Homework #1 NOTES Humphrey Glaciology Fall 2009

1 We will look at a picture in class

2 This is mainly an exercise in manipulating numbers, but it also requires thinking in 3-D in a reasonable manner. Note the full question does NOT have a correct answer.

The simplest calculation is to assume the water molecule is a cube of sides 3.6 Angstrom. Then each molecule occupies about 4 x 10^-29 m³ of space. Since there are 3.3 x 10²² molecules, the volume theoretically occupied by 1 gram of cubical water molecules is about 1.3 x 10^-6 m³ of space (1.3 cc). Since a gram of water really occupies 1 cc of space, but cubic water molecules occupy 30% more space (1.3 cc), then water molecules must be packed quite tightly, and must encroach on each other so that they occupy spaces in each other’s shapes. A somewhat better approach would be to assume the molecules are tetrahedral ( volume ~ 0.12 * a³ where a is the side length). So with tetrahedra there appears to be plenty of space.

However; to do the problem correctly, using some shape other than cubes for the molecules, requires making decisions about the packing of the shapes. Most regular shapes do not pack easily. Just to convince yourself that there is a packing problem, think of packing spheres into a box. The volume you can get in is much less than the volume of the box, and also depends heavily on the packing you use.