Question

Solve the following problems. The required atomic masses can be found in the book "Concepts of Modern Physics" by Arthur Beiser. Q1: Suppose you begin with 1.0 × 10-2g of a pure radioactive substance and 4 h later determine that 0.25%(10-2g remains. What is the half-life of the substance?

Answer 1

According to the radioactive decay law, the number of atoms after time t is
     $N(t)=N_0e^{-0.693\frac{t}{\tau_{1/2}}}$
As the mass of the substance is proportional to the number of atoms, so, in terms of the mass, the radioactive decay law is
     $M(t)=M_0e^{-0.693\frac{t}{\tau_{1/2}}}$

where, $\tau_{1/2}$ is the half life of the radioactive substance. So, we get
  
$\Rightarrow 0.693\frac{t}{\tau_{1/2}}=\ln \bigg(\frac{M_0}{M(t)}\bigg)$
  $\Rightarrow \tau_{1/2}=\frac{0.693\times t}{\ln \bigg(\frac{M_0}{M(t)}\bigg)}$
Now given
     $M_0=1.0\times 10^{-2}~g$

$M(t)=0.25\times 10^{-2}~g$
   $t=4~h$
So, we get
   $\Rightarrow \tau_{1/2}=\frac{0.693\times 4}{\ln \bigg(\frac{1}{0.25}\bigg)}~h$
  $\Rightarrow \tau_{1/2}=\frac{0.693\times 4}{\ln 4}~h$
  $\Rightarrow \tau_{1/2}=2~h$