The separation of planes in an orthorhombic cell can be determined using the formula:

d = 1 / √((h²/a²) + (k²/b²) + (l²/c²))

where:
- d is the separation distance between planes
- h, k, and l are the Miller indices of the plane
- a, b, and c are the lattice parameters of the orthorhombic cell.

Given the lattice parameters a = 0.25 nm, b = 0.5 nm, and c = 0.75 nm, we can calculate the separation of planes.

Calculation:

Let's consider the planes with Miller indices (hkl) as an example.

1. Calculate (h²/a²):

(h²/a²) = h² / (0.25 nm)²

2. Calculate (k²/b²):

(k²/b²) = k² / (0.5 nm)²

3. Calculate (l²/c²):

(l²/c²) = l² / (0.75 nm)²

4. Sum the three terms:

[(h²/a²) + (k²/b²) + (l²/c²)]

5. Take the square root and calculate the reciprocal to find the separation distance:

d = 1 / √[(h²/a²) + (k²/b²) + (l²/c²)]

This formula gives the separation distance between planes in the orthorhombic cell.

Example:

Let's consider the planes with Miller indices (111) as an example.

1. Calculate (h²/a²):

(1²/0.25 nm)² = 16 nm⁻²

2. Calculate (k²/b²):

(1²/0.5 nm)² = 4 nm⁻²

3. Calculate (l²/c²):

(1²/0.75 nm)² = 1.7778 nm⁻²

4. Sum the three terms:

16 nm⁻² + 4 nm⁻² + 1.7778 nm⁻² = 21.7778 nm⁻²

5. Take the square root and calculate the reciprocal to find the separation distance:

d = 1 / √21.7778 nm⁻² = 0.215 nm

Therefore, the separation of (111) planes in the given orthorhombic cell is approximately 0.215 nm.