Suppose the amount of a certain radioactive substance in a sample decays from 9.70mg to 3.90mg over a period of ×3.23103 years. Calculate the half life of the substance.Round your answer to 2 significant digits.
Question
Suppose the amount of a certain radioactive substance in a sample decays from 9.70mg to 3.90mg over a period of ×3.23103 years. Calculate the half life of the substance.Round your answer to 2 significant digits.
Solution
To solve this problem, we will use the formula for exponential decay, which is:
N = N0 * e^(-λt)
where: N is the final amount of the substance, N0 is the initial amount of the substance, λ is the decay constant, t is the time elapsed.
We are given that N = 3.90mg, N0 = 9.70mg, and t = 3.23103 years. We can plug these values into the formula and solve for λ:
3.90 = 9.70 * e^(-λ*3.23103)
Divide both sides by 9.70:
0.402 = e^(-λ*3.23103)
Take the natural logarithm of both sides:
ln(0.402) = -λ*3.23103
Solve for λ:
λ = -ln(0.402) / 3.23103
Now that we have λ, we can find the half-life (T) using the formula:
T = ln(2) / λ
Plug in the value of λ and solve for T. Round your answer to 2 significant digits.
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