If I understand correctly, the rate of radioactive decay depends on the amount of the radioactive element. How then can it be constant, if it depends on concentration?
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2$\begingroup$ What is constant is the fraction of the remaining atoms of the isotope in question that will decay in a given interval of time. $\endgroup$– Solomon SlowCommented Sep 14, 2023 at 11:31
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1$\begingroup$ Hi, welcome to Physics SE. What's constant is $\lambda$ in $\dot{N}=-\lambda N$, but this still implies $\dot{N}\propto N$. $\endgroup$– J.G.Commented Sep 14, 2023 at 13:54
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$\begingroup$ Thank you for your kind help. I understand that lambda is constant, but lambda is decay constant, not decay rate right? $\endgroup$– xxxCommented Sep 14, 2023 at 14:03
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1$\begingroup$ $dN/dt$ is the decay rate or activity. $\endgroup$– FarcherCommented Sep 14, 2023 at 14:09
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$\begingroup$ What we really mean when we say that the decay rate is constant is that $\frac{d \log N}{dt}$ is constant, i.e. the infinitesimal relative change in $N$ is constant. $\endgroup$– Charles HudginsCommented Sep 15, 2023 at 2:36
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2 Answers
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The rate of radioactive decay depends on the statistical probability that any random single nucleus will decay in a unit of time. That probability does not depend on the number of atoms present in a sample, which is a piece of information that the nucleus of an atom does not have any sort of access to in the first place.
But the number of decays that occur in a chunk of material in a unit of time will depend on the number of nuclei present in a sample. If you double the number of nuclei in a sample, you double the number of nuclei in that sample which will decay in a unit of time, on the average.
answered Sep 14, 2023 at 15:59
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Unfortunately, physicist are rarely exact in their wording. Thus, it would help if you added a reference. From the context it's usually clear what people mean. Here are two contexts:
- In the equation $$ N_{decay}(t) = N_{0} \cdot e^{-\lambda \cdot t} $$ the decay constant $\lambda$ has the unit of a rate, i.e. $1/time$. This is why it might be referenced as "decay rate". Thus, some people might express $\lambda = const$ by saying that the decay rate is constant.
- An alternative representation uses the half life time period $t_{1/2} = \frac{\ln(2)}{\lambda}$. E.g. here they say "The rate of decay remains constant throughout the decay process." What they trying to express is $t_{1/2} = const$.
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$\begingroup$ Thank you very much; that was insightful. $\endgroup$– xxxCommented Sep 15, 2023 at 17:35