{480} = 0.5 \]

Find the Mole Percentage of A

To find the mole percent, multiply the mole fraction of A by 100:\[ \text{Mole percent of A} = 0.5 \times 100 = 50 \% \]

Key concepts

Vapor Pressure

Vapor pressure is a key concept in understanding the behavior of liquid mixtures in closed systems. It is defined as the pressure exerted by a vapor in equilibrium with its liquid phase at a given temperature. Essentially, it indicates how much gas is above the liquid when both are in balance. This equilibrium happens when the number of molecules leaving the liquid equals the number returning to it.
For a pure liquid, the vapor pressure depends solely on temperature. Higher temperatures usually mean higher vapor pressure because more molecules have enough energy to escape into the gas phase.
In mixtures, the vapor pressure is influenced by the components and how they interact, which is critical in determining boiling points and condensation processes.

Ideal Solutions

Ideal solutions are a simplified model of mixing liquids where interactions between different molecules are uniform. This means that the attractive forces between different molecules are the same as those in the pure components. Ideal solutions obey Raoult's Law perfectly, making them a good starting point for understanding more complex behavior in real mixtures.

• In such solutions, the total vapor pressure equals the sum of the individual vapor pressures of the components, each multiplied by its mole fraction.
• Deviations from ideality in real solutions can be due to hydrogen bonding, dipole-dipole attraction, or other specific interactions.

Understanding this concept helps predict the behavior of liquid mixtures under various conditions, like boiling or condensation.

Mole Fraction

The mole fraction is a way to express the composition of a component in a mixture. It is essential for calculating properties like vapor pressure and is defined as the ratio of the number of moles of a component to the total moles in the mixture.
The formula for the mole fraction of component A, often written as \(x_A\), is given by:
\[ x_A = \frac{n_A}{n_A + n_B} \]
where \(n_A\) is the number of moles of component A and \(n_B\) is the number of moles of component B.

• The sum of the mole fractions of all components in the mixture should always be 1.
• Mole fractions provide a way to measure how much of each component is present in the mixture, which is crucial for calculating the solution's properties using laws like Raoult's.

Boiling Point of Liquid Mixtures

The boiling point of liquid mixtures depends not only on the individual boiling points of the components but also on their interactions and compositions. In mixtures, the boiling point is the temperature at which the vapor pressure equals the external pressure.
For ideal solutions, this means the boiling point can be predicted using the total vapor pressure calculated from the mole fractions and vapor pressures of the pure components.

• Solutions often boil at different temperatures than their pure components due to changes in vapor pressure.
• Understanding the boiling point is crucial for processes like distillation, where the separation of mixtures based on boiling points is performed.
• The boiling point composition diagram, for example, can help predict the boiling behavior of a specific mixture under given conditions.

Conceptually grasping how mixtures alter boiling points helps in many industrial applications, from chemical manufacturing to food processing.