`KCl` crystallizes in the same type of lattice as does `NaCl`. Given that
`(r_(Na^(o+)))/(r_(Cl^(Θ))) = 0.5` and `(r_(Na^(o+)))/(r_(K^(o+))) = 0.7`
Calculate (a) the ratio of side of the unit cell for `KCl` to that for `NaCl`, and (b) the ratio of density of `NaCl` to that `KCl`.

Explanation

(i) Both KCl and NaCl crystallize in face centred cubic arrangement in which the length is related to radii of cation and anion as :
`r_(Na^+) + r_(Cl^(-)) = a/2 and r_(K^+) + r_(Cl^-) = (a.)/2`
Now, `(r_(Na^+))/(r_(Cl^-)) = 0.52`
or `(r_(Na^+))/(r_(Cl^-)) + 1 = 0.52 + 1`
`" (adding 1 on both sides)"`
or `(r_(Na^+) + r_(Cl^-))/(r_(Cl^-)) = 1.52 " " .....(i) `
`(r_(Na^+))/(r_(K^+)) = 0.71`
But `r_(Na^+) = r_(Cl^-) xx 0.52 = 0.52 r_(Cl^(-))`
`:. (0.52 r_(Cl^-))/(r_(K^+)) = 0.71`
or `(r_(K^+))/(r_(Cl^-))) = (0.52)/(0.71)`
`(r_(K^+))/(r_(Cl^(-)) + 1) = (0.52)/(0.71) + 1`
`(r_(K^+) + r_(Cl^(-)))/(r_(Na^+) + r_(Cl^(-))) = (1.23)/(0.71) xx 1/(1.52)`
or `((a^.)/(2))/((a)/(2)) = 1.14`
`:. ("Side of unit cell for KCl")/("Side of unit cell for NaCl") = 1.14`
(ii) Density = `(Z xx M)/(a^3 xx N_A)`
Since Z and `N_A` are same for both KCl and NaCl
Density `prop M/(a^3)`
`:. (rho(KCl))/(rho(NaCl)) = (M(KCl))/(M(NaCl)) xx ([a(NaCl)]^(3))/([a(KCl)]^(3))`
`= (74.5)/(58.5) xx (1/(1.14))^(3) " " ( :. (a(NaCl))/(a(KCl)) = 1/1.14)`
`= 0.86`.