The h.c.p. and c.c.p. structures for a given crystal system of equal - sized spheres are expected to have

Options

A. the same volume, `1.33 pi r^(3)`
B. the same number of particles within the given unit cell
C. the same ratio, 0.74, of the total sphere volume to the unit - cell volume
D. All of the above

Answer

To solve the question regarding the hexagonal close packing (h.c.p.) and cubic close packing (c.c.p.) structures for a given crystal system of equal-sized spheres, we will analyze the properties of both structures step by step.

### Step-by-Step Solution:

1. **Understanding the Volume of Spheres:**
- The volume \( V \) of a single sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- For both h.c.p. and c.c.p., since the spheres are of equal size, the volume of the spheres will be the same.

**Hint:** Remember that the volume formula for a sphere is constant regardless of the packing arrangement.

2. **Number of Particles in Unit Cell:**
- In hexagonal close packing (h.c.p.), there are 6 spheres per unit cell.
- In cubic close packing (c.c.p.), there are 4 spheres per unit cell.
- Therefore, the number of particles (spheres) in the unit cell is different for h.c.p. and c.c.p.

**Hint:** Count the number of spheres in each packing arrangement to compare.

3. **Packing Fraction Calculation:**
- The packing fraction is defined as the ratio of the total volume of the spheres to the volume of the unit cell.
- For both h.c.p. and c.c.p., the packing fraction is approximately 0.74.
- This means that about 74% of the volume of the unit cell is occupied by the spheres in both arrangements.

**Hint:** Use the packing fraction formula to verify the efficiency of space utilization in both structures.

4. **Conclusion:**
- From the analysis:
- Both h.c.p. and c.c.p. have the same volume for individual spheres.
- They have different numbers of particles per unit cell (6 for h.c.p. and 4 for c.c.p.).
- Both structures have the same packing fraction of approximately 0.74.
- Therefore, the correct statements regarding h.c.p. and c.c.p. are:
- They have the same volume.
- They have different numbers of particles in the unit cell.
- They have the same packing fraction.

### Final Answer:
The h.c.p. and c.c.p. structures for a given crystal system of equal-sized spheres are expected to have:
- Same volume (True)
- Different number of particles in the unit cell (False)
- Same packing fraction (True)

Thus, the correct statements are that they have the same volume and the same packing fraction.