Problem 12 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For an ideal solution of \(\mathrm{A}\) and \(\mathrm{B}, Y_{\mathrm{A}}\) is the mole fraction of \(\mathrm{A}\) in the vapour phase at equilibrium. Which of the following plot should be linear? (a) \(P_{\text {toul }}\) vs \(Y_{\mathrm{A}}\) (b) \(P_{\text {total }} v s Y_{\mathrm{B}}\) (c) \(\frac{1}{P_{\text {total }}}\) vs \(Y_{\mathrm{A}}\) (d) \(\frac{1}{P_{\text {total }}} v s \frac{1}{Y_{\mathrm{A}}}\)

Step-by-Step Solution

Answer

The plot of \frac{1}{P_{\text {total }}} vs \frac{1}{Y_{\text{A}}}, option (d), should be linear for an ideal solution according to Raoult's Law.

Step 1: Review Raoult's Law

Raoult's Law states that for an ideal solution, the partial vapour pressure of each component is directly proportional to its mole fraction in the solution. This is expressed as: P_A = P_{A}^{*}Y_A for component A, where P_A is the partial vapour pressure of A, P_{A}^{*} is the vapour pressure of pure A, and Y_A is the mole fraction of A in the vapour phase.

Step 2: Examine the equations in the options

The question asks for a plot that should give a straight line. According to Raoult's Law, when plotting partial pressures versus mole fractions in the vapour phase for an ideal solution, we should expect a straight line if the plot reflects a direct proportionality relationship. We need to identify which option, when plotted, would yield a linear relationship based on this law.

Step 3: Analyze each option

Option (a) suggests plotting P_{total} versus Y_A, which could be influenced by the mole fraction and vapour pressure of component B as well; thus it doesn't guarantee linearity. Option (b) is the counterpart of option (a) and faces the same issue with the presence of component A. Option (c) inverts the total pressure, and since we're dealing with reciprocal relationships, this is not expected to be linear in accordance with Raoult's Law. Option (d) plots the reciprocal of total pressure against the reciprocal of the mole fraction of A, which is the only option that combines the reciprocals of both variables and thus could linearize the relationship when considering Raoult's Law in reverse for the total pressure.

Step 4: Identify the most suitable plot

Raoult’s law can be rearranged to give a linear relationship when both sides are inverted, leading to 1/P_{total} = 1/(P_{A}^{*}Y_A + P_{B}^{*}Y_B). This can be simplified further if we have a binary solution where the mole fraction of B, Y_B, equals 1 - Y_A. Substituting this in and making Y_A the subject, we result in a relationship that can be plotted in a linear fashion when we plot 1/P_{total} versus 1/Y_A.

Key Concepts

Vapour Pressure

Vapour pressure is an essential concept in understanding the behavior of liquids and solutions. It refers to the pressure exerted by a vapour when it is in equilibrium with its liquid or solid form. At this equilibrium point, the rate at which molecules evaporate from the liquid phase equals the rate at which they condense back into the liquid. For pure substances, vapour pressure is a fixed value at a given temperature, while for solutions, it varies depending on the composition of the solution.
Having a solid grasp on this concept helps us predict how a substance will behave when exposed to different environmental conditions. For instance, a liquid with a high vapour pressure at room temperature is likely to evaporate quickly—a handy piece of trivia when you're drying nail polish or watching a puddle disappear on a sunny day! In the context of an ideal solution, this concept helps us understand the role of each component in the solution and how it contributes to the overall vapour pressure.

Mole Fraction

In chemistry, the mole fraction is a way of expressing the concentration of a component in a solution. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture. Represented by the symbol 'Y' in the context of gases, it's simply a way to measure 'how much' of a component there is in a mixture.
The mole fraction is dimensionless, meaning it does not have units, making it universally applicable and immensely useful when comparing the composition of different substances regardless of their amount. In the vapour phase of an ideal solution, it becomes even more significant as it directly influences the partial vapour pressure of each component according to Raoult’s Law. A simplified way to understand mole fraction would be to think of a pizza cut into slices—each slice represents a 'mole' and the 'mole fraction' indicates the portion of particular toppings compared to the whole pizza.

Partial Vapour Pressure

Partial vapour pressure is a term that refers to the component pressure contribution of each substance in a mixture when they are present as gases. In other words, it's the pressure a single component of a mixture would exert if it alone occupied the entire volume of the mixture at the same temperature. This concept is vital for understanding how mixtures of gases interact and how they would behave if they were on their own.
The beauty of partial pressures lies in the fact that they sum up to give the total pressure of the mixture, thanks to Dalton's Law of partial pressures. In simple terms, each gas component in a mixture 'thinks' it exists alone and exerts pressure as if no other gases are present. When we talk about partial vapour pressure in the context of Raoult’s Law, we see that it is proportional to this component's mole fraction in the liquid phase of a solution—akin to saying the 'strength' of each ingredient’s aroma in a stew depends on how much of it you’ve added to the pot.

Binary Solution Equilibrium

Binary solution equilibrium pertains to a two-component system where both substances have reached a state of balance between their respective liquid and vapour phases. At this equilibrium, the rate of evaporation and condensation of each component is equal, leading to constant vapour pressures so long as temperature doesn't change.
Analyzing binary solutions is particularly simpler since we only deal with two components, making it a frequently used model in studies of vapour-liquid equilibrium. This is where Raoult's Law shines, providing us with a straightforward way of calculating the partial pressure of each component by multiplying the vapour pressure of the pure component by its mole fraction in the solution. An ideal binary solution is a theoretical simplification where interactions between different molecules are similar to those between like molecules, resulting in a directly proportional relationship between the mole fraction and partial vapour pressure. In a classroom context, think of binary solution equilibrium as a perfectly choreographed dance between two partners—each one's movements (molecules in liquid and vapour phase) are synchronized perfectly, maintaining balance and poise throughout their performance.