the half life of thorium -229 is 7340 years. how long will it take for sample of this substance to decay to 20% of its original amount
just find t such that
(1/2)^(t/7340) = 1/5
To determine how long it will take for a sample of thorium-229 to decay to 20% of its original amount, we can use the concept of half-life.
The half-life of thorium-229 is given as 7340 years. This means that after 7340 years, half of the original thorium-229 will have decayed.
Let's calculate the number of half-lives it will take for the thorium-229 sample to decay to 20% of its original amount.
1 half-life: 50% of original amount remaining
2 half-lives: (50/2)% = 25% of original amount remaining
3 half-lives: (25/2)% = 12.5% of original amount remaining
4 half-lives: (12.5/2)% = 6.25% of original amount remaining
5 half-lives: (6.25/2)% = 3.125% of original amount remaining
6 half-lives: (3.125/2)% = 1.5625% of original amount remaining
7 half-lives: (1.5625/2)% = 0.78125% of original amount remaining
8 half-lives: (0.78125/2)% = 0.390625% of original amount remaining
9 half-lives: (0.390625/2)% = 0.1953125% of original amount remaining
...
and so on.
To simplify the calculation, let's convert the 20% threshold to a decimal as 0.20.
The equation for the remaining amount after n half-lives is:
Remaining amount = (1/2)^(n)
We want to find the value of n when the remaining amount is equal to 0.20.
(1/2)^(n) = 0.20
To solve for n, we can take the logarithm of both sides. Using the natural logarithm (ln):
ln((1/2)^(n)) = ln(0.20)
n * ln(1/2) = ln(0.20)
n = ln(0.20) / ln(1/2)
Now, let's calculate the value of n:
n = ln(0.20) / ln(1/2)
n ≈ 2.3219
Therefore, it will take approximately 2.3219 half-lives for the sample of thorium-229 to decay to 20% of its original amount.
To calculate the time it will take for this to happen, we can multiply n by the half-life:
Time ≈ n * Half-life
Time ≈ 2.3219 * 7340
Time ≈ 17021.100 years
Therefore, it will take approximately 17,021.100 years for the sample of thorium-229 to decay to 20% of its original amount.
To determine how long it will take for a sample of thorium-229 to decay to 20% of its original amount, we can use the concept of half-life.
The half-life of thorium-229 is given as 7340 years, which means that every 7340 years, the amount of thorium-229 in a sample will reduce by half.
To find the answer, follow these steps:
1. Let's assume the original amount of the thorium-229 sample is "A."
2. Calculate the number of half-lives it will take for the sample to decay to 20% of its original amount:
- The first half-life reduces the amount to 50% of the original (0.5A).
- The second half-life reduces it to 25% of the original (0.25A).
- The third half-life reduces it to 12.5% of the original (0.125A).
- The fourth half-life reduces it to 6.25% of the original (0.0625A).
- The fifth half-life reduces it to 3.125% of the original (0.03125A).
- And so on...
We can see that at each half-life, the sample's amount is halved. Since we want to find the time it takes for the sample to decay to 20%, that means we need the amount to be reduced to 20% (0.2A).
3. Calculate the number of half-lives it will take for the sample to reach 20% (0.2A):
- 0.2A is equal to (1/2)^n, where "n" represents the number of half-lives.
- Solve for "n" by taking the logarithm (base 2) of both sides:
log2(0.2A) = log2(1/2)^n
log2(0.2A) = -n * log2(1/2)
n = -log2(0.2A) / log2(1/2)
4. Substitute the original amount "A" with 1 (since we are assuming an arbitrary value for the original amount):
n = -log2(0.2 * 1) / log2(1/2)
= -log2(0.2) / log2(1/2)
5. Use a calculator to determine the value of "n":
n = -log2(0.2) / log2(1/2)
≈ -(-2.3219) / (-1)
≈ 2.3219
6. Finally, multiply the number of half-lives (2.3219) by the half-life of thorium-229 (7340 years) to get the total time:
Total time = 2.3219 * 7340 years
≈ 17013 years
Therefore, it will take approximately 17013 years for a sample of thorium-229 to decay to 20% of its original amount.