In some systems we use half-life (like in radioactivity) which gives us time until a quantity changes by 50% — while in other instances (like in RC circuits) we use time constants. In both cases the rate of change of a variable over time is proportional to the instantaneous value of variable.
What is a simple intuitive way to know the difference between the kind of systems where half-life is useful, versus systems where time constants are more meaningful? (Does it have anything to do with the shape of the curve representing the change in value over time, for example?)
Electric Circuits – Time Constant vs Half-Life: When to Use Which?
capacitanceelectric-circuitselectrical-resistancehalf-liferadioactivity
Related Solutions
Do rates of nuclear decay depend on environmental factors?
There are two known environmental effects that can matter:
(1) The first one has been scientifically well established for a long time. In the process of electron capture, a proton in the nucleus combines with an inner-shell electron to produce a neutron and a neutrino. This effect does depend on the electronic environment, and in particular, the process cannot happen if the atom is completely ionized.
(2) In some exceptional examples, such as 187Re, there are beta decays with extremely low energies (in the keV range, rather than the usual MeV range). In these cases, there are significant effects due to the Pauli exclusion principle and the surrounding electron cloud. See Ionizing a beta decay nucleus causes faster decay?
Other claims of environmental effects on decay rates are crank science, often quoted by creationists in their attempts to discredit evolutionary and geological time scales.
He et al. (He 2007) claim to have detected a change in rates of beta decay of as much as 11% when samples are rotated in a centrifuge, and say that the effect varies asymmetrically with clockwise and counterclockwise rotation. He believes that there is a mysterious energy field that has both biological and nuclear effects, and that it relates to circadian rhythms. The nuclear effects were not observed when the experimental conditions were reproduced by Ding et al. [Ding 2009]
Jenkins and Fischbach (2008) claim to have observed effects on alpha decay rates at the 10^-3 level, correlated with an influence from the sun. They proposed that their results could be tested more dramatically by looking for changes in the rate of alpha decay in radioisotope thermoelectric generators aboard space probes. Such an effect turned out not to exist (Cooper 2009). Undeterred by their theory's failure to pass their own proposed test, they have gone on to publish even kookier ideas, such as a neutrino-mediated effect from solar flares, even though solar flares are a surface phenomenon, whereas neutrinos come from the sun's core. An independent study found no such link between flares and decay rates (Parkhomov 2010a). Laboratory experiments[Lindstrom 2010] have also placed limits on the sensitivity of radioactive decay to neutrino flux that rule out a neutrino-mediated effect at a level orders of magnitude less than what would be required in order to explain the variations claimed in [Jenkins 2008]. Despite this, Jenkins and Fischbach continue to speculate about a neutrino effect in [Sturrock 2012]; refusal to deal with contrary evidence is a hallmark of kook science. They admit that variations shown in their 2012 work "may be due in part to environmental influences," but don't seem to want to acknowledge that if the strength of these influences in unknown, they may explain the entire claimed effect, not just part of it.
Jenkins and Fischbach made further claims in 2010 based on experiments done decades ago by other people, so that Jenkins and Fischbach have no first-hand way of investigating possible sources of systematic error. Other attempts to reproduce the result are also plagued by systematic errors of the same size as the claimed effect. For example, an experiment by Parkhomov (2010b) shows a Fourier power spectrum in which a dozen other peaks are nearly as prominent as the claimed yearly variation.
Cardone et al. claim to have observed variations in the rate of alpha decay of thorium induced by 20 kHz ultrasound, and claim that this alpha decay occurs without the emission of gamma rays. Ericsson et al. have pointed out multiple severe problems with Cardone's experiments.
In agreement with theory, high-precision experimental tests show no detectable temperature-dependence in the rates of electron capture[Goodwin 2009] and alpha decay.[Gurevich 2008] Goodwin's results debunk a series of results from a group led by Rolfs, e.g., [Limata 2006], which used an inferior technique.
He YuJian et al., Science China 50 (2007) 170.
YouQian Ding et al., Science China 52 (2009) 690.
Jenkins and Fischbach (2008), https://arxiv.org/abs/0808.3283v1, Astropart.Phys.32:42-46,2009
Jenkins and Fischbach (2009), https://arxiv.org/abs/0808.3156, Astropart.Phys.31:407-411,2009
Jenkins and Fischbach (2010), https://arxiv.org/abs/1007.3318
Parkhomov 2010a, https://arxiv.org/abs/1006.2295
Parkhomov 2010b, https://arxiv.org/abs/1012.4174
Cooper (2009), https://arxiv.org/abs/0809.4248, Astropart.Phys.31:267-269,2009
Lindstrom et al. (2010), http://arxiv.org/abs/1006.5071 , Nuclear Instruments and Methods in Physics Research A, 622 (2010) 93-96
Sturrock 2012, https://arxiv.org/abs/1205.0205
F. Cardone, R. Mignani, A. Petrucci, Phys. Lett. A 373 (2009) 1956
Ericsson et al., Comment on "Piezonuclear decay of thorium," Phys. Lett. A 373 (2009) 1956, https://arxiv.org/abs/0907.0623
Ericsson et al., https://arxiv.org/abs/0909.2141
Goodwin, Golovko, Iacob and Hardy, "Half-life of the electron-capture decay of 97Ru: Precision measurement shows no temperature dependence" in Physical Review C (2009), 80, 045501, https://arxiv.org/abs/0910.4338
Gurevich et al., "The effect of metallic environment and low temperature on the 253Es α decay rate," Bull. Russ. Acad. Sci. 72 (2008) 315.
Limata et al., "First hints on a change of the 22Na βdecay half-life in the metal Pd," European Physical Journal A - Hadrons and Nuclei May 2006, Volume 28, Issue 2, pp 251, https://link.springer.com/article/10.1140/epja/i2006-10057-1
The short answer is no: halflives are constant.
However, let's discuss a situation in which that comment might have some kind of truth behind it. If you have a parent nucleus that decays to a radioactive daughter so that there will be two (or more) decays before stability. In general there are two possibilities for this:
- The daughter has a shorter halflife than the parent. In this case the concentration of the daughter is always $\displaystyle\frac{\tau_\text{daughter}}{\tau_\text{parent}}$ of the parent concentration. This means that the concentration of the daughter actually decays on the parent's halflife (because the daughter is constantly refreshed from the parent).
- The daughter has a longer half life than the parent. In this case the daughter will accumulate steadily as the parent decays away.
The latter case is interesting to us here because at the start the sample will register an activity that decays with the parent's (short) halflife, but after a number of those halflifes have passed the activity of the sample will be dominated by the daughter and exhibit a longer halflife.
That is something that your instructor could have meant which would not be wrong. However, the halflife of each isotope remains the same: it is only the halflife of the sample (which contains more than one isotope) that varies.
Best Answer
"What is a simple intuitive way to know the difference between the kind of systems where half-life is useful , versus systems where time constants are more meaningful."
For systems obeying an exponential decay relationship, either half life or time constant can be used. I think it's largely a matter of tradition that half life ($t_{1/2}$) is used for radioactivity and time constant ($\tau$) for C–R and L–R circuits. The relationship between them is $$t_{1/2}=(\ln 2)\tau.$$ Here are some ideas on how the traditions might have arisen...
• For C-R or L-R circuits, it was marginally easier before the days of electronic calculators to calculate $\tau =\frac LR$ or $\tau =CR$ than to calculate $t_{1/2}=(\ln 2)CR$.
• and arguably there's less motivation for knowing how long a voltage across a capacitor will take to halve than how long a radioactive activity will take to halve. For circuit behaviour a general idea of a characteristic time is usually what matters, and $\tau$ is as good as $t_{1/2}$.
• The intelligent layperson is more interested in radioactivity than in capacitor discharge and it's easier to explain the idea of half life than that of time constant, or its reciprocal, decay constant. [There was huge popular interest – see for example chapters in best sellers by Jeans and Eddington – in radioactivity in the first few decades after its discovery.]