Question

Exponentail Decay. The amount of radioactive material present at time t is given by A = A0ekt, where A0 is the initial amount, k<0 is the rate of decay. Radioactive substances are more commonly described in terms of their half-life or the time required for half of the substance to decompose. Determine the half-life of substance X if after 600 years, a sample has decayed to 85% of its original mass?

Answer 1

Answer:

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Answer 2

Given:

Amount of radioactive material present at time t, $A=A_{0}e^{kt}$

No. of years for sample X to decay = 600

Amount of sample X decayed from its original mass = 85%

To find:

Half-life of substance X.

Solution:

To determine the half-life,

$A=A_{0}e^{-kt}$

where, $A_{0}$ is initial amount and k is the rate of decay.

Half-life is the time taken by a quantity to decay exponentially to half of its original amount.

Let half-life be t = T, then $A=\frac{A_{0} }{2}$

Substitute in the formula, $A=A_{0}e^{-kt}$

$\frac{A_{0} }{2}=A_{0}e^{-kT}$

$\frac{1}{2}=e^{-kT}$

$ln(2)=-kT$

$k=\frac{-0.693}{T}$

Here, t = 600 years

$A=A_{0}e^{\frac{0.693}{T}600 }$

If substance X is decayed to 85% of its original mass, there remains 0.85 times the initial quantity.

$0.85A_{0} =A_{0}e^{\frac{-415.8}{T} }$

$0.85 =e^{\frac{-415.8}{T} }$

$ln(0.85)=\frac{-415.8}{T}$

$-0.162=\frac{-415.8}{T}$

$T=\frac{-415.8}{-0.162}$

$T=2566.67$ years ≈ 2567 years

Final Answer

Half-life of substance X after 600 years of it being decayed to 85% of its original mass is 2567 years.