Question

Cromium and tantalum both have the BCC crystal structure, and Ta forms a substitutional solid solution for all concentrations at room temperature. Compute the unit cell edge length for a 26 wt% Cr - 74 wt% Ta alloy. The room-temperature density and atomic weight of Cr are 7.19 g/cm3 and 52.00 g/mol, the room-temperature density and atomic weight of Ta are 16.6 g/cm3 and 180.95 g/mol, respectively.

Answer 1

0.3139 nm

Step-by-step explanation:

First of all, let's Calculate the average density of the Cr −Ta alloy;

Formula to get it is;

ρ_avg = 100/[(C_cr/ρ_cr) + (C_ta/ρ_ta)]

We are given;

C_cr = 26%

C_ta = 74%

ρ_cr = 7.13 g/cm³

ρ_ta = 16.6 g/cm³

Thus;

ρ_avg = 100/[(26/7.13) + (74/16.6)]

ρ_avg = 12.34 g/cm³

Similarly, let's Calculate the average atomic weight;

A_avg = 100/[(C_cr/A_cr) + (C_ta/A_ta)]

We are given;

A_cr = 52 g/mol

A_ta = 180.95 g/mol

Thus;

A_avg = 100/[(24/52) + (74/180.95)]

A_avg = 114.88 g/mol

Formula for the volume of the unit cell is;

V = n(A_avg)/(N_A × ρ_avg)

Where;

n is number of atoms in the cell which in this case = 2

N_A is avogadro's number = 6.022 × 10^(23)

Thus;

V = (2 × 114.88)/(6.022 × 10^(23) × 12.34)

V = 30.92 × 10^(-24) cm³

Length of the unit cell edge length is given by the formula;

a = ∛V

Thus;

a = ∛(30.92 × 10^(-24))

a = 0.00000003139 cm

Converting to metre, we have;

a = 0.3139 × 10^(-9) m = 0.3139 nm