Question

The half-life of a radioactive isotope represents the average time it would take half of a collection of this type of nucleus to decay. For example, you start with a sample of 1000 Oxygen-15 (¹⁵O) nuclei, which has a half-life of 122 seconds. After 122 seconds, half of the ¹⁵O nuclei will have decayed into Nitrogen-15 (¹⁵N) nuclei. After another 122s, half of the remaining Oxygen nuclei will have also decayed, and so on.
Suppose you start with 3.71×10³¹⁵O nuclei and zero ¹⁵N nuclei. How many ¹⁵O nuclei remain after 122 s has passed?

How many ¹⁵N nuclei are there after 122 s has passed?

How many ¹⁵O nuclei remain after 244 s has passed?

How many ¹⁵N nuclei are there after 244 s has passed?

Suppose you start with 6.37×10³ Carbon-14(¹⁴C) nuclei. ¹⁴C has a half-life of 5730 years and decays into Nitrogen-14(¹⁴N) via a beta decay. How much time has passed if you are left with 3.18×10³¹⁴C nuclei? (The units for years is 'yr'.)

How much time has passed if you are left with 1.59×10³¹⁴C nuclei?

Answer 1

A) 122 seconds is one half-life. After 1 half-life, half of the O-15 nuclei will have decayed into N-15.
If you start with 3710 O-15 nuclei and zero N-15, after one half-life you will have 1855 O-15 nuclei remaining. So, 1.855*10^3

B) The O-15 nuclei that have decayed will have turned into N-15. So you add the 1855 new N-15 nuclei to the zero you started with and you have 1855 N-15 nuclei. So, 1.855*10^3

C) 244 seconds is two half-lives. The first half-life results in the decay of half the original O-15 nuclei, taking us from 3710 down to 1855 . The second half-life results in the decay of half of the remaining O-15 nuclei, taking us from 1855 down to 928. So, So, 0.928*10^3

D) Since the O-15 nuclei have all turned into N-15, the number of N-15 nuclei has grown from zero to 1855 to 1855 + 928 = 2783
Alternatively, you start with 3710 total nuclei. After two half-lives, 75% of them transform from O-15 to N-15. So 75% times 3710 = 2783

E) 3180 is about half of the initial 6370 C-14 nuclei. (In fact, it's 49.92 %) So approximately one half-life has elapsed.
Using some of the fancy buttons on the calculator, you can solve this exactly:

49.92% = 2^(-t/5730)
t = 5743.24 yr