State the relation between average life and decay constant.

The normal activity of living carbon-containing matter is found to be about $15$ decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive $_{6}^{14}C$ present with the stable carbon isotope $_{6}^{12}C$. When the organism is dead,its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life ($5730$ years) of $_{6}^{14}C$,and the measured activity,the age of the specimen can be approximately estimated. This is the principle of $_{6}^{14}C$ dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of $9$ decays per minute per gram of carbon. Estimate the approximate age (in years) of the Indus-Valley civilisation.

The atomic weight of a radioactive element is $M_w \, g/mol$. The radioactivity of $m \, g$ of its mass is: ($N_A =$ Avogadro number,$\lambda =$ decay constant)

The half-life of a radioactive sample is $20 \ days$. This means that:

The half-life of $_{38}^{90} Sr$ is $28$ years. What is the disintegration rate of $15 \; mg$ of this isotope?

$A$ radioactive substance has decay constants $\lambda_{\alpha}$ and $\lambda_{\beta}$ for $\alpha$ and $\beta$ emission,respectively. If the substance emits both $\alpha$ and $\beta$ particles,find the effective half-life of the substance.