If we had a ton (2,240 lbs.) of titanium dioxide and wanted to make it
into a cube, what would be the dimensions of the cube?

Hi!!

Titanium dioxide (TiO2)group is composed of rutile, anatase, and
brookite. Their different densities are, in g/cm3:

                Rutile      Anatase     Brookite 
Density         4.2743       3.895       4.123


To make easier the calculations, we will take an average density of 4
g/cm3 for TiO2. This means that 4 grams of TiO2 have a volume of 1
cm3, we need to know how much volume has 1 ton of TiO2.

Since 1 ton = 1x10^6 g the volume occuped by 1 ton of TiO2 is:
1x10^6 / 4 = 250,000 cm3

Recall that the volume of a cube is A^3, where A is the length of each
edge of the cube.
Ok, you know the volume that the cube will have, that is 250,000 cm3, then:
A^3 = 250,000 cm3 ==> 
==> A = (250,000)^(1/3) = 62.996 cm (round this to 63 cm) = 24.8 inches.

The cube has an edge length of 24.8 inches or 63 cm.


What did we do?
Using the definition of density, and knowing the weight of the
compound we determined which volume it occupes. Then, knowing the
geometrical properties of the cube we determined the edge lenght by
isolating it from the Volume of the cube formula.

See for references the following pages:
"TiO2 Structures":
http://ruby.colorado.edu/~smyth/min/tio2.html

"Cube Volume Equation - Engineers Edge":
http://www.engineersedge.com/volume_calc/cube.htm


Seardh strategy:
"titanium dioxide" density
cube volume


I hope that this helps you. Feel free to request for a clarification
if you need it.

Regards,
livioflores-ga

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Request for Answer Clarification by renoir-ga on 26 Oct 2005 06:58 PDT

Thanks for such a speedy answer.  I'm surprised that the cube would
only be 24.8 inches long/ side.  It does not seem possible that the
cube would be so small. I'm a layman.  Could you explain in layman's
terms ( no scientific words) why the cube would be so small.  realize
this is another question,  so I will add a tip of $4.00 for this
answer.

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Clarification of Answer by livioflores-ga on 26 Oct 2005 08:09 PDT

Hi!!

The density of a material is its mass for a given volume. The volume
of a material in a container can be calculated from its density and
mass or weight.

With an aproximated density of 1 kg per liter, the water is a common
reference for relative volumes and weights. For the water 1000 liters
will weight a ton (1000 kg). You know that 1 cubic meter has 1000
liters of capacity, this means that you will need a cube of 1 meter
(39.37 inches) length each edge for 1000 liters of water.
But TiO2 has more density (about 4 kg per liter or dm3), that means
for the same cube TiO2 will weight 4 times more, in the case of the 1
meter cube, the TiO2 cube will weight 4 tons. For only one ton of TiO2
you will need a cube with only a quarter of a cubic meter of capacity,
that is a cube with an edge length that fits:
V = A^3 = 1/4 m3 ==> A = (1/4)^(1/3) m = 0.63 meters = 24.8 inches.

The layman explanation is that due its high density, 4 times the
density of the water, for a given weight the TiO2 occupes four times
the volume that the same amount of water (in weight) does. This means
that you will need an smaller container for the TiO2 than the one you
will need for the water, more precisely with only a quarter of the
capacity of the needed for the water. Since volumes have a cubic
relationship with the container's dimensions, the ratio between the
dimensions of the different containers, for regular geometric shapes
(spheres, cilinders, rectangular prisms, cubes), is:
(1/4)^(1/3) = 0.63

Note that inversely, for a given volume the TiO2 will weight four
times the weight of the same amount (in volume) of water.


For additional reference see:
"Density - Wikipedia, the free encyclopedia":
http://en.wikipedia.org/wiki/Density


Hope this helps you. Feel free to continue using the clarification
feature if you need it.

Regards, 
livioflores-ga

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Clarification of Answer by livioflores-ga on 26 Oct 2005 20:09 PDT

Thank you for the tip and good comments.

I need to correct a typo on the clarification; where it says:
"The layman explanation is that due its high density, 4 times the
density of the water, for a given weight the TiO2 occupes four times
the volume that the same amount of water (in weight) does."
It must say:
"The layman explanation is that due its high density, 4 times the
density of the water, for a given weight the WATER occupes four times
the volume that the same amount of TiO2 (in weight) does."

Thank you again and exceuse the typo.

Best regards,
livioflores-ga

Great answer.  Also answered my request for clarification.  And very quickly, also.