Animated Solution for Chemistry - Solutions: At $20^{\circ }\text{C}$, the vapour pressure of benzene is 70 torr and that of methyl benzene is 20 torr. The mole fraction of benzene in the vapour phase at $20^{\circ }\text{C}$ above an equimolar mixture of benzene and methyl benzene is...... $\times 10^{-2}$. (Nearest integer)

Imagine you are standing in front of a sealed glass container. Inside, a fascinating dynamic equilibrium is taking place. At the bottom, we have a liquid mixture of two organic compounds: benzene and methyl benzene (also known as toluene). Above this liquid, the space is filled with their vapours, constantly evaporating and condensing.

The question asks us to find the composition of this vapour phase. To do this, we need to bridge the gap between the liquid world and the vapour world.

The problem gives us a crucial clue: the liquid is an equimolar mixture. This simply means that for every molecule of benzene, there is exactly one molecule of methyl benzene.

Mathematically, the number of moles are equal ($n_A=n_B$). Therefore, their mole fractions in the liquid phase are perfectly split down the middle:

The Bridge to the Vapour

Now, how much pressure does each component exert in the vapour phase? This is where Raoult's Law comes to our rescue. It states that the partial pressure of a component is equal to its pure vapour pressure multiplied by its mole fraction in the liquid.

For benzene (let's call it component A), the pure vapour pressure is given as $70\text{ torr}$. So, its partial pressure is:

For methyl benzene (component B), the pure vapour pressure is $20\text{ torr}$. Its partial pressure is:

Analyzing the Vapour

We now know the individual pressures. According to Dalton's Law of Partial Pressures, the total pressure inside the container is just the sum of these individual pressures:

To find the mole fraction of benzene in the vapour phase (denoted as $y_A$), we take the ratio of its partial pressure to the total pressure:

Simplifying this fraction, we get:

The question asks for the answer in a very specific format: $x\times 10^{-2}$, where $x$ is the nearest integer.

Let's convert our decimal into this format:

Rounding $\mathrm{77.77...}$ to the nearest integer gives us $78$.

Before we wrap up, let's appreciate the physics here. In the liquid phase, benzene and methyl benzene were present in equal amounts ($50\%$ each). But in the vapour phase, benzene makes up nearly $78\%$ of the mixture!

Why? Because benzene has a higher pure vapour pressure ($70\text{ torr}$ vs $20\text{ torr}$), making it more volatile. This perfectly demonstrates Konovalov's First Rule: the vapour phase is always richer in the more volatile component. It's not just math; it's the beautiful reality of thermodynamics!