Answer to Question #319855 in Differential Equations for Almas

Question

Applications of Differential Equation of Exponential Reduction or Radioactivity Decay

Answer 1

The number of atoms likely to decay in a given infinitesimal time interval (dN/dt) is proportional to the number (N) of atoms present.

The rate of change is given by the equation dN/dt = −λN, where λ is the decay constant.

Integration of this equation yields N = N₀e^−λt, where N0 is the size of an initial population of radioactive atoms at time t = 0.

This shows that the population decays exponentially at a rate that depends on the decay constant. The time required for half of the original population of radioactive atoms to decay is called the half-life. The relationship between the half-life, T_1/2, and the decay constant is given by T_1/2= 0.693/λ.