## Typed Letter Signed with Numerous Notations by Hand

EINSTEIN GIVES A PROOF ON THE SUBJECT OF HIS FIRST PUBLISHED SCIENTIFIC PAPER: THE THERMODYNAMICS OF LIQUIDS, THEIR STABILITY, AND THE IMPACT OF PRESSURE.

In 1900, Einstein was granted a teaching diploma by the Federal Polytechnic Institute. He then submitted his first paper to the prestigious Annalen der Physik, which published it and the other great papers which very soon followed. Einstein was well versed and interested in thermodynamics, and the article concerned the capillary forces of a drinking straw, and was entitled “Consequences of the observations of capillarity phenomena". This was his very first publication, written to prior to his days as a clerk in the patent office.

In a previous letter to Hans M. Cassel, a lecturer at the Technical University of Berlin, from January 15, 1937 (also in the Einstein Archives), Einstein explained with equations and formulas some of the core thermodynamic principles that govern his thermodynamic field behavior, as addressed in his first paper.

Now, in this letter to Cassel, dated January 27, 1937, Einstein continues the discussion and responds to objections by Cassel, providing more detailed proofs of his theory.

He starts by stating the points on which the two scientists agree: that the pressure can be written as ln P = ln P0 + αΔ , where Δ is the externally applied pressure on the membrane. He writes, “1. We agree...If there is an overload of outer pressure on the semi permeable membrane which is surrounded by drops, then the vapor pressure increases which creates the drop forming liquid from p.o to p according to the formula [set forth above].” Einstein then writes “Because it is a matter of minor modifications, we can calculate [as follows]. He then uses a property of logarithms, and some algebra to solve for P [Pressure]: ln P− ln Po = ln P/P0 = ln P0 + P− P0/P0 = ln 1+P− P0/P0. Since ln(1 + x) ≈ x, ln P− ln P0 ≈ P− P0/P0 ≈ αΔ.Therefore, P≈ P+ αPΔ." Einstein then defines A = α P0.

Einstein continues, “2. We further agree, that for Δ, the capillary pressure, 2G/p is used so that we get P ≈ P0 + 2A σ/r.” Here the external pressure Δ is the Young-Laplace pressure. Einstein now turns to their points of disagreement. Einstein requires that if the radius of a droplet increases to r′ = r + Δ r, the pressure P should increase. Furthermore, if r′ = r − Δ r, the pressure should decrease. In other words, dP /dr > 0. Cassel’s requirement for stability is that dP /dr = 0 – which implies that the pressure does not depend on the size of the droplet. Einstein believes this is false because it would lead to a neutral equilibrium – any droplet size would be allowed since there is no pressure to prevent the droplet from growing (or shrinking). In contrast, Einstein requires a stable equilibrium, where any change (increase in size or decrease in size) is unfavorable and returns the drop to the original size. If the droplet becomes larger, liquid will evaporate from the surface and cause the droplet to shrink. In the event that the droplet shrinks, vapor will condense onto the droplet’s surface and cause it to grow. Because P0 is a constant, Einstein states that for a droplet to be stable, d/dr (σ /r > 0), which is equivalent to dP /dr > 0.

Einstein writes, “3. Now, there is a serious point where I do not understand you, namely the requirements for the stability of the drop. I claim that the drop is stable, when a) (1b) is satisfied, and b) When for a drop with a larger radius (r+ Δ r) the added vapor pressure is larger than p, and for a drop with a smaller radius (r- Δ f) the vapor pressure is smaller than p. Then the drop radius r will be restored with evaporation or in certain cases condensation...I can not understand that you require that dp/dr must disappear. In this case the equilibrium would be neutral. But we are searching for the requirements of a stable equilibrium.”

After setting forth these proofs, Einstein seems shocked that Cassel seems to believe that the membrane surface depends on the droplet size. “4. Now I will come to the main point of our disagreement. So that drip stability is possible according to inequality (2) and equation (1b), [the following formula] must be used - d/dr (σ/r) > 0. You rightfully say that the fulfillment of the inequality needs a dependency of σ to r. You claim that this dependency in fact is there because of the existence of the capillary-active substance. I dispute this [claim]. The arguments you use are absolutely incomprehensible to me. Do you really want to claim that the area density T from the active substance, aside from its particle pressure, also depends on the radius of the drop?”

Einstein then uses the Gibb’s equation to further show that σ should depend only on P. And, since σ depends only on P, then d/dr(σ /r) < 0. “He says, “If not, then it follows from the Gibbs equation that σ depends alone on p’. One only needs to integrate T as a function of p’ in the Gibbs equation. But if σ is independent of r, then the requirement for the drop stability is never satisfied.” And since σ depends only on P, d/dr(σ /r) < 0 still. Therefore, Cassel’s reasoning cannot be correct. Einstein continues, “The entire pressure in the inside of the drop, however, absolutely fulfills now dP=dp+dp’+d (2σ/r). But P has nothing to do with the stability of the drop, and I can not concur with your conclusion on this.”

Einstein also worked on a similar field of physics with his law on viscosity and in developing the Bose-
Einstein Constant. The specific science discussed in this letter is of great value in research over a broad
spectrum. Today its concepts are used in nanotechnology, and will eventually permit drug delivery systems using that technology.

Filled with equations and formulas, the letter shows Einstein at work and communicating his results to a fellow scientist.

Three 8.5x11 inch pages. Princeton: January 27, 1937. Signed “A. Einstein” and with numerous notations and additions by hand. Text in German. Usual folds; very good condition.

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