# Delete if easily shocked by maths

David Micklem dmicklem at cmgm.nospam.invalid
Mon Sep 10 12:38:08 EST 2001

```In article <KSfMxPA\$fNn7EAsg at genesys.demon.co.uk>, Dr. Duncan Clark
<news@[127.0.0.1]> wrote:

>In article <Pine.SGI.4.33.0109101536470.6683511-100000 at mole.bio.cam.ac.u
>k>, the eminent Michael Witty at University of Cambridge, England wrote
>>The other has 1mm diameter beads (approximately).
>>
>>What is the difference in surface area (which could bind protein) for,
>>let us say, 10ml of each of the beads?
>>
>>All this is so I have a rough idea about how much bigger to make the
>>
>>Formulae about the surface area of spheres don't seem to help.  I know
>>because I have tried and retired with a headache.
>
Well, I'll have a stab at this, using formulae for the area (A) and
volume (V) of a sphere:

A= 4pi*r^2
V= 4/3*pi*r^3

so A/V = 3/r.

So, for a given volume, the surface area is proportional to 1/r. The
larger beads have 10* the radius of the smaller ones and will thus have
one tenth the area.

To double check this in a more roundabout way:

First, calculate the number of spheres per ml of material:

Spheres pack randomly with a density of roughly 0.64, so approximately

The volume of a single sphere is  (4*pi*r^3)/3, so the number of
spheres in 1ml will be:

640/((4*pi*r^3)/3) = 152/r^3 (roughly).

So there will be roughly 1.2 *10^3 of your larger beads (r=0.5mm) per
ml, and 1.2 * 10^6 of the smaller (r=0.05mm) beads.

Sanity check: volume increases as the cube of radius. The larger beads
are 10 times bigger, so 1000* volume -> you should need 1000 * more of
the smaller beads for the same volume. Excellent.

The area of each bead is 4pi*r^2. So the total area in 1ml of the
4pi*r^2 * 640/((4*pi*r^3)/3) = 1920/r  = 3840 (sq mm)

Sanity check: area of one bead is 4*pi*0.5^2 = pi. 1200 beads/ml ->
3769, which is close enough given that the 1200 was only rough.

For the smaller beads the total area is 1920/0.05 = 38400 (sq mm).

ie For 1ml starting volume, the smaller beads provide 38400/3840 = 10
times the area than the large ones do.

HTH,

David

--
D.R. Micklem,              Time flies like an arrow...
Dept. of Anatomy,          Fruit flies like a banana.
Cambridge University,      Email:dmicklem at cmgm.stanford.edu
Cambridge, UK              Phone: +44 (1223) 333776
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```