We replace 7.5 atomic percent of the chromium atoms in its...

We replace 7.5 atomic percent of the chromium atoms in its BCC crystal with tantalum. X-ray diffraction shows that the lattice parameter is $0.29158 \mathrm{~nm}$. Calculate the density of the alloy.

Key Concepts

Substitutional Alloy and Average Atomic Mass

In substitutional alloys, some atoms in the host material are replaced by atoms of a different element. This alters the average atomic mass of the unit cell. Calculating the overall average atomic mass involves weighting the atomic masses of the constituent elements by their respective atomic percentages, which is crucial in subsequent density computations.

X-ray Diffraction and Lattice Parameter Determination

X-ray diffraction is a key experimental technique for accurately determining the lattice parameters of crystalline materials. The lattice parameter obtained from XRD directly influences the calculation of the unit cell volume, thereby affecting the overall density calculation of the alloy.

Density Calculation in Crystalline Solids

This concept involves calculating the density of a crystalline material by determining the mass of all atoms in a unit cell and dividing it by the volume of that cell. The formula used typically is density = (number of atoms per unit cell × average atomic mass) / (Avogadro's number × unit cell volume). This approach bridges the atomic scale properties with the macroscopic density measurement.

Crystal Structure and Unit Cell Volume

Understanding the crystal structure is vital because it tells us the arrangement and number of atoms per unit cell. In a cubic system like BCC (body?centered cubic), which has a characteristic lattice parameter 'a', the unit cell volume is calculated as a³. The specific atomic arrangement (two atoms per unit cell in BCC) is integral in computing the density.

Transcript

00:01 We know that the density of our structure is 17 grams per cubic centimeter.

00:06 But that's the same thing as saying, the density is 17 times 10 to the third kilograms per meter cubed.

00:13 We should also know that our molar mass is 180 .95 times 10 to the negative third kilograms per mole.

00:23 Our crystal is body -centered cubic.

00:26 So that means if we have eight corners and every one quarter we have one quarter, we have one eighth of an atom, we have one atom, but if we want to find the total, we'll take one plus one to find two.

00:39 So we'll have a total of two atoms in our cell.

00:43 We know that volume is equivalent to z times m divided by n a times row, where v is our volume, z is our total number of atoms, m is our molar mass, and a is alvigradra's number, and row is density.

01:00 But we know in solid state chemistry, more specifically for a body -centered cubic, volume is equivalent to our side length cubed.

01:12 So if we want to find our side length, side length is given by a is equivalent to z times m, divided by avragadro's number times density, raised to the one -third power.

01:27 So now we can find a.

01:29 A is equivalent to 2 times 180 .95 times 10 to the negative third, divided by avrogadro's number multiplied by 17 times 10 to the third.

01:41 We'll get that our side length a is 3 .282 times 10 to the negative 10 meters, but that is also the same thing as saying that our side length a is 328 .2 pica meters...

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