Chemical kinetics
That the life of an electrolytic capacitor is halved for every 10 °C is not an isolated empirical rule: it is the Arrhenius equation with an activation energy of 0.83 electronvolts.
01How fast, not whether it happens
Thermodynamics says whether a reaction can occur; kinetics says how long it takes. They are independent questions, and confusing them leads to absurd conclusions. The oxidation of silicon in air is spontaneous and highly exothermic, and yet a wafer does not fall apart: the native layer grows two nanometers and stops, because at room temperature the rate is negligible.
That last warning is important. The coefficients come from balancing; the orders, from measuring. They only coincide when the reaction occurs in a single elementary step, which is the exception.
| Order | Integrated law | Half-life | What plots as a straight line |
|---|---|---|---|
| 0 | \( [A] = [A]_0 - kt \) | \( [A]_0 / 2k \) | \( [A] \) against \( t \) |
| 1 | \( [A] = [A]_0 e^{-kt} \) | \( \ln 2 / k \) | \( \ln[A] \) against \( t \) |
| 2 | \( 1/[A] = 1/[A]_0 + kt \) | \( 1 / k[A]_0 \) | \( 1/[A] \) against \( t \) |
The last column is the working method: all three plots are tried and the one that gives a straight line reveals the order. The slope gives \( k \). Only first order has a constant half-life, which is how it is recognized immediately.
02The Arrhenius equation
For two molecules to react it is not enough that they collide: they have to collide with enough energy to overcome a barrier, the activation energy. And since the fraction of molecules that overcome a barrier follows an exponential, so does the rate constant:
From here comes the rule that everyone who repairs power supplies knows: for every 10 °C increase, the life of the electrolytic capacitor is cut in half. It is not a separate rule, it is Arrhenius written in a different way. With an activation energy of 0.83 eV, a 10 °C step around 100 °C gives exactly a factor of 2.
Four failure mechanisms with their measured activation energy. Move the operating temperature and the estimated life appears, the acceleration factor relative to the test and how many degrees it takes to double the life.
Nobody can wait ten years to find out whether a product lasts ten years. It is subjected to a much higher temperature for weeks, the time of failure is measured and the result is extrapolated with Arrhenius to the real temperature of use. The whole edifice rests on two assumptions: that \( E_a \) does not change over that range and that the failure mechanism remains the same. If heat triggers a new mechanism, the extrapolation is worth nothing.
03Catalysts
A catalyst offers an alternative path with a lower activation energy. It does not change the end point —neither the enthalpy nor the equilibrium constant moves— and it is not consumed either: it changes only how long it takes.
| Process | Typical \( E_a \) | Where it matters |
|---|---|---|
| Reaction in solution at room temperature | 0.4 to 0.8 eV | Chemical etching, developing |
| Electromigration in aluminum | 0.6 eV | Life of the traces of a chip |
| Thermal oxidation of silicon, linear regime | 2.0 eV | Growth of the gate oxide |
| Diffusion of phosphorus in silicon | 3.7 eV | Why dopants do not migrate on their own at 25 °C |
The last row explains something that is taken for granted: a transistor keeps its doping profile for decades because, with a 3.7 eV barrier, at room temperature diffusion is on the order of one atom every several ages of the universe. The same process at 1000 °C takes minutes, and that is what is used to manufacture it.
04In the lab
Follow the fading of permanganate with oxalic acid by measuring absorbance or color against time. Plot \( [A] \), \( \ln[A] \) and \( 1/[A] \) against \( t \), and determine the order by which of the three gives a straight line.
Repeat the same reaction at four temperatures between 10 and 50 °C. Plot \( \ln k \) against \( 1/T \), obtain \( E_a \) from the slope and compare with the tabulated values.
Look up the datasheets of three electrolytic capacitors of different quality and note their rated life and their test temperature. Extrapolate with Arrhenius to 40 °C and compare: the difference in price is usually explained there.
05Common mistakes
- Reading the order from the balanced equation. The order is measured; the coefficients are balanced.
- Believing that a catalyst shifts the equilibrium. It speeds up the forward and reverse reactions equally: the end point is the same.
- Confusing spontaneous with fast. Diamond is unstable relative to graphite and nothing happens.
- Extrapolating with Arrhenius outside the tested range. It holds as long as the failure mechanism does not change.
- Using degrees Celsius in the exponent. \( T \) goes in kelvin, and so does \( 1/T \).
- Assuming a constant half-life at any order. Only first order has one.
06Self-assessment
How is the order of a reaction determined?
Experimentally: you try which of the three plots —\( [A] \), \( \ln[A] \) or \( 1/[A] \) against \( t \)— gives a straight line.
What is obtained from the slope of \( \ln k \) against \( 1/T \)?
The activation energy, with its sign changed and divided by the Boltzmann constant.
Why is the life of an electrolytic capacitor cut in half every 10 °C?
Because its degradation mechanism follows Arrhenius with \( E_a \approx 0.83 \) eV, and with that barrier ten degrees give exactly a factor of 2 around 100 °C.
Does a catalyst change \( \Delta H \)?
No. It changes the path and the activation energy; the starting point and the end point stay where they were.
Why does the doping of a transistor not diffuse on its own?
Because the diffusion of phosphorus in silicon has an activation energy of 3.7 eV: at room temperature the rate is indistinguishable from zero.
What does an accelerated test assume?
That the activation energy does not change over the tested range and that the failure mechanism remains the same.
07Further reading
- Raymond Chang and Kenneth A. Goldsby. Chemistry. 12th ed. (Spanish edition, “Química”), McGraw-Hill, 2017. Chapter 13: rate laws, orders, mechanisms, Arrhenius and catalysis.
- Peter Atkins and Julio de Paula. Physical Chemistry. 10th ed. (Spanish edition, “Química física”), Editorial Médica Panamericana, 2018. The complete treatment of reaction mechanisms and transition state theory.
- Michael Pecht et al. Physics-of-Failure-Based Reliability Prediction. Wiley, in Electronic Reliability Handbook. The Arrhenius, Peck and Black models applied to the service life of electronic components.