Density of the unit cell when atomic radius and atomic weight are given
Answers
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We see that there is 1 atom per unit cell (
1
8
atom
at each corner) and that the edge length of the cell (
a
) is twice the atomic radius (
r
).
The volume of one atom is
V
atom
=
4
3
π
r
3
.
The atom occupies the whole cube, so its effective volume is
V
eff
=
a
3
=
(
2
r
)
3
=
8
r
3
The packing efficiency is
V
atom
V
eff
=
4
3
π
r
3
8
r
3
=
4
3
π
8
=
π
6
=
0.5236
That is, the atom fills only 52.36 % of the unit cell.
Experimentally, the effective volume of an atom is
V
eff
=
1 cm
3
l
Po
9.32
g Po
×
209
g Po
1
mol Po
×
1
mol Po
6.022
×
10
23
l
atoms Po
=
3.723
×
10
-23
l
cm
3
/atom Po
But
V
atom
V
eff
=
0.5236
∴
V
atom
=
0.5236
V
eff
=
0.5236
×
3.723
×
10
-23
l
cm
3
=
1.950
×
10
-23
l
cm
3
4
3
π
r
3
=
1.950
×
10
-23
l
cm
3
r
3
=
3
4
π
×
1.950
×
10
-23
l
cm
3
=
4.655
×
10
-24
l
cm
3
r
=
1.67
×
10
-8
l
cm
=
1.67
×
10
-10
l
m
=
167
×
10
-12
l
m
=
167 pm
The radius of a polonium atom is 167 pm.