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.

0.3139 nm

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

part 1: the half-life of the element is 22 years.

part 2: it will take 132.02 year for the sample to decay from 308.0 g to 4.8125 g.

explanation:

part 1: there are two ways to determine the half life time.

first way: the half life time can be determined directly from its definition as it is the time required for the sample to decay to its half concentration.

from the data the initial concentration was 45.0 g and decayed to its half value (22.5 g) in 22.0 years.

second way: the radio active decay is considered first order reaction so we can use the laws of first order reactions.

k = (1/t) ln (a/a-x), where k is the rate constant of the reaction,

t is the time of the reaction

a is the initial concentration of the reactant

a-x is the remaining concentration at time (t)

a = 45.0 g the initial concentration at t = 0 year.

at t = 22 year, a-x = 22.5 g

then k = (1/22 year) ln (45.0 g/ 22.5 g) = 0.0315 year-1.

the half-life time (t1/2) = ln 2/ k = 0.693/ k =0.693/ 0.0315 = 22 years.

part 2:

using the same law k = (1/t) ln (a/a-x),

using the given data; a = 308.0 g and a-x = 4.8125 g and the calculated k from the part 1 (k = 0.0315 year-1)

t = (1/k) ln (a/a-x) = (1/0.0315) ln (308.0/ 4.8125) = 132.02 years