Unit Cell Problem: Find Volume Unoccupied | Help Solving Entrance Exam Question

[astrophysics12]

Homework Statement

The fraction of volume unoccupied in the unit cell of the body centered cubics lattice is?

Homework Equations

The Attempt at a Solution

I got the volume occupied by the atoms as (8/3)πr³. I am not sure if it is right. I just assumed that all the atoms were of same element.
I also don't know about the dimensions of the cube.
Can somebody help me? It is from an old entrance exam question paper.

 

[ehild]

Homework Statement

The fraction of volume unoccupied in the unit cell of the body centered cubics lattice is?

Homework Equations

The Attempt at a Solution

I got the volume occupied by the atoms as (8/3)πr³. I am not sure if it is right. I just assumed that all the atoms were of same element.
I also don't know about the dimensions of the cube.
Can somebody help me? It is from an old entrance exam question paper.

What is r in your result, and how did you get that formula?
The problem asks the fraction of volume unoccupied. What do you think it means?

ehild

 

[CWatters]

I got the volume occupied by the atoms as (8/3)πr³

You should show/explain your working but I believe that's correct (if r is the radius of the atom).

I also don't know about the dimensions of the cube.

Assume the atoms are spherical and touching the one in the middle. eg so the diagonal of the cube is 4r.

 

[NascentOxygen]

Did the paper not give you a figure to base your calculations on?

I could only refer to online references, e.g., http://departments.kings.edu/chemlab/animation/bcc.html

I can't comment on similar/dissimilar atoms.

 

[astrophysics12]

What is r in your result, and how did you get that formula?
The problem asks the fraction of volume unoccupied. What do you think it means?

ehild

r is the radius of the atom.
Volume unoccupied is the free volume. It is the difference between total volume of cubic cell and the volume occupied by the atoms. Am I right?

 

[astrophysics12]

You should show/explain your working but I believe that's correct (if r is the radius of the atom).
Assume the atoms are spherical and touching the one in the middle. eg so the diagonal of the cube is 4r.

Thanks. The side should be 4/√3

 

[BruceW]

... Assume the atoms are spherical and touching the one in the middle. eg so the diagonal of the cube is 4r.

yes. Although the OP should also check that the corner spheres do not overlap with each other in this case. (Maybe it is intuitively clear to some people, but I need to check these things with pen and paper, to convince myself).

 

[BruceW]

Thanks. The side should be 4/√3

looks good. but there is an 'r' missing from here. Anyway, keep going, you have the unit cell sides and the occupied volume, so you are pretty close to the final answer now.

 

[ehild]

r is the radius of the atom.
Volume unoccupied is the free volume. It is the difference between total volume of cubic cell and the volume occupied by the atoms. Am I right?

Correct so far.
You need the ratio of the unoccupied volume to the whole volume of the unit cell. You got that the occupied volume is 8pi/3 r³, and the side of the cubic unit cell is 4r/√3.

ehild