Question

The \[{\text{KCl}}\] crystallizes in the same type of lattice as does \[{\text{NaCl}}\] . Given that
\[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = 0.5\] and
\[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}}}}{{{{\text{r}}_{{{\text{K}}^ + }}}}} = 0.7\]
Find the ratio of the sides of the unit cell for \[{\text{KCl}}\] to that for \[{\text{NaCl}}\] :

Answer 1

Hint:We first need to calculate the relation between the ionic radius and edge length of the \[{\text{NaCl}}\] type structure. The \[{\text{NaCl}}\] type structure is face centred structure. The length of the edges can be calculated using Pythagoras theorem.

Complete step by step answer:
In sodium chloride structure or FCC that is face centred cubic structure the atoms are present on the corners of the cube as well as the centre of the each face of the cube. The atoms in case of FCC structure touches along the face diagonal. The length of the cube with edge length “a” has a face diagonal as calculated from the Pythagoras theorem is \[\sqrt 2 {\text{a}}\] . Now in case of a face centred cube the atoms touch along the face diagonal so the relation between the ionic radius and edge length will be calculated using the face diagonal length. If the cationic radius is \[{{\text{r}}_ + }\] and anionic radius is \[{{\text{r}}_ - }\] . Then the relation between the face diagonal and ionic radius is:
\[{\text{2}}\left( {{{\text{r}}_ + } + {{\text{r}}_ - }} \right) = \sqrt 2 {\text{a}}\]
In the question we need to calculate the ratio as follow because 2 will cut in both numerator and denominator:
\[\dfrac{{{{\left( {{{\text{r}}_ + } + {{\text{r}}_ - }} \right)}_{{\text{KCl}}}}}}{{{{\left( {{{\text{r}}_ + } + {{\text{r}}_ - }} \right)}_{{\text{NaCl}}}}}} = ?\]
We have been given:
\[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = 0.5\]
We will add 1 to the both side and hence we will get:
\[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} + 1 = 0.5 + 1\]
Taking LCM we will get,
\[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}} + {{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = 1.5\]
We have been given
\[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}}}}{{{{\text{r}}_{{{\text{K}}^ + }}}}} = 0.7\] and \[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = 0.5\]
Dividing both of them we will get,
\[\dfrac{{{{\text{r}}_{{{\text{K}}^ + }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = \dfrac{{0.5}}{{0.7}}\]
Following the same process we will get,
\[\dfrac{{{{\text{r}}_{{{\text{K}}^ + }}} + {{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = \dfrac{{0.5}}{{0.7}}\]
We have two equations as:
\[\dfrac{{{{\text{r}}_{{{\text{K}}^ + }}} + {{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = \dfrac{{1.2}}{{0.7}}\] and \[\dfrac{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}} + {{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}}{{{{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = 1.5\]
We will divide both of them and we will get the following ratio:
\[\dfrac{{{{\text{r}}_{{{\text{K}}^ + }}} + {{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}}{{{{\text{r}}_{{\text{N}}{{\text{a}}^ + }}} + {{\text{r}}_{{\text{C}}{{\text{l}}^ - }}}}} = \dfrac{{1.2}}{{1.5 \times 0.7}} = 1.143\]
Hence, the ratio between edges between the is \[1.143\].

Note:
In ionic crystals the similarly charged ions never touch each other due to the electrostatic repulsion that occurs between similarly charged ions which decreases the stability of the crystal. Always ions with opposite charges touch each other so that attraction increases and crystal formed is stable.