Step-by-step Derivation:

1. Identify the given information:

   • Crystal structure: Face-centered cubic (FCC)
   • Cell edge length, a = 400 pm
   • Density of the substance, ρ = 8 g cm⁻³
   • Total mass of the crystal, m = 256 g
   • The total number of atoms is given in the format N × 10²⁴.

2. Convert the cell edge length to cm: The density is given in g cm⁻³, so we need the cell edge length in cm. We know that 1 pm = 10⁻¹⁰ cm. $a = 400 \, \text{pm} = 400 \times 10^{-10} \, \text{cm} = 4 \times 10^{-8} \, \text{cm}$

3. Calculate the volume of the unit cell: The unit cell is cubic, so its volume V is a³. $V = a^3 = (4 \times 10^{-8} \, \text{cm})^3$ $V = 64 \times 10^{-24} \, \text{cm}^3$

4. Calculate the mass of one unit cell: The mass of a unit cell can be calculated using the density and volume. $\text{Mass of unit cell} = \text{Density} \times \text{Volume}$ $m_{\text{cell}} = ρ \times V$ $m_{\text{cell}} = (8 \, \text{g cm}^{-3}) \times (64 \times 10^{-24} \, \text{cm}^3)$ $m_{\text{cell}} = 512 \times 10^{-24} \, \text{g}$

5. Calculate the number of unit cells in 256 g of the crystal: The total number of unit cells is the total mass of the crystal divided by the mass of a single unit cell. $\text{Number of unit cells} = \frac{\text{Total mass}}{m_{\text{cell}}}$ $\text{Number of unit cells} = \frac{256 \, \text{g}}{512 \times 10^{-24} \, \text{g}}$ $\text{Number of unit cells} = \frac{1}{2} \times 10^{24} = 0.5 \times 10^{24}$

6. Calculate the total number of atoms: For a face-centered cubic (FCC) structure, the number of atoms per unit cell (Z) is 4. The total number of atoms is the number of unit cells multiplied by the number of atoms per unit cell. $\text{Total atoms} = (\text{Number of unit cells}) \times Z$ $\text{Total atoms} = (0.5 \times 10^{24}) \times 4$ $\text{Total atoms} = 2 \times 10^{24}$

7. Determine the value of N: The problem states that the number of atoms is N × 10²⁴. Comparing this with our calculated value: $N \times 10^{24} = 2 \times 10^{24}$ $N = 2$

The value of N is 2.