Abstract
For electronic components, the rule of thumb that “life halves for every ten degrees of temperature” is widely used. This paper counts the lifetime ratio per ten degrees under the Arrhenius law, where life depends exponentially on inverse temperature, for four activation energies and four temperature bands, and sets it beside the view that writes the exponential in temperature itself. No new theorem or law is claimed.
First, with one activation energy fixed, the ten-degree ratio changes with the temperature band: at $0.7$ eV it is $2.420937$ for $25\to35$ degrees Celsius, $2.079348$ for $55\to65$, $1.851651$ for $85\to95$ and $\mathbf{1.648510}$ for $125\to135$, and for all four activation energies it shrank as temperature rose (Section 2).
Second, the activation energy giving exactly a factor of two per ten degrees differs by band. It was $\mathbf{0.548776}$ eV for $25\to35$ degrees, $0.662796$ eV for $55\to65$, $0.787568$ eV for $85\to95$ and $\mathbf{0.970655}$ eV for $125\to135$ (Section 2).
Third, stacking “ten degrees halves” misses by an order of magnitude: using $0.548776$ eV, which doubles for $25\to35$ degrees, the lifetime ratio from $25$ to $125$ degrees is $\mathbf{213.7152}$, about a fifth of the $1024$ from halving ten times. Using the value that doubles for $85\to95$ degrees gives $2206.1160$ (Section 3).
Fourth, with the exponential in temperature itself, the ratio does not depend on temperature: under $Q_{10}=2$, that is $Z=33.219281$ degrees, the ratio is $2.000000$ for every ten degrees and $1024.0$ from $25$ to $125$ degrees (Section 3).
Fifth, the separator is whether the exponent is taken in inverse temperature or in temperature itself: in the Arrhenius law, which takes it in inverse temperature, the same ten degrees makes a smaller difference in inverse temperature at higher temperature, so the ratio shrinks. Taken in temperature itself, the ratio is constant, and “ten degrees halves” is a rule of thumb of that form (Section 4).
Keywords: Arrhenius law, activation energy, acceleration factor, temperature coefficient, component lifetime, Z value.
Scope (what is not claimed)
No new theorem or law is claimed: the Arrhenius law, obtaining acceleration factors from activation energies, and the $Q_{10}$ and Z-value views are all known. Which law is right is not judged: whether component life follows the Arrhenius law or an exponential in temperature itself is not stated; it is only counted that the two forms give different ratios for the same ten degrees. Activation energies are set values: $0.3$, $0.5$, $0.7$ and $1.0$ eV are values for comparison, not those of particular components or failures. Failure mechanisms are not entered. What chemical or physical process sets the life is not treated. Relation to earlier papers: Paper 717 showed that, trading temperature for time in the Z-value view, the rate of exchange does not depend on absolute temperature. Here it is counted that writing the same exchange with the Arrhenius law makes the ratio shrink with absolute temperature. Paper 523 showed that even sharing the same exponential, the prefactor comes from three different roots. Here it is not the prefactor but the form of the exponent that is changed, and the effect on the ten-degree ratio is examined. Paper 457 showed that trading flavour for speed works only when two temperature coefficients differ. Here it is counted that even a single temperature coefficient takes different values across bands depending on how it is written. What is added: setting out the ten-degree ratio for four activation energies and four bands, finding for each band the activation energy that gives exactly a factor of two, finding the miss when the rule is stacked ten times, setting it beside the ratio with the exponential in temperature itself, and placing the separator on “the exponent in inverse temperature, or in temperature itself”.
1. Notation and the Question
Setting: when life follows the Arrhenius law, the ratio of lifetimes (the acceleration factor) on raising the absolute temperature from $T_1$ to $T_2$ is as follows. In the view writing the exponential in temperature itself, the ratio per ten degrees is a constant $Q_{10}$.
$$\mathrm{AF}=\exp\!\left[\frac{E_a}{k_B}\left(\frac{1}{T_1}-\frac{1}{T_2}\right)\right],\qquad \mathrm{AF}_{Q_{10}}=Q_{10}^{(T_2-T_1)/10}$$The activation energy giving exactly a factor of two per ten degrees is $E_a=k_B\ln2/(1/T_1-1/T_2)$, and $Q_{10}=2$ corresponds to a Z value of $10/\log_{10}2$ degrees. The check uses $k_B=8.617333262\times10^{-5}$ eV/K.
Question: does “ten degrees halves the life” hold alike at every temperature?
2. The Ten-Degree Ratio Under the Arrhenius Law
Computation 2.1: lifetime ratio per ten degrees.
| activation energy | $25\to35$ C | $55\to65$ C | $85\to95$ C | $125\to135$ C |
|---|---|---|---|---|
| $0.3$ eV | $1.460711$ | $1.368531$ | $1.302171$ | $1.238909$ |
| $0.5$ eV | $1.880502$ | $1.686906$ | $1.552793$ | $1.429109$ |
| $0.7$ eV | $2.420937$ | $2.079348$ | $1.851651$ | $1.648510$ |
| $1.0$ eV | $3.536289$ | $2.845651$ | $2.411167$ | $2.042353$ |
Computation 2.2: the activation energy giving exactly two per ten degrees is $0.548776$ eV for $25\to35$ degrees, $0.662796$ eV for $55\to65$, $0.787568$ eV for $85\to95$ and $0.970655$ eV for $125\to135$, and with these values the ratio differed from $2$ by at most $2.2\times10^{-16}$.
Observation 2.3 with one activation energy fixed, the ten-degree ratio shrinks as temperature rises. For all four activation energies the ratio was largest for $25\to35$ degrees and smallest for $125\to135$: to keep “two per ten degrees”, the activation energy would have to change by band from $0.548776$ to $0.970655$ eV.
3. The Miss When Stacked, and the Exponential in Temperature Itself
Computation 3.1: using $0.548776$ eV, which doubles for $25\to35$ degrees, gives $1.620904$ for $85\to95$ degrees, $1.479764$ for $125\to135$, and a lifetime ratio of $213.7152$ from $25$ to $125$ degrees. Using $0.787568$ eV, which doubles for $85\to95$, gives $2.704075$ for $25\to35$ and $2206.1160$ from $25$ to $125$.
Computation 3.2: under $Q_{10}=2$ ($Z=33.219281$ degrees), the ratio was $2.000000$ for every ten degrees and $1024.0$ from $25$ to $125$ degrees.
Observation 3.3 stacking “ten degrees halves” ten times can miss the Arrhenius estimate by an order of magnitude. Against $1024$, an activation energy fitted in the low band gives $213.7152$, one fitted in the high band $2206.1160$: the same hundred-degree extrapolation differs by more than tenfold depending on which band the rule was fitted in.
4. The Separator
Observation 4.1 the separator is whether the exponent is taken in inverse temperature or in temperature itself. The Arrhenius law takes the exponent in the difference of inverse temperatures, so the same ten degrees gives a smaller difference at higher temperature and the ratio shrinks: in the $Q_{10}$ and Z-value views, which take the exponent in the difference of temperatures themselves, the ten-degree ratio is the same at every temperature. “Ten degrees halves” is a rule of thumb of that form.
Observation 4.2 what is not said. Which law component life follows, activation energies of real components, and failure mechanisms are not treated. What is shown is that the two forms give different ratios for the same ten degrees, and that under the Arrhenius law the ratio changes by band.
5. Roll Call and Closing
- $\mathrm{AF}=\exp[E_a/k_B\cdot(1/T_1-1/T_2)]$ (Section 1): exponential in inverse temperature.
- at $0.7$ eV, from $2.420937$ to $\mathbf{1.648510}$ (Computation 2.1): shrinks by band.
- activation energy for a factor of two, $\mathbf{0.548776}$ to $\mathbf{0.970655}$ eV (Computation 2.2): differs by band.
- $25\to125$ degrees: $\mathbf{213.7152}$ or $2206.1160$ (Computation 3.1): can miss the stacked $1024$ by an order.
- $Q_{10}=2$: $2.000000$ for every ten degrees (Computation 3.2): independent of temperature.
- inverse temperature, or temperature itself (Observation 4.1): the separator; Papers 717, 523, 457.
Closing: “Ten degrees halves the life” holds, under the Arrhenius law, only in one temperature band for one activation energy. With one activation energy fixed, the ten-degree ratio shrank as temperature rose, and the value giving exactly two moved from $0.548776$ eV for $25\to35$ degrees to $0.970655$ eV for $125\to135$: fitted in the low band, the lifetime ratio from $25$ to $125$ degrees was $213.7152$, about a fifth of the $1024$ from stacking ten times. The ratio is independent of temperature only when written as an exponential in temperature itself, as with $Q_{10}$ or Z values: the separator is whether the exponent is taken in inverse temperature or in temperature itself. Placed among the earlier papers: the rate of exchange shown by Paper 717 to be temperature-independent in the Z-value view was rewritten with the Arrhenius law and counted as shrinking, and beside the prefactor of Paper 523 and the two temperature coefficients of Paper 457, another dividing line, the form of the exponent, was set. To be honest, which a component follows is not said. On which form the rule of thumb stands, and no further.
References
- Laidler, K. J. (1987). Chemical Kinetics. Harper & Row.
- Ohring, M. (1998). Reliability and Failure of Electronic Materials and Devices. Academic Press.
- Yamagishi, Y. (2026). Trading Flavour for Speed Works Only When Two Temperature Coefficients Differ [Paper 457]. YOI: https://www.yamagishiyuuki.com/yoi/457; DOI: 10.5281/zenodo.22733066
- Yamagishi, Y. (2026). The Exponential Is Shared, but the Prefactor Comes from Three Different Roots [Paper 523]. YOI: https://www.yamagishiyuuki.com/yoi/523; DOI: 10.5281/zenodo.22867544
- Yamagishi, Y. (2026). Time and Temperature Can Be Traded, but Because the Rate of Exchange Is Exponential Only the Shortening Side Gains Orders [Paper 717]. YOI: https://www.yamagishiyuuki.com/yoi/717; DOI: 10.5281/zenodo.23014987
Reference
Yamagishi, Y. (2026). “Ten Degrees Halves the Life” Holds, Under the Arrhenius Law, Only in One Temperature Band for One Activation Energy [Paper 930]. YOI: https://www.yamagishiyuuki.com/yoi/930
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 930. This paper is a preprint and has not been peer reviewed.