Question

\[\text{AgCl}\] has the same structure as that of \[\text{NaCl}\]. The edge length of the unit cell of \[\text{AgCl}\]is found to be 555pm and density of \[\text{AgCl}\]is $\text{5}\text{.561 g c}{{\text{m}}^{-3}}$. If the percentage of sites that unoccupied is x, the value of 100x is:

Answer 1

Hint: By using the relationship between density (d), number of the atoms (z), edge length (a) and atomic mass (M) of the unit cell i.e. $\text{d = }\dfrac{\text{z }\times \text{ M}}{{{a}^{3}}\text{ }\times \text{ }{{\text{N}}_{\text{A}}}}\ \text{Kg }{{\text{m}}^{-3}}$ , by putting all the value here, we can calculate the value of 100x.

Complete Step-by-Step Answer:
- In the given question, we have to calculate the value of 100x for which the density and edge length of silver chloride is given.
- As we know that that the relationship between density (d), number of the atoms (z), edge length (a) and atomic mass (M) of the unit cell is given by
$\text{d = }\dfrac{\text{z }\times \text{ M}}{{{a}^{3}}\text{ }\times \text{ }{{\text{N}}_{\text{A}}}}\ \text{Kg }{{\text{m}}^{-3}}$ ….. (1)
- So, firstly, we have to calculate the molar mass of the silver chloride i.e. $\text{AgCl = 107}\text{.86 + 35}\text{.5 = 143}\text{.36 g/mol}$
- It is given that the density and edge length of the silver chloride is $\text{5}\text{.561 g c}{{\text{m}}^{-3}}$ and 555pm respectively.
- Also, we know that the Avogadro's number is $\text{6}\text{.022 }\times \text{ 1}{{\text{0}}^{23}}$, so by putting all the value in equation (1) we will get,
$\text{d = }\dfrac{\text{z }\times \text{ M}}{{{a}^{3}}\text{ }\times \text{ }{{\text{N}}_{\text{A}}}}$ = $\text{5}\text{.561= }\dfrac{\text{z }\times \text{ 143}\text{.36}}{{{(555\text{ }\times \text{ 1}{{\text{0}}^{3-}})}^{3}}\text{ }\times \text{ 6}\text{.022 }\times \text{ 1}{{\text{0}}^{23}}}$
$\text{z = 3}\text{.995}$
- Now, we know that the sodium chloride has FCC or Face Centred Cubic lattice structure so the total number of atoms will be 4.
- But in AgCl the number of atoms is 3.995.
- Thus, the percentage i.e. unoccupied by x is:
$\dfrac{\text{4 - 3}\text{.995}}{4}\text{ }\times \text{ 100}$ = 0.12%.
- So, 100x will be equal to $100\text{ }\times \text{ 0}\text{.12 = 12}$

Therefore, the value of the 100x is 12.

Note: FCC or Face Centred Cubic unit cell is the arrangement of atoms in a cube in which the atoms are present at the corners (anion) and the face diagonal (cation). The contribution of each atom at the corner is 1/8 and the face diagonal it is, 1/2.