The Question is from neet-chemistry the-solid-state , and it was created by the author P SRAVAN KUMAR

Question

A hypothetical crystalline substance exhibits a unique crystal structure with the following unit cell dimensions: $a = b 
eq c$, $\alpha = \beta = 90^\circ$, and $\gamma = 120^\circ$.  This substance most likely belongs to which crystal system?

P SRAVAN KUMAR · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
 **Primary Concept**: Crystal Systems and Unit Cell Parameters
 **Related Topics**: Crystallography, Bravais Lattices
 **Prerequisites**: Understanding of unit cell dimensions ($a, b, c$) and interaxial angles ($\alpha, \beta, \gamma$)
## SOLUTION PATHWAY
**Given Information**:
  $a = b 
eq c$
  $\alpha = \beta = 90^\circ$
  $\gamma = 120^\circ$
**Method & Steps**:
 1. Identify the relationships between unit cell parameters.
  Reasoning:  We need to compare the given parameters ($a, b, c, \alpha, \beta, \gamma$) to the defining characteristics of different crystal systems.
 2. Match the parameters to the crystal systems.
  Working: The given parameters show that two axes ($a$ and $b$) are equal in length and are at a right angle to each other ($\alpha = \beta = 90^\circ$).  The third axis ($c$) has a different length, and the angle between $a$ and $b$ ($\gamma$) is $120^\circ$. This matches the defining characteristics of the hexagonal crystal system.
  Key Point:  The hexagonal system is characterized by $a = b 
eq c$ and $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$.
 3. Determine the crystal system.
  Result: Hexagonal
  Verification:  Double-check the parameters against the defining characteristics of the hexagonal system.
 ## OPTION ANALYSIS
A :x: Trigonal systems have $a = b = c$ or $a = b 
eq c$, but with different angle constraints than those given.
B :x: Orthorhombic systems have $a 
eq b 
eq c$ and $\alpha = \beta = \gamma = 90^\circ$.
C :x: Monoclinic systems have $a 
eq b 
eq c$ and $\alpha = \gamma = 90^\circ$, $\beta 
eq 90^\circ$.
D ✓: Hexagonal systems are defined by $a = b 
eq c$, $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing hexagonal and trigonal systems: Pay close attention to the interaxial angles.
  2.Misinterpreting the meaning of  $
eq$: Remember that $a = b 
eq c$ means $a$ and $b$ are equal, but different from $c$.
 **Quick Tips**:
  1.Create a table summarizing the characteristics of different crystal systems to aid memorization.
  2.Visualize the unit cell structures to understand the relationships between the parameters.

P SRAVAN KUMAR · Answer

Trigonal

P SRAVAN KUMAR · Answer

Orthorhombic

P SRAVAN KUMAR · Answer

Monoclinic

P SRAVAN KUMAR · Answer

The correct Option is "Hexagonal"

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