Question

Two liquids \(A\) and \(B\) have vapour pressure in the ratio \(P_{A}^{\circ}: P_{B}^{\circ}=1: 3\) at a certain temperature. Assume \(A\) and \(B\) form an ideal solution and the ratio of mole fractions of \(A\) to \(B\) in the vapour phase is \(4: 3\). Then the mole fraction of \(B\) in the solution at the same temperature is : (a) \(\frac{1}{5}\) (b) \(\frac{2}{3}\) (c) \(\frac{4}{5}\) (d) \(\frac{1}{4}\)

Short answer

The mole fraction of B in the solution is (b) \frac{2}{3}.

Step-by-step solution

Understanding the problem

The problem gives the ratio of the vapour pressure of components A and B, and the ratio of their mole fractions in the vapour phase. According to Raoult's Law for ideal solutions, the vapour pressure of a component is proportional to its mole fraction in the solution. We will use the given ratios and Raoult's Law to find the mole fraction of B in the solution.

Express the variables

Let the mole fraction of A in the liquid phase be represented by XA and that of B by XB. Since the solution is binary, XB = 1 - XA. The given ratios are P(A)_0/P(B)_0 = 1/3 and (mole fraction of A in vapour phase)/(mole fraction of B in vapour phase) = 4/3.

Using Dalton's Law for the partial pressures

According to Dalton's Law, the partial pressure of A in the vapour phase (PA) is equal to the product of the mole fraction of A in the liquid phase (XA) and its pure component vapour pressure (P(A)_0). Likewise for B, PB = XB * P(B)_0. The total vapour pressure is the sum PA + PB.

Setting up the mole fraction ratio in vapour phase

According to the given ratios, we have PA/PB = 4/3, and because PA = XA * P(A)_0 and PB = (1 - XA) * P(B)_0, we can substitute and get the equation: (XA * P(A)_0) / ((1 - XA) * P(B)_0) = 4/3.

Substitute the known pressure ratio

Substitute the given ratio P(A)_0/P(B)_0 = 1/3 into the equation to solve for XA: (XA * 1) / ((1 - XA) * 3) = 4/3.

Solving for the mole fraction of A

We simplify the equation: 3XA = 4 - 4XA, 7XA = 4, XA = 4/7.

Determining the mole fraction of B

Using XB = 1 - XA, we get XB = 1 - 4/7, thus XB = 3/7.

Converting the mole fraction to a ratio of the options

Express the mole fraction of B as a decimal or fraction equivalent to one of the given options. XB = 3/7 is equivalent to the decimal 0.4286, which does not match exactly with any options. We can eliminate options (a) and (c) because 3/7 is clearly within the range of option (b) 2/3 and not as low as 1/5 or as high as 4/5, and option (d) 1/4 is also too low.

Identify the correct option

The mole fraction of B in the solution is closest to the mole fraction given in option (b) 2/3 when expressed as a decimal.

Key concepts

Vapour Pressure

Vapour pressure is a crucial concept when discussing solutions and their properties. It refers to the pressure exerted by a vapor that is in equilibrium with its liquid or solid form at a given temperature. Think of it as the amount of 'push' the molecules in the vapour phase give at the surface of the liquid. It's an indication of a liquid's evaporation rate; the higher the vapour pressure, the more readily the liquid transforms into a gas.

When a solution is made up of different liquids, as in our textbook problem, each component contributes to the overall vapour pressure of the solution. This is where Raoult's Law comes into play, which states that the vapour pressure of each component of an ideal solution is directly proportional to its mole fraction in the solution. Meaning if we have a larger amount of a certain component, it will have a larger impact on the total vapour pressure.

Mole Fraction

In the realm of chemistry, the mole fraction is a way of expressing the concentration of a component in a mixture or solution. It's defined as the ratio of the number of moles of a particular substance to the total number of moles of all substances present. The beauty of mole fraction is that it's a pure number, without units, which makes it incredibly versatile in calculations.

The mole fraction is essential when applying Raoult's Law. When dealing with ideal solutions, the vapour pressure of each component is directly proportional to its mole fraction in the solution. This core concept helps us deduce that if the component's proportion in the solution changes, so will its contribution to the total vapour pressure. In our textbook exercise, understanding mole fraction is key to determining the composition of the vapor phase and, subsequently, the liquid solution.

Ideal Solution

What exactly is an ideal solution? It's a mixture where the intermolecular forces of attraction between the molecules of different components are nearly the same as the forces of attraction between the molecules of the same component. Because of this intermolecular harmony, the physical properties of an ideal solution like vapour pressure, boiling point, and freezing point can be predicted by Raoult's Law with precision.

Ideal solutions follow the rule that the change in enthalpy (or heat content) upon mixing is zero; there is no heat gained or lost. In context, our problem assumes an ideal solution between liquids A and B. This assumption allows us to calculate the vapour pressures and mole fractions of each component with confidence that they will adhere to the predicted behavior outlined by Raoult's law.

Dalton's Law

Moving on to Dalton's Law, this principle is named after John Dalton, a pioneer in the study of gases. Dalton's Law of partial pressures tells us that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas. This law applies perfectly to vapour mixtures over ideal solutions.

In the context of our exercise, Dalton's Law is key to understanding how the vapour pressures of components A and B combine to create the total pressure above the solution. It says that if you know the mole fraction and the vapour pressure of each component in its pure state, you can calculate the contribution each makes to the total vapour pressure. And in practice, this becomes a powerful tool for deducing the composition of both the vapour and the solution, by combining Raoult's and Dalton's laws to analyze the provided ratios and ultimately solve for the mole fraction of component B.