The atomic radius of metal X is 125 picometers (pm) and a crystal of metal X has a unit cell that is face-centered cubic. Calculate the density of metal X (atomic weight = 42.3 g/mol)

Answer 1

With face centered cubic, there is:

1/8 of an atom in 8 corners: 8(1/8)=1atom

and

1/2 of an atom on 6 sides: 6(1/2)=3 atoms

So, in one unit cell which is face-centered cubic, there are 3+1=4atoms

Using a reference table or your notes, you find that the length of a cube of a unit cell for face centered cubic is:

a=√8) r

where

a=side length of cube

r=radius

So,

r=125pm

a=√8) 125pm

a=353.55pm

So, the volume is

V=LWH

V=(353.55pm)³

V=4.41929×10⁷pm³

They probably want the volume in cubic centimeters and density in g/cm³

Convert pm³ to cm³ with

1cm=10¹⁰ pm, so 1cm³=(10¹⁰)³pm³=10³⁰pm³

So,4.41929×10⁷pm³=4.41929×10^-23cm³

This is the volume.

Now, what's the mass?

What's the mass of 4 atoms X?

42.3 g/mol = 1 mol X = 6.02×10²³ atoms X

So, the mass of 4 atoms is

4atomsX (42.3g X / 6.02×10²³ atoms X)=2.81×10^-22 g X

Now, use the density formula

D=m/v

m=2.81×10^-22 g X

V=4.41929×10^-23cm³

d=(2.81×10^-22 g X) / (4.41929×10^-23cm³)

d=6.358487 g/cm³

So, this would be 6.36 g/mL