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What is the distance between Na^(+) and ...

What is the distance between `Na^(+)` and `Cl^(-)` ions in NaCl crystal if its density is 2.165g `cm^(-3)` ? NaCl crystallizes in fcc lattice.

Text Solution

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Let the edge length of unit cell = a cm
For fcc structure, number of molecules per unit cell, Z=4
Gram formula mass of NaCl, M = 23 +35.5=58.5g `mol^(-1)`
Mass of unit cell = `(Z xx" Gram formula mass")/("Avogadro.s number")`
= `(4 x (58.5 g mol^(-1)))/((6.022 xx 10^(23) mol^(-1)) = 3.886 xx 10^(-22)` g
Density of unit cell = `("Mass")/("Volume")`
or `2.165 g cm^(-3) = (3.886 xx 10^(-22)g)/(a^3)`
`:. a^3 =((3.886 xx 10^(-22)g))/((2.165 g cm^(-3)) = 1.795 xx 10^(-22) cm^3`
`= 179.5 xx 10^(-24) cm^3`
Edge length, `a=(179.5)^(1//3) xx 10^(-8) cm^3`
= `5.64 xx 10^(-8)` cm=564 pm
Edge length of `Na^(+)Cl^(-)` unit cell = `2 (""^(r)N_a^(+) + ""^(r)Cl_a^(-))`=564 pm
or `(""^(r)Na_a^(+) + ""^(r)Cl_a^(-)) =(564)/(2) = 282` pm
Distance between `Na^+` and `Cl^(-)` ions = 282 pm
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