If 70% of a radioactive element remains radioactive after 400 million years , then what percent remains radioactive after 600 million years? What is the half-life of this element
If the half-life is k years, then after t years,
(1/2)^(t/k)
remains. Thus, to find k, solve
(1/2)^(400/k) = 0.7
600 = 3/2 * 400, or 1.5 half-lives. So, the amount remaining after 600 mega years is
(1/2)^(3/2 * 400/k)
= (1/2)^(400/k)^(3/2)
= 0.7^(3/2)
= ?
To find the percent of the element that remains radioactive after 600 million years, we need to determine the rate at which it decays over time.
Given that 70% of the element remains radioactive after 400 million years, we can calculate the decay constant using the formula:
N = N₀ * e^(-λ*t)
Where:
N₀ is the initial quantity of the radioactive element,
N is the remaining quantity after time t,
e is the base of the natural logarithm (approximately 2.71828), and
λ is the decay constant.
Let's assume N₀ = 100 (to simplify calculations).
Using the given values, we have:
70 = 100 * e^(-λ * 400)
To solve for λ, we can take the natural logarithm of both sides and solve for λ:
ln(70/100) = -λ * 400
λ ≈ -0.04879
Now we can find the remaining percentage after 600 million years. Using the same formula and substituting t = 600:
N = 100 * e^(-0.04879 * 600)
N ≈ 35.356
Therefore, approximately 35.36% of the element remains radioactive after 600 million years.
To calculate the half-life of this element, we can use the following formula:
t₁/₂ = ln(2) / λ
Where:
t₁/₂ is the half-life, and
λ is the decay constant we calculated earlier.
Substituting the value of λ into the formula:
t₁/₂ = ln(2) / (-0.04879)
t₁/₂ ≈ 14.23 million years
Therefore, the half-life of this element is approximately 14.23 million years.
To find the percentage of the radioactive element remaining after 600 million years, we can use the concept of half-life. The half-life of a radioactive element is the time it takes for half of the radioactive atoms to decay.
Given that 70% of the element remains after 400 million years, it means that 30% of the radioactive atoms have decayed. Therefore, 70% of the original atoms are still radioactive.
We can now find the half-life of the element:
1. Start with 100% of the original radioactive atoms.
2. After 1 half-life, only 50% of the original atoms will remain.
3. After 2 half-lives, 25% of the original atoms will remain (50% from the previous step times 50% again).
4. After 3 half-lives, 12.5% of the original atoms will remain (25% from the previous step times 50% again).
5. After 4 half-lives, 6.25% of the original atoms will remain (12.5% from the previous step times 50% again).
From this pattern, we can observe that it takes approximately 4 half-lives for the radioactive element to decay to less than 10% of the original amount. Therefore, we can estimate that the half-life of this element is around 400 million years.
Now, let's determine the percentage of the radioactive element remaining after 600 million years:
Since the half-life is 400 million years, after 600 million years, there will be 1.5 half-lives (600 million years divided by 400 million years). For each half-life, the percentage of the element remaining is halved.
So, for 1.5 half-lives, the remaining percentage will be (50% from the first half-life times 50% from the second half-life times 100% from the initial amount) = 25%.
Therefore, approximately 25% of the radioactive element will remain after 600 million years.