Question

Suppose that the mass of a single Ag-atom is ’m’. Ag metal crystallizes in fcc lattice with a unit cell of length ‘a’. The density of Ag metal in terms of ‘a’ and ‘m’ is:
[A] $\dfrac{4m}{{{a}^{3}}}$
[B] $\dfrac{2m}{{{a}^{3}}}$
[C] $\dfrac{m}{{{a}^{3}}}$
[D] $\dfrac{m}{4{{a}^{3}}}$

Answer 1

We can calculate this by using the formula, Density = Mass$\div$Volume. For mass, we need to calculate the number of atoms in each unit cell in a face centred cubic lattice and put it in the above formula to get the answer.

Complete answer:

Fcc lattice is the abbreviation used for Face Centred Cubic lattice. In a face centred cubic lattice, atoms are arranged in each corner and at the centre of each face.
To calculate the density, we need to find out the number of atoms in each unit cell of the face centred cubic lattice.

As we know, there are 8 corners in a cube. Therefore, it gives us a total of 8 atoms in the corners.

Each corner atom is shared with 8 other unit cells i.e. contribution from each corner atom is $\dfrac{1}{8}$. Therefore, total contribution of the 8 corner atoms= $8\times \dfrac{1}{8}$ = 1.

Now, as we know there are 6 faces in a cube, so we have 6 face centred atoms. But each face centred atom is shared among 2 unit cells i.e. contribution from each face centred atom is $\dfrac{1}{2}$. Therefore, total contribution of the 6 face centred atoms = $6\times \dfrac{1}{2}$ = 3

Therefore, the total number of atoms in each unit cell = 4.

As we can see it is given in the question, the mass of a single Ag atom is ‘m’. And Ag metal crystallizes in the fcc lattice i.e. it has 4 atoms per unit cell. Therefore, mass of 4 Ag atoms= 4m

The length of the unit cell is given as ‘a’. Since, it is a cube its volume will be ${{a}^{3}}$.

As we know, Density = Mass$\div$Volume.

Substituting the value of mass and volume from the above calculation in this formula, we get

Density = $\dfrac{4m}{{{a}^{3}}}$.

Therefore, the correct answer is the option [A] $\dfrac{4m}{{{a}^{3}}}$.__

Note:
We must remember the placement of atoms in different lattices and the contribution by each atom to find out the number of atoms in each unit cell of that lattice. It is important here to remember that a face centred lattice does not have an atom at its body centre like the body centred cubic lattice. From the name itself it is clear that it will contain atoms at the face centre and the corner atom is counter in every lattice.