Although most natural potassium nuclei are stable 39K (93.3%) or 41K (6.7%) nuclei, about 0.0117% are radioactive 40K nuclei, which have a half-life of 1.26 billion years. If you were worried about 40K radioactivity and wanted to wait for 97 % of the 40K in the environment to decay away, how long would you have to wait?
To calculate how long you would have to wait for 97% of the 40K nuclei to decay away, we can use the concept of the half-life.
The half-life of 40K is given as 1.26 billion years. This means that in 1.26 billion years, half of the 40K nuclei would have decayed.
So, let's calculate the number of half-lives required for 97% of the 40K nuclei to decay:
1. Start with the initial amount of 40K, which is 0.0117%.
- Convert it to the decimal form: 0.0117% = 0.0117/100 = 0.000117
2. Calculate the remaining amount of 40K after one half-life:
- Half of the initial amount would have decayed, so the remaining amount would be 0.000117 * 0.5 = 0.0000585
3. Repeat step 2 for each subsequent half-life until you reach 97% decay:
- Half-life 2: 0.0000585 * 0.5 = 0.00002925
- Half-life 3: 0.00002925 * 0.5 = 0.000014625
- ...
- Half-life n: amount * 0.5 (keep halving until the amount is less than 0.03%, i.e., 97% decayed)
4. Keep calculating the half-life until the remaining amount is less than 0.00002925. This represents 97% decay, as we started with 0.0117%.
Now, let's calculate the number of half-lives required to reach 97% decay:
0.00002925 = 0.000117 * 0.5^n
Divide both sides by 0.000117:
0.00002925 / 0.000117 = (0.000117 * 0.5^n) / 0.000117
0.25 = 0.5^n
Take the logarithm of both sides:
log(0.25) = log(0.5^n)
-2 = n * log(0.5)
Solve for n:
n = -2 / log(0.5)
Using a logarithm calculator or mathematical software, we find:
n ≈ 2.3219
So, you would need approximately 2.3219 half-lives for 97% of the 40K to decay.
To find the total time required, multiply the number of half-lives by the half-life of 40K:
Total time = n * half-life
Total time = 2.3219 * 1.26 billion years
Total time ≈ 2.93 billion years
Therefore, you would have to wait approximately 2.93 billion years for 97% of the 40K nuclei in the environment to decay away.