Calculate the decay constant and half-life of a radioactive isotope that decays from 1000 Bq to 500 Bq in 10 seconds.
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To calculate the decay constant (λ), we can use the formula:
N = N0 * e^(-λt)
where:
N = final amount of the radioactive isotope (500 Bq)
N0 = initial amount of the radioactive isotope (1000 Bq)
t = time elapsed (10 seconds)
e = Euler's number (approximately 2.71828)
First, let's rearrange the formula to solve for the decay constant (λ):
λ = - (1/t) * ln(N/N0)
Plugging in the given values:
λ = - (1/10) * ln(500/1000)
Now we can evaluate the right side of the equation:
λ = - (1/10) * ln(0.5)
≈ - (1/10) * (-0.69315)
≈ 0.06931 s^(-1)
The decay constant (λ) is approximately 0.06931 s^(-1).
Next, we can calculate the half-life (T), which is the time it takes for the amount of the radioactive isotope to decrease by half. The half-life is related to the decay constant by the following formula:
T = ln(2) / λ
Plugging in the value of λ:
T = ln(2) / 0.06931
≈ 10.002 s
The half-life (T) is approximately 10.002 seconds.
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