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The Reality Of Molecules


More notes from Subtle is the Lord: the science and the life of Albert Einstein.

About the Nineteenth Century, Briefly

1. Chemistry. The first edition of the Encyclopedia Britannica was completed in 1771, and contained the following entry: ‘Atom. In philosophy, a particle of matter, so minute as to admit no division.’ However, other than the omission of the idea of a primal matter from which all things are made (πρώτη ϋλη, próti ýli), little had changed in the understanding of matter since the time of the Ancient Greeks.

Things started to change by 1808, when John Dalton introduced the modern science of Chemistry, but Avogadro’s principle, introduced in 1811, which made explicit the idea that molecules are real things, was only gradually, and very reluctantly, accepted by chemists. Even by 1869, atomic theory was considered something that ”chemists of high authority . . . would gladly dispense with, and which they are ashamed of using.”

2. Kinetic Theory. Physicists, on the other hand, had little choice but to confront the particulate model of matter: Planck felt that the increase of Entropy as an absolute law “is incompatible with the assumption of finite atoms.” An idea must die.

In 1895, the lines were drawn. Boltzmann on one side, along with the younger mathematicians, advocating the statistical model, with luminaries like Ostwald and Ernst Mach insisting that the atomic model was nothing more than a mathematical tool.

Many of these arguments revolved around the determination of Avogadro’s number, \(N\).

3. The End of Indivisibility. Marie Curie wrote in 1900 that the explanation of radioactivity in terms of the expulsion of subatomic particles “seriously undermines the principles of chemistry.”

4. The End of Invisibility. Chemists and physicists at least agreed that atoms, if they did exist, were too small to be seen. In 1828 a botanist named Robert Brown reported seeing the random motion of particles suspended in water. Einstein claimed this to be the result of water molecules pushing against the suspended objects, thereby making the molecules visible, if indirectly.

The Pots of Pfeffer and the Laws of van ’t Hoff

Wilhelm Pfeffer, renowned for his work in botany and plant physiology, came up with a technique to construct semipermeable membranes. A decade later van ’t Hoff detected that highly concentrated solutions moved with considerable pressure though such a membrane to a dilute region. He correctly deduced that this pressure is similar in nature to the behavior of ideal gases, and therefore subject to the same laws discovered by Boltzmann.

This was the basis of Einstein’s doctoral thesis.

The Doctoral Thesis

Einstein submitted his thesis on July 20, 1905, even though he had finished it by April 30. Pais explains the delay was because he was “rather busy with other things” — in those months he also completed his papers on Brownian Motion and Special Relativity.

His thesis was of more widespread practical interest than any other paper he ever wrote, and of all papers published before 1912, it is the most widely cited paper in the period between 1961 and 1975.

It also provided yet another formula for determining Avogadro’s number.

Eleven Days Later: Brownian Motion

1. Another Bit of Nineteenth Century History. The mechanisms involved were bitterly contested, but it did not take long before it was suggested that the motion of the particles was the result of collisions with individual water molecules.

Poincaré saw the phenomenon of heat being converted to motion as being contrary to Carnot’s principle.

2. The Overdetermination of \(N\). Einstein was unfamiliar with the history of Brownian motion, and the principles he used were based on those discussed in his previous paper.

His paper, called ’On the Motion Required by the Molecular Kinetic Theory of Heat of Particles Suspended in Fluids at Rest,’ was published on May 11, 1905. It was the third time in less than two months that he published a paper containing a fundamental discovery about the determination of Avogadro’s number.

3. Einstein’s First Paper on Brownian Motion. Einstein argued that van ’t Hoff’s discoveries related to solutions applied equally to suspensions.

4. Diffusion as a Markovian Process. Einstein linked the random walk of suspended particles as a statistical (Markov) process, resulting from the diffusion of many microscopic particles.

5. The Later Papers. He wrote four more papers between 1905 and 1908 on Brownian motion. In 1909, he wrote “I had believed it to be impossible to measure Brownian motion so precisely.” By this time he was more occupied with quantum theory, and expressed concerns that the 15 percent difference in Planck’s determination of Avogadro’s number was ‘disquieting, since one must say that the theoretical foundation of Planck‘s formula is fictitious.’

Einstein and Smoluchowski; Critical Opalescence

Why is the sky blue? The scattering of light through a gas increases sharply at temperatures near the critical point. Smoluchowski explained this as a consequence of the mean square of the particle number fluctuations, \(\overline {\delta^2}\)

\[\begin{equation} \overline{\delta^2} = \frac{R \, T}{N V(-\delta p / \delta V)_T} \end{equation}\]

But near the critical point \((\delta p / \delta V)_T = (\delta^2 p / \delta^2 V)_T = 0\), and under the assumptions of the kinetic theory of gases, the equation blows up with a pronounced maximal value.

Smoluchowski saw this as the true cause not only of critical opalescense, but also as an explanation of the blue sky during midday, and the red sky at sunset. John Tyndall had explained the blue sky as a result of the scattering of light by dust particles or droplets, but it was Rayleigh’s belief that the air molecules themselves were responsible for the “Tyndall phenomenon.” However, Smoluchowski did not produce a detailed scattering calculation because it would have involved “far-reaching” changes to Rayleigh’s work.

But in 1910, Einstein produced the following formula for monochromatic polarized light in an ideal gas:

\[\begin{equation} r = \frac{R \, T}{N} \frac{(n^2 - 1)^2}{p} \left( \frac{2\pi}{4} \right)^4 \frac{\Phi}{(4\pi\Delta)^2} cos^2 \vartheta \end{equation}\]

where \(r\) is the ratio of the scattered to primary intensity, \(n\) the index of refraction, \(\lambda\) the incident wavelength, \(\Phi\) the irradiated gas volume, \(\Delta\) the distance of observation, and \(\vartheta\) the scattering angle.

Not only did he show the quantitative relationship between critical opalescence and Rayleigh scattering, but he also came up with yet another mechanism to obtain Avogadro’s number.

This was Einstein’s last major paper on classical statistical physics. Ostwald conceded in 1908 that the evidence of the atomic theory of matter was overwhelming, however Mach was still unconvinced when he died in 1916.