If 50% of a radioactive element remains after 4000 years, what is the half-life?
a. 1 half-life
b. 2 half-lives
c. 3 half-lives
d. 4 half-lives
c. 3 half-lives
To determine the half-life of a radioactive element, you can use the formula:
t₁/2 = (ln 2) / λ
where t₁/2 represents the half-life, and λ represents the decay constant.
In this case, we are given that 50% of the element remains after 4000 years. This means that the remaining fraction (R) is 0.5, and the elapsed time (t) is 4000 years. The original fraction (N₀) is 1.
Using the equation for radioactive decay:
R = N/N₀ = e^(-λt),
we can solve for the decay constant (λ). Taking the natural logarithm of both sides:
ln R = ln e^(-λt)
ln R = -λt
Substituting the given values:
ln 0.5 = -λ * 4000,
we can solve for λ:
λ = -(ln 0.5) / 4000
Now we can plug this value of λ into the equation for the half-life:
t₁/2 = (ln 2) / λ
t₁/2 = (ln 2) / (-(ln 0.5) / 4000)
Simplifying this equation:
t₁/2 = (ln 2) * (4000 / (ln 0.5))
Calculating this expression:
t₁/2 ≈ 2771.02 years
Since this value is between 2000 and 4000, we can conclude that the half-life is approximately 2 half-lives. Therefore, the answer is b. 2 half-lives.
To determine the half-life of a radioactive element, we need to determine how many times it takes for the remaining amount to halve.
In this case, after 4000 years, 50% of the element remains. This means the remaining amount is half of the initial amount.
To find the half-life, we need to find how many times the initial amount is halved to reach 50%.
Let's go through the options:
a. 1 half-life: If the half-life is 1, then after 4000 years, 50% of the element would remain, which matches the given scenario.
b. 2 half-lives: If the half-life is 2, then after the first half-life (2000 years), 50% of the initial amount would remain. However, after the second half-life (another 2000 years), 25% of the initial amount would remain, which doesn't match our scenario.
c. 3 half-lives: If the half-life is 3, then after the first half-life (1333.33 years), less than 50% of the initial amount would remain.
d. 4 half-lives: If the half-life is 4, then after the first half-life (1000 years), 50% of the initial amount would remain.
From the options provided, it seems that the correct answer is:
a. 1 half-life