Animated Solution for Chemistry - States of Matter: The parameters of the unit cell of a substance are $a=2.5,b=3.0,c=4.0$, $\alpha =90^{\circ },\beta =120^{\circ },\gamma =90^{\circ }$. The crystal system of the substance is

A Journey into the Monoclinic Unit Cell

The solid state of matter is a beautiful display of geometric perfection. At the heart of this perfection lies the unit cell, the smallest repeating building block of a crystal lattice. By simply measuring the edge lengths and the angles between them, we can classify any crystal into one of the seven fundamental crystal systems. Let's dive into this problem and decode the geometry of our mystery substance.

The problem provides us with the exact dimensions of a unit cell. We are given three edge lengths: $a=2.5$, $b=3.0$, and $c=4.0$. We are also given the three axial angles: $\alpha =90^{\circ }$, $\beta =120^{\circ }$, and $\gamma =90^{\circ }$.

Our first step is to translate these raw numbers into mathematical relationships. Looking at the edge lengths, it is immediately clear that none of them are equal. We can write this as the inequality $a eq b eq c$.

Next, we examine the angles. We see that two of the angles, $\alpha$ and $\gamma$, are perfectly orthogonal, meaning $\alpha =\gamma =90^{\circ }$. However, the third angle, $\beta$, is $120^{\circ }$, which means $\beta eq 90^\circ$. These two sets of conditions are the unique fingerprint of our crystal system.

Now, we must match our fingerprint against the standard seven crystal systems. Let's test a few possibilities:

Could it be Hexagonal? A hexagonal system does feature a $120^{\circ }$ angle (specifically $\gamma =120^{\circ }$), but it strictly requires two of its edge lengths to be equal ($a = b eq c$). Since our edge lengths are all different, the hexagonal system is ruled out.

What about Orthorhombic? An orthorhombic system does have unequal edge lengths ($a eq b eq c$), but it requires all three angles to be exactly $90^{\circ }$ ($\alpha =\beta =\gamma =90^{\circ }$). Because our $\beta$ is $120^{\circ }$, this is also incorrect.

Could it be Triclinic? The triclinic system is the most unsymmetrical of all, where nothing is equal. It requires $a eq b eq c$ and $\alpha eq \beta eq \gamma eq 90^\circ$. Since we have two $90^{\circ }$ angles, the triclinic system is out of the question.

By eliminating the others, we arrive at the Monoclinic system. Let's verify its standard parameters. A monoclinic unit cell is defined by having all three edge lengths unequal ($a eq b eq c$), two angles equal to $90^{\circ }$, and one angle not equal to $90^{\circ }$ ($\alpha = \gamma = 90^\circ, \beta eq 90^\circ$).

Our derived conditions match this definition flawlessly! Therefore, the substance crystallizes in a monoclinic system.

Pro-Tip for JEE: Questions like this are highly scoring if you have the parameters memorized. Use the mnemonic CTOMHRT (Cubic, Tetragonal, Orthorhombic, Monoclinic, Hexagonal, Rhombohedral, Triclinic) to remember the systems in order of decreasing symmetry. And always remember, in a monoclinic system, it is conventionally the $\beta$ angle that deviates from $90^{\circ }$.