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In chemical kinetics, the rate constant for a first-order reaction depends on temperature according to the Arrhenius equation, $$ k=Ae^{\frac{-E_a}{RT}}, $$ due to the activation energy barrier. Radioactive decay also follows first-order kinetics, with a rate given by $$ \frac{\mathrm{d}N}{\mathrm{d}t}=- \lambda N, $$ where $\lambda$ is the decay constant. However, $\lambda$ is said to be independent of temperature, unlike chemical rate constants. Why doesn’t the decay constant exhibit a similar temperature dependence, given that both processes are first-order? Is there a fundamental difference in the underlying mechanisms, or does the nuclear energy scale simply render thermal effects negligible?

I'm new to radioactive decays so, hoping a high-school level explanation of this topic.

Qmechanic
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Vansh Patel
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2 Answers2

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It much depends on what you put for $E_\mathrm{a}$.

In both cases there is a metastable object with some activation energy. To assess the effect of temperature you need to compare the activation energy $E_\mathrm{a}$ to the energy of thermal fluctuations $E_\mathrm{t} = k_\mathrm{B}T$. That is what the Arrhenius formula does.

At room temperature (300 K) you have $E_\mathrm{t}\approx 30 \ \mathrm{meV}$ which is not that far from the typical values for the activation energy in case of chemical complexes (say 0.1 - 1 eV). But in case of the nuclei the energy is going to be from hundreds of keV upwards - the 30 meV thermal fluctuations at 300 K have no effect. If you want to see thermal effects on nuclear physics you need to have appropriately high temperatures - centre of a star, particle accelerator etc.

Maybe another way to put it is that the temperature dependence may as well be there (as per your first equation), but you expect it to be so flat (due to high value of $E_\mathrm{a}$) that that you can just as well neglect it at temperatures where ordinary matter exists.

Karel
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Chemical processes occur as interactions between electron shells, and the available energy is affected by the relative speeds of atoms as they collide. Increasing the temperature increases these relative speeds and also increases the vibrational energy within molecules. As a result collisions become more likely to push groups of atoms over activation barriers, and so chemical reaction rates increase.

At the centres of stars there are collisions between nuclei, which can affect their internal energy and can cause nuclei to react together. Interactions between fast-moving nuclei and other particles are studied by some physicists, and the speed (temperature) of the interaction is important. The speeds involved are much higher than the speeds of atoms in chemical reactions. In "normal" earth-like situations these interactions do not occur and the nucleus of an atom is affected very little by things happening around it. In collisions between electron shells a nucleus may change its velocity, but this does not add internal energy to the nucleus, so it does not affect its decay.

Peter
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