Question: "Plutonium isotope Pu-239 has a half-life of 24,100 years. Suppose that 10 grams of Pu-239 was released in the Chernobyl nuclear accident. How long will it take for the 10 grams to decay to 1 gram?"
I know that I am suppose to use
y= ye^kt , to get the equation and then plug in the info. to get the time, but I'm not sure how to start, please help! Thanks!
y = a e^(kt) where a is the initial amount
so y = 10(e^(kt))
when t = 24100, y = 5 ---> you should have 1/2 of the original 10 g left.
0.5 = e^(24100k)
24100k = ln(0.5)
k = ln(.5)/24100
so y = 10(e^(t)(ln.5/24100))
so you want y to be 1
1 = 10 (e^(t)(ln.5/24100))
0.1 = (e^(t)(ln.5/24100))
(t)(ln.5/24100)) = ln .1
t = 80059
so about 80,000 years !!!!
Ohhh, I get it!! Thanks so much, you helped me a lot! :)
To solve this problem, you can use the exponential decay formula:
N = N₀ * (1/2)^(t / T)
Where:
- N is the final amount (1 gram in this case)
- N₀ is the initial amount (10 grams in this case)
- t is the time it takes to decay (which we want to find)
- T is the half-life of the substance (24,100 years in this case)
To solve for t, we can rearrange the formula:
(1/2)^(t / T) = N / N₀
Taking the logarithm on both sides (to solve for t):
log((1/2)^(t / T)) = log(N / N₀)
Using the logarithm properties, we can bring down the exponent:
(t / T) * log(1/2) = log(N / N₀)
Now, we can solve for t by isolating it on one side:
t = (T * log(N / N₀)) / log(1/2)
Let's plug in the values:
N = 1 gram
N₀ = 10 grams
T = 24,100 years
t = (24,100 * log(1 / 10)) / log(1/2)
Using the natural logarithm (ln) function in most calculators:
t ≈ (24,100 * ln(1 / 10)) / ln(1/2)
Calculating this, we find:
t ≈ (24,100 * -2.3026) / -0.6931
t ≈ 79,076.85 years
Therefore, it will take approximately 79,077 years for 10 grams of Pu-239 to decay to 1 gram.
To solve this problem, you can use the equation you mentioned, which is the exponential decay formula:
y = ye^kt
Here's how you can start:
1. Start with the initial amount of Plutonium-239 (Pu-239), which is 10 grams. This will be your "y" value in the equation.
y = 10 grams
2. Identify the half-life of Pu-239, which is 24,100 years. The half-life is the time it takes for half of the substance to decay.
t₁/₂ = 24,100 years
3. To find the decay constant (k), use the equation:
k = ln(2)/t₁/₂
Substituting the half-life value:
k = ln(2)/24,100 years
Calculate the value of k.
4. Now, plug in the known values into the exponential decay equation to determine the time it takes for the 10 grams to decay to 1 gram.
y = ye^kt
Substitute the initial amount, final amount (1 gram), and the decay constant (k) into the equation:
1 = 10e^(kt)
5. Rearrange the equation to solve for "t", the time it takes for 10 grams to decay to 1 gram:
e^(kt) = 1/10
Take the natural logarithm (ln) of both sides of the equation to get rid of e:
ln(e^(kt)) = ln(1/10)
Use the property of logarithms to simplify the equation:
kt * ln(e) = ln(1/10)
kt = ln(1/10)
Solve for "t" by dividing both sides of the equation by k:
t = ln(1/10) / k
6. Substitute the value of k that you calculated in step 3 into the equation, and solve for "t". This will give you the time it takes for the 10 grams to decay to 1 gram.
Keep in mind that all values used in these calculations are approximate, as the decay rate of Pu-239 may vary in reality.