calculate the mass of a unit cell copper crystal atomic mass of
Answers
Explanation:
A unit cell is the smallest group of atoms in a crystal lattice. A cubic unit cell has 12 edges, 8 corners, 6 faces, and 1 center where an atom may be found occupying a fraction of volume of the cell. There are three main types of cubic unit cell: primitive cubic, body centered cubic (BCC), and face centered cubic (FCC). Each one has a different arrangement of atoms and the mass of one unit can be calculated from the total number of atoms inside the cell.
Answer and Explanation:
We are asked to determine the mass of one unit of copper which crystallized in a face-centered cubic structure (FCC). To find the mass, we must determine the number of copper atoms inside a unit cell and perform mole conversions.
Step 1: Determine the number of copper atoms inside a unit cell
Considering the structure of a cubic unit cell, the fraction of an atom inside the cell depends on its position:
Edge: one-fourth of the atom is inside the cell
Corner: one-eighth of the atom is inside the cell
Face: one-half of the atom is inside the cell
Center: one-whole atom is inside the cell
Next, consider the structure of an FCC:
Note: atoms placed on corners are highlighted green while those at faces are highlighted blue
From the structure, we can see that an FCC has 1 atom in each of the 8 corners and 1 atom in each of the 6 faces. Calculate the number of atoms in total by taking into account only the fraction inside the cube
Atoms from corners
=
8
c
o
r
n
e
r
s
×
1
8
a
t
o
m
s
1
c
o
r
n
e
r
=
1
C
u
a
t
o
m
Atoms from faces
=
6
f
a
c
e
s
×
1
2
a
t
o
m
s
1
f
a
c
e
s
=
3
C
u
a
t
o
m
s
Total number of atoms
=
Atoms from corners + Atoms from faces
=
1
C
u
a
t
o
m
+
3
C
u
a
t
o
m
s
=
4
Cu atoms per unit cell
Step 2: Perform mole conversions to get mass
In order to perform mole conversions, we must first determine the conversion and constants needed:
From number of atoms, we can calculate the number of moles using Avogadro's number
From the number of moles, we can calculate the mass of Cu using its molar mass
Avogadro's number
=
6.022
×
10
23
a
t
o
m
s
Molar mass of Cu
=
63.546
g
/
m
o
l
Note: Avogadro's number gives the number of particles/ atoms in 1 mole
Perform mole conversion:
Mass of 1 unit cell
=
4
a
t
o
m
s
;
×
1
m
o
l
6.022
×
10
23
a
t
o
m
s
×
63.546
g
m
o
l
Mass of 1 unit cell
=
4.22
×
10
−
22
grams Cu per unit cell
The mass of a copper crystallized in a face-centered cubic unit cell is
4.22
×
10
−
22
grams
.
Answer:
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