Solid State: Real-World Scenarios (8)

Question

Solid State: real-world scenarios for JEE+NEET

doubts.ai · Accepted Answer

Solution — Step by Step In an FCC lattice, atoms touch along the face diagonal — not along the edge. Three atoms (centre of face + two corners) line up along the diagonal. So face diagonal = 4r, where r is the atomic radius. Face diagonal of a cube with edge a is a2. a2 = 4r r = a2/4 = a/(22) For a = 400 \, pm: r = 400/(22) = 200/2 = 1002 141.4 \, 	ext{pm}. Atomic radius r 141.4 \, 	ext{pm}. Why This Works The relationship between edge length and atomic radius depends entirely on which direction the atoms touch. For each lattice type, the touching direction is fixed: - Simple cubic: atoms touch along edge → a = 2r - BCC: atoms touch along body diagonal → a3 = 4r, so r = a3/4 - FCC: atoms touch along face diagonal → a2 = 4r, so r = a2/4 These three relations cover 95% of solid-state problems in JEE/NEET. Memory hook: "Simple cubic — edge. BCC — body diagonal (think 3D). FCC — face diagonal (think 2D face)." Once you remember which diagonal, the geometry follows. Alternative Method — Use Packing Fraction For FCC, packing fraction = 0.74. Number of atoms per unit cell = 4. Volume of atoms = 4 (4/3)\pi r^3. Volume of cube = a^3. 0.74 = 4 (4/3)\pi r^3 / a^3 r = a (0.74 \cdot 3/(4 \cdot 4\pi))^{1/3} This gives the same answer but with much more arithmetic. Not recommended for exam pacing. Common Mistake Students often use a = 4r for FCC, confusing it with BCC's body diagonal relation. Always verify the touching direction first. Another classic: forgetting 2 in the face diagonal. Students sometimes write face diagonal = 2a (which is wrong — that's the perimeter half). JEE Main asks 1-2 solid-state problems per shift. NEET typically asks one. The three lattice formulas (SC, BCC, FCC) plus packing fraction and density formula cover most questions. CBSE Class 12 boards include this template every alternate year for 4 marks. The follow-up usually asks for density: = (Z \cdot M)/(N A \cdot a^3) where Z = atoms per cell, M = molar mass.

Question

Solution — Step by Step

Why This Works

Alternative Method — Use Packing Fraction

Common Mistake

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