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Chapter 12: Problem 28

What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice? $(\mathbf{a}) 1,(\mathbf{b}) 2,($ (c) 3 (d) 4 (e) 5.

Short Answer

Expert verified
In a face-centered cubic (FCC) lattice, there are 8 corner atoms and 6 face atoms. The corner atoms contribute 1 atom to the unit cell, while the face atoms contribute 3 atoms to the unit cell. Therefore, the minimum number of atoms in the unit cell of an element with a face-centered cubic lattice is \(1 + 3 = 4\). The correct answer is (d) 4.

Step by step solution

01

The face-centered cubic lattice structure

An FCC lattice has atoms positioned at every corner of the cube and at the center of each face. Each unit cell in an FCC lattice has 8 corner atoms and 6 face atoms. However, since the corner atoms are shared by 8 adjacent unit cells, and the face atoms are shared by 2 adjacent unit cells, we need to determine their contributions to the unit cell.
02

Calculate the contribution of the corner atoms

Each corner atom is shared by 8 unit cells. So, the contribution of a corner atom to a single unit cell is \( \frac{1}{8}\). Since there are 8 corner atoms, their total contribution to the unit cell is: \[8 \times \frac{1}{8} = 1\]
03

Calculate the contribution of the face atoms

Each face atom is shared by 2 unit cells. So, the contribution of a face atom to a single unit cell is \( \frac{1}{2}\). Since there are 6 face atoms, their total contribution to the unit cell is: \[6 \times \frac{1}{2} = 3\]
04

Determine the total number of atoms in the unit cell

Now we need to add the contributions from both the corner atoms and the face atoms to find the total number of atoms in the unit cell: \[1 + 3 = 4\] So, the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice is 4. The correct answer is (d) 4.

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Most popular questions from this chapter

An ideal quantum dot for use in TVs does not contain any cadmium due to concerns about disposal. One potential material for this purpose is InP, which adopts the zinc blende (ZnS) structure (face-centered cubic). The unit cell edge length is \(586.9 \mathrm{pm} .(\mathbf{a})\) If the quantum dot is shaped like a cube, how many of each type of atom are there in a cubic crystal with an edge length of \(3.00 \mathrm{nm} ? 5.00 \mathrm{nm} ?(\mathbf{b})\) If one of the nanoparticles in part (a) emits blue light and the other emits orange light, which color is emitted by the crystal with the \(3.00-\mathrm{nm}\) edge length? With the \(5.00-\mathrm{nm}\) edge length?

What molecular structural features cause high-density polyethylene to be denser than low-density polyethylene?

Imagine the primitive cubic lattice. Now imagine grabbing the top of it and stretching it straight up. All angles remain \(90^{\circ} .\) What kind of primitive lattice have you made?

Silver chloride (AgCl) adopts the rock salt structure. The density of \(\mathrm{AgCl}\) at \(25^{\circ} \mathrm{C}\) is $5.56 \mathrm{~g} / \mathrm{cm}^{3}$. Calculate the length of an edge of the AgCl unit cell.

(a) What is a monomer? (b) Which of these molecules can be used as a monomer: ethanol, ethene (also called ethylene), methane?
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