The half-life of a radioactive isotope represents the average time it would take

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.
1. Suppose you start with 4.52×10315O nuclei and zero 15N nuclei. How many 15O nuclei remain after 122 s has passed?

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

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

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

5. Suppose you start with 5.22×103 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 2.61×10314C nuclei? (The units for years is 'yr'.)

6. How much time has passed if you are left with 1.30×10314C nuclei?

Explanation / Answer

As mentioned here, 122 seconds is one half-life which means half of whatever O-15 nuclei was present will be decayed to N-15.

1) If you start with 4520 O-15 nuclei and zero N-15 nuclei. After 122sec you will have 2260 O-15 nuclei remaining.

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

- Whatever O-15 nuclei decay after 1 hal-flife will be turning to N-15 nuclei.

So, you will have 2260 N-15 nuclei

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

-244 seconds means two half-life

After first 122 seconds we were left with 2260 O-15 nuclei.

After next 122 seconds, decay of half of present O-15 nuclei occurs so, we will have 1130 O-15 nuclei

4) How many 15N nuclei remain after 244 s has passed?

-All of the O-15 nuclei that have decayed in two half-lives will get converted to N-15.

So, after first half-life(122sec) the N-15 will be 2260 and after next half-life(122sec) the N-15 will increase more by 1130 and will become 3390.

hence after 244sec N-15 nuclei passed=3390

5) Suppose you start with 5.22×103 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 2.61×103 14C nuclei? (The units for years is 'yr'.)

-2.61 * 103 is half of 5.22 * 103 So, half of the carbon 14C have decayed. So, time passed will be equal to half-life

hence timepassed T=5730 years

6) How much time has passed if you are left with 1.30×103 14C nuclei?

-1.30 X 103 nuclei = 24.90% of original

log(0.2490)=-(t/ Thalf-life)* log2

-0.6038=-(t/5730) * 0.3010

t=11494 years approx

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