Question

pls help me, is very urgent. three quarters of radioactive sample decayed after three days, what is the half-life of the element

Answer 1

Use the equation \[N(t)=N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}\], Where \(N(t)\) is the number of atoms with half life \(t_{1/2}\) from an original number of atoms \(N_0\) after time \(t\). Hence the fraction remaining will be \[\frac{N(t)}{N_0}=\left(\frac{1}{2}\right)^{t/t_{1/2}}\]. Taking logs of both sides we get \[\log\left(\frac{N(t)}{N_0}\right)=\frac{t}{t_{1/2}}\log\left(\frac{1}{2}\right)\].
We know that \[\frac{N(t)}{N_0}=\frac{1}{4}\] since 3/4 has decayed leaving 1/4 left, and that \(t=3\). so plugging in, and rearranging we get
\[t _{1/2} =\frac{3\log(1/2)}{\log(1/4)}=1.5\] so the half life is 1.5 days.