radioactive isotope decays with a rate proportional to the original amountof material present. If originally there is 50mg of the material present and aftertwo hours it has lost 10% of its mass, determine the half-life, the time at which itis half its original mass.
Question
radioactive isotope decays with a rate proportional to the original amountof material present. If originally there is 50mg of the material present and aftertwo hours it has lost 10% of its mass, determine the half-life, the time at which itis half its original mass.
Solution
The decay of a radioactive isotope can be modeled by the exponential decay equation:
N(t) = N0 * e^(-λt)
where: N(t) is the amount of the isotope at time t, N0 is the original amount of the isotope, λ is the decay constant, t is the time.
Given that the original amount of the material is 50mg and after two hours it has lost 10% of its mass, we can write:
40mg = 50mg * e^(-2λ)
Solving for λ gives:
λ = -ln(0.8) / 2 = 0.11157 per hour
The half-life (T) of a radioactive isotope is given by the formula:
T = ln(2) / λ
Substituting the value of λ into this equation gives:
T = ln(2) / 0.11157 = 6.21 hours
So, the half-life of the isotope is approximately 6.21 hours.
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