_The_Natural_Base_e_
An accident in 1986 at the Chernobyl nuculear plant in the Ukraine released a large amount of plutonium (Pu-239)into the atmosphere. The half-life of Pu-239 is about 24,110 years. Find the decay constant. Use the function N (t) = N 0 e -kt to find what remains of an intial 20 grams of Pu-239 after 5000 years. How long will it take for these 20 grams to decay to 1 gram?
17g
A=(.5)t/k
To find the decay constant (λ) from the half-life (T1/2), we can use the following formula:
λ = ln(2) / T1/2
In this case, the half-life of Pu-239 is T1/2 = 24,110 years. Let's calculate the decay constant:
λ = ln(2) / 24,110
≈ 2.8728 × 10^-5 per year
Now, to find what remains of an initial 20 grams of Pu-239 after 5000 years, we can use the formula N(t) = N0 * e^(-λt), where N0 is the initial amount, t is the time passed, and e is Euler's number (approximately 2.71828).
N(t) = 20 * e^(-2.8728 × 10^-5 * 5000)
≈ 15.855 grams
Therefore, after 5000 years, approximately 15.855 grams of Pu-239 will remain.
To determine how long it will take for these 20 grams to decay to 1 gram, we can rearrange the formula to solve for t:
N(t) = N0 * e^(-λt)
Rearranging gives us:
t = ln(N(t) / N0) / -λ
Substituting the given values, we have:
t = ln(1 / 20) / -2.8728 × 10^-5
≈ 27678.43 years
Therefore, it will take approximately 27,678.43 years for the 20 grams of Pu-239 to decay to 1 gram.
To find the decay constant (k) of Pu-239, we can use the formula for half-life. The half-life (t1/2) of a substance is defined as the amount of time it takes for the substance to decay to half of its initial value. In this case, the half-life of Pu-239 is given as 24,110 years.
The formula for the decay constant is derived from the half-life equation. It is given by k = (ln(2))/t1/2, where ln is the natural logarithm. Substituting the given half-life of Pu-239, we can calculate the decay constant as follows:
k = ln(2)/t1/2
k = ln(2)/24110 years
Next, we can use the formula N(t) = N0 * e^(-kt) to find the remaining amount of Pu-239 after 5000 years, given an initial amount of 20 grams (N0).
N(t) = N0 * e^(-kt)
N(5000) = 20 * e^(-k * 5000)
To find the time it takes for the 20 grams to decay to 1 gram, we need to solve for the time t when N(t) = 1.
1 = 20 * e^(-k * t)
To solve this equation, we need the value of the decay constant (k) determined earlier.
Finally, we can substitute the derived value of k into the equation and solve for t to find the time it takes for the 20 grams to decay to 1 gram.