Wilhelmy’s birthplace – Stargard, Pomerania – in less happy times

The mid-point of the 19th century – 1850 – was a milestone year for the neophyte science of thermodynamics. In that year, Rudolf Clausius in Germany gave the first clear joint statement of the first and second laws, upon which Josiah Willard Gibbs in America would develop chemical thermodynamics. 1850 was also the year that the allied discipline of chemical kinetics was born, thanks to the pioneering work of Ludwig Wilhelmy.

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Ludwig Ferdinand Wilhelmy was born on Christmas Day 1812 in Stargard, Pomerania (now Poland). After completing his schooling, he studied pharmacy and subsequently bought an apothecary shop. In 1843, at the age of 31, he sold the shop in order to pursue research interests at university, where he made the acquaintance of Rudolf Clausius and Hermann von Helmholtz. In 1846, Wilhelmy received his doctorate from Heidelberg University, and it was here in 1850 that he conducted the first quantitative experiments in chemical kinetics, using a polarimeter to study the rate of inversion of sucrose by acid-mediated hydrolysis.

Wilhelmy’s work had a seminal quality to it because – apart from being a talented individual – he observed that great guiding principle when commencing an exploration of the unknown: he kept things simple.

He chose a monomolecular decomposition reaction, used a large volume of water to keep the acid concentration unchanged during the experiment, maintained constant temperature and followed the inversion process with a polarimeter, which did not physically disturb the conditions of the system under study. By rigorously limiting system variables, Wilhelmy discovered a simple truth: the rate of change of sucrose concentration at any moment is proportional to the sucrose concentration at that moment.

Now it just so happened that in Wilhelmy’s earlier doctoral studies, he had become familiar with utilizing differential equations. So it was a straightforward task for him to model his new discovery as an initial value problem, which he wrote as

where Z is the concentration of sucrose, T is time, S is the acid concentration (presumed unchanging throughout the reaction), and M is a constant today called the reaction velocity constant. Wilhelmy integrated this equation to

where C is the constant of integration. Recognising that when T = 0 the sucrose concentration is its initial value Z_{0}, he wrote

or

He then proceded to show that this equation was consistent with his experimental results, and thus became the first to put chemical kinetics on a theoretical foundation.

A page from Wilhelmy’s pioneering work *“Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet”*, published in *Annalen der Physik und Chemie 81 (1850), 413–433, 499–526*

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**Inversion of sucrose**

Sucrose has a dextrorotatory effect on polarized light, but on acid hydrolysis the resulting mixture of glucose and fructose is levorotatory, because the levorotatory fructose has a greater molar rotation than the dextrorotatory glucose. As the sucrose is used up and the glucose-fructose mixture is formed, the angle of rotation to the right (as the observer looks into the polarimeter tube) becomes less and less. It can be demonstrated that the angle of rotation is directly proportional to the sucrose concentration at any moment during the inversion process.

A Laurent polarimeter from around 1900.

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**A neglected pioneer**

Ludwig Wilhelmy’s groundbreaking research into the kinetics of sucrose inversion was published in *Annalen der Physik und Chemie* in 1850, but it failed to garner attention and in 1854 he left Heidelberg, retiring to private life in Berlin at the age of 42. He died ten years later in 1864, his pioneering work still unrecognized.

It was not until 1884, twenty years after Wilhelmy’s death and thirty four years after his great work, that Wilhelm Ostwald – one of the founding fathers of physical chemisty – called attention to Wilhelmy’s paper. Among those who took notice was the talented Dutch theoretian JH van ‘t Hoff, who in 1884 was engaged on kinetic studies of his own, soon to be published in the milestone monograph *Études de dynamique chimique* (Studies in Chemical Dynamics). In this book, van ‘t Hoff extended and generalized the mathematical analysis that had originally been given by Wilhelmy.

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**A digression on half-life**

In his study of sucrose inversion, Ludwig Wilhelmy showed that the instantaneous reaction rate was proportional to the sucrose concentration at that moment, a result he expressed mathematically as

In his published paper, Wilhelmy did not mention the fact that the fraction of sucrose consumed in a given time is independent of the initial amount. But he might well have noticed that by expressing Z as a fraction of Z_{0}, the left hand side of the equation simply becomes the logarithm of a dimensionless number.

For example the half-life time (T_{0.5}) i.e. the time at which half of the substance present at T_{0} has been consumed

JH van ‘t Hoff was well aware of this fact – he derived a half-life expression on page 3 of the *Études*. And in all likelihood he was also aware that this kinetic truth produced a conflict with the thermodynamic necessity for a chemical reaction to reach equilibrium.

**Reconciling kinetics and thermodynamics**

The starting concentration of sucrose in Wilhelmy’s inversion experiment is Z_{0}. So after the half life period T_{0.5} has elapsed, the sucrose concentration will be Z_{0}/2. After further successive half life periods the concentrations will be Z_{0}/4, Z_{0}/8, Z_{0}/16 and so on. The fraction of sucrose consumed after n half lives is

This is a convergent series whose sum is Z_{0} – corresponding to the total consumption of the sucrose and the end of the reaction. The problem with this formula is that it implies that Wilhelmy’s sucrose inversion reaction – or any first order reaction – will take an infinitely long time to complete. This is not consistent with the fact that chemical reactions are observed to attain thermodynamic equilibrium in finite timescales.

In the *Études*, van ‘t Hoff successfully reconciled kinetic truth and thermodynamic necessity by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant K_{c} is in fact the quotient of the kinetic velocity constants for the forward (k_{1}) and reverse (k_{-1}) reactions

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**Wilhelmy’s legacy**

Wilhelmy’s pioneering work may not have been recognised in his lifetime, but the science of chemical kinetics which began with him developed into a major branch of physical chemistry, involving many famous names along the way.

In the 1880s, van ‘t Hoff and the Swedish physical chemist Svante Arrhenius – both winners of the Nobel Prize – made important theoretical advances regarding the temperature dependence of reaction rates, which proved a difficult problem to crack.

In 1899, DL Chapman proposed his theory of detonation. The chemical kinetics of explosive reactions was then taken forward by Jens Anton Christiansen, whose idea of chain reactions was further developed by Nikolaj Semyonov and Cyril Norman Hinshelwood, both of whom won the Nobel Prize for their development of the concept of branching chain reactions, and the factors that influence initiation and termination.

Several other Nobel Prize winners have their names associated with chemical kinetics, including Walther Nernst, Irving Langmuir, George Porter and JC Polanyi. The work of all these illustrious men has enriched this important subject.

But for now, we must return to Ludwig Wilhelmy in Heidelberg. It is 1854, and having failed to garner any interest in his seminal studies, he has packed his bags at the university, handed in his keys at the porter’s lodge, and is ready to begin the long journey home to Berlin.

Heidelberg, Germany in the 1850s

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**Mouse-over links to works referred to in this post**

Jacobus Henricus van ‘t Hoff **Studies in Chemical Dynamics**

Ludwig Wilhelmy **“Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet” , published in Annalen der Physik und Chemie 81 (1850), 413–433, 499–526**

P Mander April 2016

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