A certain isotope decays radioactively so that the mass satisfies the equation m′=−0.01m. Here the time is measured in days. What is the half life? [hint: solve the equation then find at what time your solution gives half the inital mass]
The given equation is m' = -0.01m where m = the mass at time t, days.
Assume that m(0) = m₀, the initial mass. The solution for the differential equation is as follows: [tex] \frac{dm}{dt} = -0.01m \\\\ \frac{dm}{m} = -0.01dt \\\\ \int _{m_{0}}^{m} \frac{dm}{m} = -0.01 \int_{0}^{t} dt \\\\ ln \frac{m}{m_{0}} =-0.01t \\\\ m(t)=m_{0} e^{-0.01t} [/tex]
At half-life, the time is given by [tex]e^{-0.01t} = \frac{m_{0}/2}{m_{0}} = \frac{1}{2} \\\\ -0.01t = ln(0.5) \\\\ t = - \frac{ln(0.5)x}{-0.01} = 69.315[/tex]
