A rock containing a newly discovered fossil is found to contain 5 mg of an unstable form of potassium and 5 mg of the stable element formed from its decay. If the half-life of the unstable form of potassium is 1.3 billion years, how old is the rock? What can you infer about the age of the fossil?

well now, it's a bit unclear.

If we are to infer that half of the unstable potassium has decayed into the stable form, then that means one half-life has elapsed, and the rock is about 1.3 gigayears old.

However, the stable element ha a lower atomic weight, so 5mg of the stable element represents more than half of the original unstable potassium. Picky, but that's how things are.

On the other hand, we're dealing with such a long time span that the slight variation probably won't affect much.

Naturally, the fossil is going to be older than the surrounding rock, but probably not by a lot, or the bones would have decayed before being fossilized.

mmmhh, that's a tough one.

the fossil is 1.3 billion years old

I think the decay is either to Ca-40 or Ar-40 and both have atomic masses so close to K it probably isn't worth worrying about in relation to 1.3 billion years.

To determine the age of the rock, you can use the concept of radioactive decay and the ratio of the unstable potassium isotope (parent isotope) to the stable element (daughter isotope).

The half-life of the unstable potassium isotope is given as 1.3 billion years. This means that in 1.3 billion years, half of the parent isotope will decay into the stable daughter isotope.

In the given scenario, the rock contains 5 mg of both the unstable and stable forms of potassium. Since the half-life is 1.3 billion years, this means that the rock started with 10 mg of the unstable potassium isotope.

You can calculate the number of half-lives that have occurred by dividing the starting amount of the unstable isotope (10 mg) by the amount of the parent isotope that remains (5 mg). In this case, you get 10 mg / 5 mg = 2 half-lives.

Since each half-life represents 1.3 billion years, you can multiply the number of half-lives (2) by the length of each half-life to determine the age of the rock. 2 half-lives * 1.3 billion years/half-life = 2.6 billion years.

Therefore, the rock is approximately 2.6 billion years old.

As for the fossil, the fact that it was found within the rock suggests that the fossil is also around the same age as the rock itself. The age of the fossil can be inferred to be approximately 2.6 billion years as well.