### Arrhenius Law

September 13, 2021

While the human body has difficulty performing tasks such as lawn mowing and gardening in the summer's heat, heat enhances the performance of machinery operating as heat engines. According to the second law of thermodynamics, a temperature difference is required to perform work, and the equation for the maximum efficiency of a heat engine, the Carnot efficiency η, reveals that the larger the temperature difference, the better.
ηCarnot = (Thot-Tcold)/Thot

One of the more famous high school yearbook portraits - Sadi Carnot (1796-1832) in 1813 at age 17 in his uniform for the École Polytechnique.

Carnot, who is called the father of thermodynamics, served as a mechanical engineer in the French Army. He ppublished a book in 1824, "Reflections on the Motive Power of Fire," that contained his theory of the maximum efficiency of heat engines that became the basis for the second law of thermodynamics and its concept of entropy.

One fundamental concept of his book was that heat engines are more efficient when the temperature of their heat reservoir is increased.

(An engraving by Louis-Léopold Boilly (1761–1845) via Wikimedia Commons. Click for larger image.)

Gas turbines, such as those for jet engines, are heat engines for which temperatures are pushed to extremes to gain efficiency. The thermodynamic cycle for these is the Brayton cycle. In such turbine engines, air is compressed, mixed with fuel, and combusted. The resultant high temperature flow is used to drive the compressor, and rotary devices, such as electrical generators, to create useful work. In an aircraft, the hot exhaust will produce thrust. The following figure illustrates the maximum efficiency of a Brayton cycle engine as a function of the temperature between the inlet (at 298 K) and the outlet.

The maximum efficiency, ηB, of a Brayton cycle engine as a function of the difference in temperature between the exit and inlet when the inlet temperature is 298 K.

Materials scientists work to create turbine blade materials and coatings that allow operation at higher temperatures. At a difference temperature of 1000 K (an operating temperature of about 1025 °C) the slope is 0.02%/K.

(Created with Gnumeric. Click for larger image.)

I've done many calculations in which precise data are unavailable, and for these I've used various rules of thumb, such as Trouton's rule that the entropy of vaporization is almost always 85 J/K-mol) for a liquid at its boiling point. One common rule of thumb in chemistry is that reaction rates double for every temperature increase of ten degrees Celsius (or kelvin). This rule derives from an important equation, the Arrhenius equation, derived from statistical mechanics by physical chemist, Svante Arrhenius (1859-1927), who was awarded the 1903 Nobel Prize in Chemistry.

The Arrhenius equation allows calculation of the rate constant of a chemical reaction based on an activation energy and the absolute temperature. It appears here in two equivalent forms, depending on the units used:

in which k is the rate constant, T is the absolute temperature (K), A is a constant for each particular reaction, Ea is the activation energy for the reaction, R is the gas constant, and kB is the Boltzmann constant. Taking ratios at room temperature shows a doubling of rate if the activation energy is 52.9 kJ/mol. Not all reactions will have this value, but it's likely a good average.

The equation was conceived in 1889 by Arrhenius, who reasoned that reactants need to have sufficient kinetic energy in order to merge with each other to form a product. Through statistical mechanics, a given temperature will produce a fraction of molecules with kinetic energy greater than the activation energy, and that fraction is the rate constant in a reversible reaction.

Physical chemist, Svante Arrhenius (1859-1927), who was awarded the 1903 Nobel Prize in Chemistry.

Arrhenius was the first to realize that atmospheric carbon dioxide would increase Earth's surface temperature.

The lunar crater, Arrhenius, and the Martian crater, Arrhenius, are named in his honor.

(Portion of a Wikimedia Commons image from Zeitschrift für Physikalische Chemie (1909). Click for larger image.)

In pre-computer times, chemists would deduce the activation energy from a plot of the logarithm of the rate constant against inverse temperature, 1/T. In this plot, called an Arrhenius plot, the slope is (Ea/R) and the y-intercept at 1/T = 0 is the log of the scale factor, A. Computer applications now give these values by a linear regression of the data.

To demonstrate the ubiquity of the Arrhenius law, Keith J. Laidler (1916-2003) of the University of Ottawa (Ottawa, Canada) published a paper in the Journal of Chemical Education in 1972 in which he included data showing that the chirping frequency of tree crickets (Oecanthus) follows the Arrhenius Law. The data, plotted in the figure, gives an activation energy of 12.2 kcal/mol (51 kJ/mol).

Arrhenius law for the chirping frequency of tree crickets (Oecanthus).

The frequency of flashing of fireflies gives the same activation energy, 12.2 kcal/mol, but with more scatter of data points.[1]

(Created using Gnumeric from data if ref. 1. Click for larger image.)

Axel Lorke, a professor of physics at the University of Duisburg-Essen, has recently published an article about the Arrhenius law in Physics Today.[2] In that article he shows that egg yoke gelation, the point at which the viscous and elastic moduli are equal, obeys the Arrhenius law. He was fortunate in obtaining gelation data from a decade old paper by researchers who were willing to wait a hundred hours for some data (see graph).[3]

"I'll have my eggs sous vide - cook for a hundred hours at 54 °C."

Egg yoke gelation follows the Arrhenius law.

(Created using Gnumeric from data in refs. 2-3. Click for larger image.)

### References:

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