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.

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