Density of a crystal is given by :

Options

A. `(a^(3)xxM)/(zxxN_(0))`
B. `(N_(0)xxM)/(zxxa^(3))`
C. `(zxxM)/(a^(3)xxN_(0))`
D. `(A^(3)xxN^(0))/(zxxM)`

Answer

To find the correct expression for the density of a crystal, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Density**:
Density (ρ) is defined as the mass (m) of a substance divided by its volume (V).
\[ \rho = \frac{m}{V} \]

2. **Consider a Cubic Unit Cell**:
For a cubic unit cell, the volume can be expressed as:
\[ V = a^3 \]
where \( a \) is the length of one side of the cube.

3. **Mass of the Unit Cell**:
The mass of the unit cell is determined by the number of atoms (or particles) it contains. Let \( z \) be the number of atoms per unit cell. The mass of one atom can be represented as \( m \). Therefore, the total mass of the unit cell is:
\[ \text{Mass of unit cell} = z \times m \]

4. **Substituting Mass and Volume into Density Formula**:
Now, substituting the expressions for mass and volume into the density formula:
\[ \rho = \frac{z \times m}{a^3} \]

5. **Relating Atomic Mass to Density**:
The mass of one atom can be expressed in terms of the atomic mass (M) and Avogadro's number (\( N_0 \)):
\[ m = \frac{M}{N_0} \]
where \( M \) is the atomic mass (or molecular mass for compounds).

6. **Final Expression for Density**:
Substituting this expression for \( m \) into the density formula gives:
\[ \rho = \frac{z \times \left(\frac{M}{N_0}\right)}{a^3} \]
This can be rearranged to:
\[ \rho = \frac{z \times M}{N_0 \times a^3} \]

7. **Conclusion**:
The correct expression for the density of a crystal is:
\[ \rho = \frac{z \times M}{N_0 \times a^3} \]