exponential decay law

The exponential decay law describes how the number of atoms in a radioactive sample decreases over time. This process is statistical and occurs at an exponential rate. The rate of decay is proportional to the number of atoms present, represented by the equation (dN/dt = -\lambda N), where (N) is the number of atoms and (\lambda) is the decay constant.12

Integrating this equation gives (N = N_0 e^{-\lambda t}), showing that the population decays exponentially. The time it takes for half of the atoms to decay is called the half-life ((T_{1/2})), which is related to the decay constant by (T_{1/2} = 0.693/\lambda). This law is fundamental in understanding radioactive processes and is an example of an exponential function, which can model phenomena where the rate of change depends on the current value.123