1. After 42 days a 2.0 g sample of phosphorus-32 contains only 0.25 g of the isotope. What is the half-life of phosphorus-32?

Half-life means the time for a given sample to decrease in mass by half.

So let's say H is the half-life of phosphorus-32 in days:

After H days, the 2.0 g sample becomes 1.0 g (shrinks by half).

After 2H days (one more half-life), the sample becomes 0.5 g (because it shrinks by half again).

After 3H days (tack on another half-life), the sample becomes 0.25 g (shrinks in half one more time).

So we know that after 3H days, the 2.0 g sample will become 0.25 g.

We also know that after 42 days, the 2.0 g sample becomes 0.25 g.

We can therefore set this equality:

3H = 42 days

Divide both sides by 3:

3H / 3 = 42 days / 3

H = 14 days

The half-life of phosphorus-32 is 14 days.