Animated Solution for Chemistry - States of Matter: Consider the following table.\begin{array}{ccc} \hline \textbf{Gas} & \mathbf{a / (kPa\ dm^6\ mol^{-1})} & \mathbf{b / (dm^3\ mol^{-1})} \\ \hline A & 642.32 & 0.05196 \\ B & 155.21 & 0.04136 \\ C & 431.91 & 0.05196 \\ D & 155.21 & 0.4382 \\ \hline \end{array}$a$ and $b$ are van der Waals' constants. The correct statement about the gases is

Attraction vs. Size

Imagine you are a tiny observer, shrunk down to the size of a nanometer, floating inside a sealed container of gas. If you were observing an ideal gas, the molecules would be like ghostly billiard balls—they take up absolutely zero space and pass right through each other without any attraction. But in the real world, gases are not ideal. They have character. They have size. And they feel the pull of their neighbors.

This beautiful, messy reality is captured by the van der Waals equation, a mathematical masterpiece that corrects the ideal gas law by introducing two crucial constants: $a$ and $b$.

The constant $a$ is the gravitational pull of the molecular world. It represents the magnitude of the intermolecular attractive forces. When molecules fly past each other, they feel a slight tug. A higher value of $a$ means this tug is stronger.

The constant $b$, on the other hand, is the stubbornness of matter. It represents the excluded volume, or the actual physical size of the gas molecules. Matter cannot be crushed into nothingness; a higher value of $b$ means the molecules are bulkier and take up more space.

In our problem, we are given a table of four gases ($A$, $B$, $C$, and $D$) along with their respective van der Waals constants. Our mission is to compare their volumes and compressibilities. To do this systematically, we must isolate our variables.

Let's first look at gases $A$ and $C$. Notice something interesting? They have the exact same value for constant $b$ ($0.05196{\text{ dm}}^3{\text{ mol}}^{-1}$). This means the molecules of gas $A$ and gas $C$ are physically the same size. The playing field is level.

However, their $a$ values are vastly different. Gas $A$ has an $a$ value of $642.32$, while gas $C$ has an $a$ value of $431.91$.

Volume

Because $a_A>a_C$, the molecules in gas $A$ experience a much stronger attractive pull towards one another. Imagine a crowd of people holding hands and pulling each other inward; the crowd naturally shrinks and occupies less space.

Similarly, the stronger intermolecular forces in gas $A$ pull its molecules closer together, causing the gas to contract. Therefore, under the same conditions of temperature and pressure, gas $A$ will occupy a lesser volume than gas $C$. Conversely, we can say that gas $C$ will occupy more volume than gas $A$.

Compressibility

Now, let's shift our focus to gases $B$ and $D$. This time, the $a$ values are identical ($155.21$), meaning their intermolecular attractive forces are exactly the same. But their $b$ values differ significantly.

Gas $B$ has a tiny $b$ value of $0.04136$, while gas $D$ has a much larger $b$ value of $0.4382$.

What does this mean for compressibility? Compressibility is simply how easily a gas can be squeezed into a smaller volume. If a gas has very large, bulky molecules (a high $b$ value), there is less "empty space" between them. When you try to compress gas $D$, its large molecules quickly bump into each other, and their electron clouds repel fiercely. It fights back against compression.

Gas $B$, with its much smaller molecules, has plenty of room to be squeezed. Therefore, gas $B$ is more compressible than gas $D$.

By breaking the problem down and analyzing the physical meaning of the constants, the answer reveals itself elegantly. Gas $C$ occupies more volume than gas $A$ because it lacks the strong inward pull of intermolecular forces. Gas $B$ is more compressible than gas $D$ because its molecules are smaller and less stubborn.

Combining these two profound insights, we find that they perfectly match option (b). The van der Waals equation isn't just algebra; it's the story of how molecules interact in the real world!