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Question
The unit cell of Na is bcc and its density is 0.97 g/cm3. What is the radius of a sodium atom if the molar mass of Na is 23 g/mol?
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Solution
Given: Density = 0.97 g/cm3, Molar mass (M) = 23 g/mol
To find: Radius of sodium atom (r)
Formula:
1. Density (ρ) = `"M n"/("a"^3 "N"_"A")`
2. For bcc unit cell, r = `(sqrt3 "a")/4`
Calculation: For a bcc lattice, number of atoms per unit cell is 2.
∴ n = 2
From formula (i),
`"a"^3 = "M n"/(rho "N"_"A")`
`"a"^3 = (23 "g mol"^-1 xx 2 "atom")/(0.97 "g cm"^-3 xx 6.022 xx 10^23 "atom mol"^-1) = 7.875 xx 10^-23 "cm"^3`
a = `root(3)(7.875 xx 10^-23 "cm"^3) = 4.286 xx 10^-8`cm
From formula (ii),
r = `(sqrt3 "a")/4 = (sqrt3 xx 4.286 xx 10^-8)/4 = 1.86 xx 10^-8` cm
Radius of sodium atom is 1.86 × 10-8 cm.
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