# Exploring Albert Einstein's Ph.D. Thesis: A Historical Perspective

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Einstein completed his doctoral thesis in 1905 under the supervision of Professor Alfred Kleiner, an experimental physicist at the University of Zürich. The dissertation, titled “A New Determination of Molecular Dimensions,” was submitted to the University of Zürich, as ETH (where he had earned his previous degree) was not permitted to confer PhDs at that time. Students from ETH were allowed to present their dissertations at the University of Zürich until 1909.

The year 1905 is famously referred to as Einstein's *annus mirabilis*, or "marvelous year," during which he published four pivotal research papers that transformed the field of physics. Among these was his work on the photoelectric effect, which ultimately earned him the Nobel Prize in Physics in 1921. The other significant papers addressed Brownian motion, special relativity, and introduced the mass-energy equivalence formula, E=mc².

While his initial papers garnered significant acclaim, his doctoral thesis did not receive the same level of recognition at first, prompting this exploration into its content and impact.

The thesis was published in Bern on April 30, 1905, and is notable for its brevity—comprising only 24 pages, a length considered unusually short for a doctoral dissertation today.

In this work, Einstein aimed to develop a new theoretical approach for calculating molecular sizes. He derived Avogadro’s number using viscosity data from a sugar-water solution and its experimental diffusion rate. Although his initial estimate was off by a factor of three, subsequent calculations brought him closer to the currently accepted value.

Einstein dedicated his thesis to Marcel Grossmann, a close friend and talented mathematician who provided him with course notes when he missed classes. Grossmann also played a key role in recommending Einstein for a position at the patent office, demonstrating his steadfast support during difficult times.

The core of the thesis revolved around hydrodynamics and the relationship between viscosity coefficients. Einstein examined the stationary flow of a homogeneous, incompressible fluid, using the Navier-Stokes equations to describe its motion.

In his calculations, Einstein assumed a suspension of numerous identical spherical particles within the fluid, ensuring that their total volume was less than that of the liquid. He made several key assumptions: no external forces were acting on the particles, they did not influence each other, hydrodynamic stresses were applicable only at the particle surfaces, and the flow velocity at the sphere surfaces was zero.

Einstein demonstrated that the flow could still be characterized by the same equations with a modified value of viscosity, now referred to as 'effective viscosity,' which he denoted as ?*.

He derived a relationship linking N (Avogadro's number) with the radius of a molecule and the mass of the solute per unit volume.

He concluded that “One gram of sugar dissolved in water has the same effect on the coefficient of viscosity as do small suspended rigid spheres of a total volume of 0.98 cm³.” Einstein also formulated a remarkable equation for the diffusion constant of suspended particles, applying Van’t Hoff’s law for osmotic pressure and Stokes’ law for particle mobility. He used these findings to analyze the sugar solution, yielding an approximate value for Avogadro's number and an estimation of sugar molecule radii.

In 1909, French physicist Jean Perrin independently derived a significantly different value for Avogadro’s number through careful studies of Brownian motion. Einstein shared his hydrodynamic approach with Perrin, who had a student, Jacques Bancelin, verify Einstein’s calculations and identify discrepancies in his viscosity formula.

On December 27, 1910, Einstein reached out to his former collaborator Ludwig Hopf regarding this confusion.

I have verified my previous calculations and arguments and found no errors. I would appreciate it if you could carefully recheck my work. There may be an error in my findings, or the volume of Perrin’s suspended substance may be greater than he believes.

Hopf identified a mistake in the differentiation technique, prompting Einstein to revise his thesis. He published the corrections in a 1911 article with the help of Paul Drude, editor of *Annalen der Physik*, which included 18 pages of recalculations and estimated Avogadro’s number as N = 4.15 x 10²³.

Despite subsequent attempts yielding differing results, Einstein's final estimation using better data arrived at N = 6.56 x 10²³, a value that was relatively accurate.

Currently, the accepted value of Avogadro’s number is N = 6.022 x 10²³, with modern experimental data for sugar solutions confirming this.

A precise determination of molecular sizes is crucial, as it allows for more accurate testing of Planck’s radiation formula than through radiation measurements.

Dr. Kleiner, Einstein’s dissertation advisor, offered a favorable review of his work.

The calculations presented are among the most challenging in hydrodynamics, requiring a keen understanding of mathematical and physical problems. Mr. Einstein has demonstrated his capability to tackle scientific challenges, and I recommend acceptance of the dissertation.

Kleiner sought a second opinion from colleague Heinrich Burkhardt, a mathematics professor.

The treatment of the subject reflects a fundamental mastery of relevant mathematical techniques. I found no errors in my checks.

Interestingly, in his biography *Albert Einstein: A Documentary Biography*, Carl Seelig recounts that Einstein humorously noted his dissertation was initially returned by Kleiner with feedback that it was too concise. After adding a single sentence, it was accepted without further comment.

Correspondence with his wife Mileva Mari? suggests that Einstein engaged in diverse discussions with Professor Kleiner. He had previously attempted to submit an incomplete dissertation in 1901 but withdrew it after three months due to a lack of clarity and completion.

Einstein’s motivation to finish his doctorate waned for a time, as he expressed to his friend Michele Besso that “the whole comedy has become tiresome for me.” Fortunately, by the following year, his perspective had shifted.

Thus, it is accurate to state that Einstein earned his doctorate for his dissertation “A New Determination of Molecular Dimensions,” which later became one of his most cited works.

This research has proven to have a broad range of practical applications, distinguishing it from other contributions by Einstein. The study of citations reveals that four of Einstein's papers were among the most cited from 1961 to 1975, with his thesis ranking first—cited four times more frequently than his paper on the general theory of relativity, which brought him fame.

While relative citation frequencies may not always matter, they highlight the distinction of creating foundational work that, despite being less cited, remains significant—akin to a paper introducing a fundamental concept of gravity.

This illustrates that even Einstein's thesis was not without flaws, serving as encouragement for graduate students worldwide who often labor under immense pressure to produce lengthy dissertations.

A copy of Einstein’s dissertation is archived in ETH’s research collection, available in German.

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