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Question: The edge length of the unit cell of KCl(fcc) is \( 6.28\,\mathop A\limits^o \) . Assuming anion-cati...

The edge length of the unit cell of KCl(fcc) is 6.28 Ao6.28\,\mathop A\limits^o . Assuming anion-cation contact along the cell edge, calculate the radius in ( A0\mathop A\limits^0 ) of the potassium ion.
(A) 0.90.9
(B) 1.31.3
(C) 1.81.8
(D) None of the above

Explanation

Solution

The chemical name of KClKCl is potassium chloride. It is a metal halide salt formed by the potassium cation and chloride anion. It is a white crystalline powder that has an FCC lattice. It is soluble in water and is used in industries for various purposes.

Complete step by step solution:
Here, we are given an FCC lattice. In a Face-centred cubic unit cell, atoms are placed at each corner and the centre of all the faces of the cell. In FCC there are total 88 corners and 66 face centre. So the total number of atoms in an FCC structure is 44 . The relation between the lattice parameter and the radius is given by the formula:
a=4r2a = \dfrac{{4r}}{{\sqrt 2 }}
Where a=a = edge or lattice parameter
r=r = radius of the lattice
Now, we are given the edge length =6.28 Ao= 6.28\,\mathop A\limits^o and . Now, we will calculate the radius of Cl−C{l^ - } ion.
We know that the ratio of the radius of the cation to anion in an FCC lattice is 0.7310.731 .
Also, the sum of the radius of cation and anion in FCC is given as equal to half of its edge length. It is represented as:
r++r−=a2{r^ + } + {r^ - } = \dfrac{a}{2}
Where, r+={r^ + } = cation and r−={r^ - } = anion. In the given salt the cation is K+{K^ + } and the anion is Cl−C{l^ - } .
Putting all the values in the given relation we get:
rK++rCl−=a2{r_{{K^ + }}} + {r_{C{l^ - }}} = \dfrac{a}{2}
rK++1.8=6.282{r_{{K^ + }}} + 1.8 = \dfrac{{6.28}}{2}
rK++1.8=3.14=1.34{r_{{K^ + }}} + 1.8 = 3.14 = 1.34
rK+=1.34 A0{r_{{K^ + }}} = 1.34\,\mathop A\limits^0
Hence the radius of the Potassium ion will be 1.34 A01.34\,\mathop A\limits^0 .
Therefore, option (B) is correct.

Note:
Every crystal lattice is formed from a unit cell. A unit cell is the basic entity of a lattice. A unit cell is the smallest repeating unit of the cell which when repeated over and over gives the crystal lattice. A lattice constant or parameter is used to define the physical dimension of unit cells in a crystal lattice.