Chemistry · Ch 13 — Nuclear Chemistry and Radioactivity

Graphical Representation of Decay

13.5.5

Graphical Representation of Decay

The exponential decay law can be turned into a straight-line graph, which is often more practical to work with than the raw exponential curve. Taking $\log_{10}$ of both sides of $N = N_0 e^{-\lambda t}$ gives $\log_{10} N = -\frac{\lambda}{2.303}t + \log_{10}N_0$ -- a straight line in $\log_{10}N$ versus $t$ , with slope $-\frac{\lambda}{2.303}$ and intercept $\log_{10}N_0$ .

In practice, the number of undecayed nuclei $N$ itself is rarely something that can be measured directly; what a detector actually measures is the activity, $\frac{dN}{dt}$ , which is itself directly proportional to $N$ . So instead of plotting $\log_{10}N$ against $t$ , the same straight-line relationship is plotted as $\log_{10}(\text{activity})$ against $t$ , and it has the identical negative slope, $-\frac{\lambda}{2.303}$ -- meaning the decay constant $\lambda$ can be read directly off a real, measured activity-versus-time dataset without ever needing to know $N_0$ or the absolute number of atoms present. …

Figure 13.3Plot of log10(activity) versus time

What this figure shows. A straight-line graph with time t on the horizontal axis and log10(activity, in dps) on the vertical axis. Because N (and hence dN/dt, the activity) decays exponentially with time, taking log10 of both sides of the decay law converts the curve into a straight line of the form log10(N) = -(λ/2.303) x t + log10(N0); since activity is proportional to N, the same straight-line form holds for log10(activity) versus t. The line has a negative slope equal to -λ/2.303, so the decay constant λ can be read directly off the graph from the slope, wi …

Figure 13.4 (activity vs time)Plot of activity versus time

What this figure shows. A curve (not a straight line) with time on the horizontal axis and activity (rate of decay, in dps) on the vertical axis, starting high and falling off ever more slowly as time increases -- the classic exponential-decay shape, mirroring the fall in the number of undecayed radioactive atoms with time, since activity is directly proportional to the number of undecayed nuclei present at each instant. FIDELITY NOTE: the source reuses the figure label 'Fig. 13.4' twice more later in the same chapter for two unrelated nuclear-reactor diagrams (Fig. 13.4(a) and Fig. 13.4(b), under section 13.8.2) -- a book numbering duplication disclosed here for transparency, matching how …

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