Below are listed the atomic weight, density and atomic radius for three hypothetical metal alloys. For each determine whether its crystal structure is FCC, BCC, or simple cubic and then justify your determination. Alloy
A
B
C
Atomic
Weight
(g/mol)
43.1
184.4
91.6
Density
(g/cm
3)
6.40
12.30
9.60
Atomic
Radius
(nm)
0.122
0.146
0.137

Alloy A is a BCC structure

Alloy B is SC structure

Alloy C is BCC structure

Explanation:

To solve for each of these three alloys we will need to do so by trial and error method to calculate the density and compare it to the value given in the problem. The density formula is;

ρ = nA/(V_c × N).

Where;

n is 1,2 and 4 for SC, BCC & FCC crystal structures respectively.

A is atomic weight

V_c is volume expressed as a³.

a is given by 2R,4R/√3, 2R√2 for SC, BCC & FCC crystal structures respectively with R being the atomic radius.

N is avogadros number of atoms and has a constant value of 6.022 × 10^(23) atoms/mol

A) Thus,

For metal alloy A, we are given;

A = 43.1 g/mol

If we assume it's a BCC structure, then;

n = 2

V_c = a³ = (4R/√3)³ = (4 × 1.22 × 10^(-8) × 1/√3)³ = 22.4756 × 10^(-24) cm³

N = 6.022 × 10^(23) atoms/mol

Plugging in the relevant values into the density formula, we have;

ρ = (2 × 43.1)/(22.4756 × 10^(-24) × 6.022 × 10^(23)) ≈ 6.4 g/cm³

This value is same as the one in the question, thus metal alloy A is a BCC crystal structure

B) For metal alloy B, we are given;

A = 184.4 g/mol

If we assume it's a SC structure, then;

n = 1

Thus;

V_c = a³ = (2 × 1.46 × 10^(-8))³ = 24.897 × 10^(-24) cm³

Plugging in the relevant values into the density formula, we have;

ρ = (1 × 184.4)/(24.897 × 10^(-24) × 6.022 × 10^(23)) ≈ 12.3 g/cm³

This value is same as the one in the question, thus metal alloy B is an SC crystal structure

C) For metal alloy C, we are given;

A = 91.6 g/mol

If we assume it's a BCC structure, then;

n = 2

V_c = a³ = (4R/√3)³ = (4 × 1.37 × 10^(-8) × 1/√3)³ = 31.67 × 10^(-24) cm³

Plugging in the relevant values into the density formula, we have;

ρ = (2 × 91.6)/(31.67 × 10^(-24) × 6.022 × 10^(23)) ≈ 9.6 g/cm³

This value is same as the one in the question, thus metal alloy C is an SC crystal structure

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