Law of Decay

In order to do some calculations with radioactive decays, we want to assume that the decay rate $\mathrm{d}N/\mathrm{d}t$ is proportional to the number of remaining nuclei $N$. Calling the proportional constant $\lambda$ gives the following differential equation: $$\frac{\mathrm{d}N}{\mathrm{d}t} = -\lambda N$$ The symbol $\lambda$ is called decay constant. In the next step, we can easily separate the variables and integrate both sides: $$\int_{N_0}^N\frac{\mathrm{d}N}{N} = -\int_0^{t}\lambda\,\mathrm{d}t$$ The limits are chosen to $N = N_0$ at a start time $t=0$ and the remaining nuclei $N$ at an arbitrary time $t$. Solving these integrals finally leads to $$\ln N - \ln N_0 = -\lambda t$$ Now we can use the laws for logarithmic calculations and obtain $$\ln\left(\frac{N}{N_0}\right) = -\lambda t$$ Solving this equation to $N$ finally leads to $$\boxed{N = N_0 \mathrm{e}^{-\lambda t}}$$ This very important formula describes the number of remaining nuclei $N$ with respect to the elapsed time $t$. It is sometimes useful to replace the decay constant $\lambda$ with the decay time $\tau$ by using the following definition: $$\boxed{\tau = \frac{1}{\lambda}}$$ In some cases, the half-life $T_{1/2}$ of radioactive material is given instead of the decay time. This can be calculated from the parameters given above by setting $N$ to $1/2\,N$ which leads to the equation $$\frac{1}{2}N_0 = N_0 \mathrm{e}^{-\frac{T_{1/2}}{\tau}}$$ Canceling out $N_0$ and solving this equation to $T_{1/2}$ finally results in $$\boxed{T_{1/2} = \tau \ln 2 = \frac{\ln2}{\lambda}}$$ This formula gives the possibility to convert the half-life, time constant, and decay constant into the other possibilities.