Expression for Decay Constant

Starting from the rate law $-\frac{dN}{dt} = \lambda N$ (section 13.5.2), rearranging gives $\frac{dN}{N} = -\lambda\, dt$. Integrating both sides, $\int \frac{dN}{N} = -\int \lambda\, dt$, gives $\ln N = -\lambda t + C$, where $C$ is a constant of integration that is pinned down using the starting condition of the sample.

Let $N_0$ be the number of undecayed nuclei present at some arbitrary starting time $t=0$; then at $t=0$, $N = N_0$, so substituting into $\ln N = -\lambda t + C$ gives $\ln N_0 = C$. Feeding this value of $C$ back in gives $\ln N = -\lambda t + \ln N_0$, which rearranges to $\lambda t = \ln \frac{N_0}{N}$, or $\lambda = \frac{1}{t}\ln\frac{N_0}{N}$. Converting from natural log to base-10 log (using $\ln x = 2.303 \log_{10} x$) gives the practical working form used throughout the worked problems in this chapter: $\lambda = \frac{2.303}{t}\log_{10}\frac{N_0}{N}$. …