‘Teachers don’t find it interesting, nor do students.’ That’s the only way chemical technologist Maarten Biesheuvel (see photo) can explain that a theory about the activity of ions in solution dating from 1923 still holds, even though it rattles all over. But his own theory has recently been published online on arXiv, and it no longer contains a square root of the ion concentration, but a cube root.

For Biesheuvel, who is a researcher at the public-private water technology centre Wetsus in Leeuwarden, it is a spin-off from a textbook on electrochemical processes that he is working on with Wageningen assistant professor Jouke Dykstra. ‘Thanks to the corona crisis, I finally had the time and the motivation.’ While writing, he decided to elaborate on his dormant thoughts about ion activity.

Heavyweight

As early as 1916, the Dane Niels Bjerrum incorporated that cube root into a formula that excellently covered earlier experiments with freezing-point depression. ‘In those days, they were already able to measure it with millikelvin accuracy’, Biesheuvel explains. Two years later, the Indian Jnan Chandra Ghosh provided a theoretical foundation.

Ghosh’s assumption that ions are distributed evenly, as in a crystal lattice, was a weak point. This gave Peter Debye from the Dutch province Limburg, who was then connected to the ETH in Zürich, and his German assistant Ernst Hückel all the space necessary for an alternative, which Debye published in Chemisch Weekblad among other magazines. He reasoned from a single ion with a specific diameter, which the other ions surrounded as point charges. The Gouy-Chapman diffuse charge model of electrical double layers served as a basis, and Biesheuvel suggests that Debye clung too strongly to this theory to be able to come up with something new. In any case, the derivation wasn’t the prettiest.

The result was not a cube root, but a square root. Only at extremely low concentrations did this give a somewhat acceptable result. Correction factors did bring some improvement afterwards, but only for salts with univalent ions; it still was somewhat of a makeshift solution. Nevertheless, the square root became leading in the field. Biesheuvel suspects that Bjerrum and Ghosh could not compete with a heavyweight like Debye, who would even go on receiving the Nobel Prize – though not for this work.

Shell

Biesheuvel takes a fundamentally different approach to the problem. He envelops ions with a shell that ends where another ion’s shell begins. ‘I use a Laplace equation to describe this interaction. When I was still at Wageningen University, I used a similar approach for protein molecules together with Martien Cohen Stuart and Saskia Lindhoud.’

Surprisingly, Biesheuvel automatically arrives at a cube root. And again the fit with experiments turns out to be excellent. In the first version, this only applied to univalent ions, but a modification – in which each shell does not contain a separate ion but an ion pair – also makes it applicable to bivalent and trivalent ions.

One day, Biesheuvel hopes to submit his theory to a journal. But there is no rush: at this stage of his career he doesn’t need citation scores, he has his doubts about the accuracy of peer reviews in 2021 and he does not want to saddle Wetsus with an open access bill. ‘And I dare say that preprint servers produce better thought-out manuscripts. You can submit a second version, but even the first remains online forever. You really have to think twice before you press publish.’

‘Teachers don’t find it interesting, nor do students.’ That’s the only way chemical technologist Maarten Biesheuvel (see photo) can explain that a theory about the activity of ions in solution dating from 1923 still holds, even though it rattles all over. But his own theory has recently been published online on arXiv, and it no longer contains a square root of the ion concentration, but a cube root.

For Biesheuvel, who is a researcher at the public-private water technology centre Wetsus in Leeuwarden, it is a spin-off from a textbook on electrochemical processes that he is working on with Wageningen assistant professor Jouke Dykstra. ‘Thanks to the corona crisis, I finally had the time and the motivation.’ While writing, he decided to elaborate on his dormant thoughts about ion activity.

Heavyweight

As early as 1916, the Dane Niels Bjerrum incorporated that cube root into a formula that excellently covered earlier experiments with freezing-point depression. ‘In those days, they were already able to measure it with millikelvin accuracy’, Biesheuvel explains. Two years later, the Indian Jnan Chandra Ghosh provided a theoretical foundation.

Ghosh’s assumption that ions are distributed evenly, as in a crystal lattice, was a weak point. This gave Peter Debye from the Dutch province Limburg, who was then connected to the ETH in Zürich, and his German assistant Ernst Hückel all the space necessary for an alternative, which Debye published in Chemisch Weekblad among other magazines. He reasoned from a single ion with a specific diameter, which the other ions surrounded as point charges. The Gouy-Chapman diffuse charge model of electrical double layers served as a basis, and Biesheuvel suggests that Debye clung too strongly to this theory to be able to come up with something new. In any case, the derivation wasn’t the prettiest.

The result was not a cube root, but a square root. Only at extremely low concentrations did this give a somewhat acceptable result. Correction factors did bring some improvement afterwards, but only for salts with univalent ions; it still was somewhat of a makeshift solution. Nevertheless, the square root became leading in the field. Biesheuvel suspects that Bjerrum and Ghosh could not compete with a heavyweight like Debye, who would even go on receiving the Nobel Prize – though not for this work.

Shell

Biesheuvel takes a fundamentally different approach to the problem. He envelops ions with a shell that ends where another ion’s shell begins. ‘I use a Laplace equation to describe this interaction. When I was still at Wageningen University, I used a similar approach for protein molecules together with Martien Cohen Stuart and Saskia Lindhoud.’

Surprisingly, Biesheuvel automatically arrives at a cube root. And again the fit with experiments turns out to be excellent. In the first version, this only applied to univalent ions, but a modification – in which each shell does not contain a separate ion but an ion pair – also makes it applicable to bivalent and trivalent ions.

One day, Biesheuvel hopes to submit his theory to a journal. But there is no rush: at this stage of his career he doesn’t need citation scores, he has his doubts about the accuracy of peer reviews in 2021 and he does not want to saddle Wetsus with an open access bill. ‘And I dare say that preprint servers produce better thought-out manuscripts. You can submit a second version, but even the first remains online forever. You really have to think twice before you press publish.’