Definition of decay constant Calculation of half life Calculation of initial number of nuclei at $t=0$ The decay constant $(\lambda)$ of a radioactive nucleus equals the ratio of the instantaneous rate of decay $\left(\frac{\Delta N}{\Delta t}\right)$ to the corresponding instantaneous number of radioactive nuclei. Alternatively, The decay constant $(\lambda)$ of a radioactive nucleus is the constant of proportionality in the relation between its rate of decay and number of its nuclei at any given instant. Alternatively, $\left(\frac{\Delta N}{\Delta t}\right) \propto N$ $\left(\frac{\Delta N}{\Delta t}\right) = \lambda N$ The constant $(\lambda)$ is known as the decay constant Alternatively, The decay constant equals the reciprocal of the mean life of a given radioactive nucleus. $\lambda = \frac{1}{\tau}$ where, $\tau = \text{ mean life }$ Alternatively, The decay constant equal the ratio of $\ln 2$ to the half life of the given radioactive element. $\lambda = \frac{\ln 2}{T_{1/2}}$ where $T_{1/2}=$ Half life Alternatively, The decay constant of a radioactive element, is the reciprocal of the time in which the number of its nuclei reduces to $\frac{1}{e}$ of its original number. We have$R = \lambda N$$R(20\text{ hrs}) = 10000 = \lambda N_{20}$$R(30\text{ hrs}) = 5000 = \lambda N_{30}$$\therefore \frac{N_{20}}{N_{30}} = 2$ This means that the number of nuclei, of the given radioactive nucleus, gets halved in a time of $(30-20)$ hours $=10$ hours $\therefore$ Half life $=10$ hours This means that in 20 hours ( $=2$ half lives), the original number of nuclei must have gone down by a factor of 4 . Hence Rate of decay at $t=0$ $\lambda N_0 = 4\lambda N_{20}$ $=4 \times 10000 = 40,000$ disintegrations per second [Note : Award full marks of the last part of this question even if student does not calculate initial number of nuclei and calculates correctly rate of disintegration at $t=0$ ] i.e., $R_0 = 40,000$ disintegrations per second $N_0 = \frac{40000}{\lambda} = \frac{40000}{\ln 2}$ $\times 10 \times 60 \times 60$ $N_0 = \frac{144 \times 10^7}{0.693}$ $= 2.08 \times 10^9 \text{ nuclei }$