Which one of the following statements about packing in solids is incorrect?

We are asked to identify which statement about packing in solids is incorrect. Let's evaluate each option step by step.

First, consider option A: "Coordination number in bcc mode of packing is 8." In body-centered cubic (bcc) packing, each atom is located at the corners of a cube and one atom is at the center. The central atom touches all eight corner atoms. Since each corner atom is shared by eight adjacent unit cells, the central atom has eight nearest neighbors. Therefore, the coordination number for bcc is indeed 8. This statement is correct.

Next, option B: "Coordination number in hcp mode of packing is 12." Hexagonal close packing (hcp) is a close-packed structure. In such structures, each atom is surrounded by six atoms in its own layer, three atoms in the layer above, and three atoms in the layer below, totaling 12 nearest neighbors. Thus, the coordination number for hcp is 12. This statement is correct.

Now, option C: "Void space in hcp mode of packing is 32%." Void space is the percentage of the unit cell volume not occupied by atoms. For close-packed structures like hcp and ccp (cubic close packing, also known as fcc), the packing efficiency is identical. The packing efficiency is calculated as the fraction of space occupied by atoms. In hcp, the atoms are arranged in layers with an ABAB... sequence. The packing efficiency for hcp is given by the formula: $$\text{Packing efficiency} = \frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} \times 100\%$$

For hcp, the unit cell contains two atoms. The volume occupied by atoms is $$2 \times \frac{4}{3}\pi r^3$$ where $$r$$ is the atomic radius. The volume of the unit cell can be derived from the geometry. The height $$c$$ of the unit cell is related to the atomic radius by $$c = \frac{4\sqrt{6}}{3}r$$, and the base area is $$2 \times \frac{\sqrt{3}}{4} \times (2r)^2 = 2\sqrt{3}r^2$$ (since the side length $$a = 2r$$). Thus, the volume of the unit cell is: $$\text{Volume} = \text{base area} \times \text{height} = (2\sqrt{3}r^2) \times \left(\frac{4\sqrt{6}}{3}r\right) = \frac{8\sqrt{18}}{3}r^3 = \frac{8 \times 3\sqrt{2}}{3}r^3 = 8\sqrt{2}r^3$$

Now, the volume occupied by atoms is: $$2 \times \frac{4}{3}\pi r^3 = \frac{8}{3}\pi r^3$$

Therefore, the packing efficiency is: $$\frac{\frac{8}{3}\pi r^3}{8\sqrt{2}r^3} \times 100\% = \frac{\pi}{3\sqrt{2}} \times 100\% \approx \frac{3.1416}{3 \times 1.4142} \times 100\% \approx \frac{3.1416}{4.2426} \times 100\% \approx 0.7405 \times 100\% = 74.05\%$$

Thus, the void space is $$100\% - 74.05\% = 25.95\% \approx 26\%$$. The statement claims void space is 32%, which does not match. Therefore, this statement is incorrect.

Finally, option D: "Void space in ccp mode of packing is 26%." Cubic close packing (ccp), also known as face-centered cubic (fcc), has the same packing efficiency as hcp. The unit cell contains four atoms. The volume occupied by atoms is $$4 \times \frac{4}{3}\pi r^3 = \frac{16}{3}\pi r^3$$. The unit cell edge length $$a$$ relates to the atomic radius by $$a = 2\sqrt{2}r$$, so the volume is $$a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3$$. The packing efficiency is: $$\frac{\frac{16}{3}\pi r^3}{16\sqrt{2}r^3} \times 100\% = \frac{\pi}{3\sqrt{2}} \times 100\% \approx 74.05\%$$ Thus, void space is approximately 26%, matching the statement. This statement is correct.

Since option C states an incorrect void space percentage for hcp, it is the incorrect statement. Hence, the correct answer is Option C.