Question 31.21: A sample contains an isotope of half-life t1/2 . a Show that...
A sample contains an isotope of half-life t_{1/2} .
a Show that the fraction f of nuclei in the sample which remain undecayed after a time t is given by the equation:
f = (\frac{1}{2})^{n} when n = \frac{t}{t_{1/2}}
b Calculate the fraction f after each of the following times:
i t_{1/2} ii 2t_{1/2}
iii 2.5t_{1/2} iv 8.3t_{1/2}
a We need to find an expression for the decay constant λ so that we can substitute it into the decay equation.
The quantity f is the ratio of atoms remaining to decay to the original number of atoms in the sample: f = \frac{N}{N_{0}}
When t = t_{1/2}, f = \frac{1}{2} = e^{–λt_{1/2}}
Taking logarithms of both sides, ln \left(\frac{1}{2} \right) = –λt_{1/2}
Therefore λ = – \frac{ln \left(\frac{1}{2}\right)}{t_{1/2}}
Then the equation f = e^{–λt} becomes f = e^{ln \left(\frac{1}{2}\right) \frac{t}{t_{1/2}}}
Remember that e^{ln \left(\frac{1}{2} \right)} = \frac{1}{2}
So f = \frac{1}{2}^{\frac{t}{t_{1/2}}}
b i 0.50
ii 0.25
iii 0.177 ≈ 0.18
iv 0.0032