Albert Einstein Describes the Thermodynamics of Liquids, Their Pressures and Equilibrium – the Subject of His First Published Scientific Paper – Using Equations and a Force-Balance Diagram

Scientists today are exploring the relationship between this type of work – the stability and equilibrium of liquids – to apply to black holes

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 first publication was so early it was written to prior to his days as a clerk in the patent office.

In physics, the Young–Laplace equation is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension (although usage on the latter is only applicable if assuming that the wall is very thin). The Young-Laplace equation is also applied to droplets of liquids. Knowledge of capillary pressure is important in medicine, as it relates to treatments for cardiac and respiratory issues.

Hans M. Cassel was a lecturer at the Technical University of Berlin, and is on the list of “Displaced Scholars” who fled Germany for the United States after Hitler took power. He was a bio-chemist interested in liquids, and among his publications were “The Stability of Emulsions” and “The Dimensions of Soap Molecules As Supposedly Deduced From Surface Tension Measurements.” In the mid-1930s, knowing of Einstein’s work in the field, Cassel engaged in a discussion with him of the problem of the thermodynamic stability of a droplet of liquid that forms on the surface of a permeable membrane. A good analogy to this physical problem would be that of a leaky water balloon under pressure. Under what conditions are droplets formed on the outside of the balloon’s surface, and what is the size of the droplets?

Einstein engaged with Cassel on this subject, describing a liquid (“phase 1”) in contact with the atmosphere / a gas (“phase 2”). Dissolved in the liquid are droplets of a substance that have a radius R. The Young—Laplace equation is written 2*gamma/R or 2G/r. It is the Young-Laplace equation, with gamma representing the surface tension of the droplet.

At equilibrium, the amount of material leaving the liquid phase must equal the amount of gas entering the liquid phase. The pressure at equilibrium going up must equal that going down or it would no longer be at equilibrium. Specific volume is an intrinsic value, representing the volume per particle, per gram of substance (i.e., it is a value that is independent of the amount of the substance).

In this unpublished letter, Einstein sets up the description of the above problem. He says,  “If one now makes a circle progress between the two phases…” (if one starts with the dissolved substance in the liquid phase, and traces the mathematics through of evaporating the liquid and dissolved substance, and then condensing it back into the liquid phase, then the work must be zero). The work is PdV (pressure multiplied by the change in volume), so going around a cyclic process returns you to the volume you started with, meaning dV = 0, so work is 0. The origin of V1dp1 = V2dp2 is from the Gibbs free energy, a thermodynamic potential that measures the maximum or reversible work that may be performed by a thermodynamic system at a constant temperature.

In describing this, Cassel figured that the total pressure the liquid phase is exerting on the vapor phase should be its vapor pressure (p1) plus the Young-Laplace pressure from the dissolved substance.

As Einstein points out, “In truth, p1 will be independent of the excess pressure…”, meaning that the pressure the liquid exerts upwards (its vapor pressure) is independent of the size of the droplet if it is dissolved completely, such as with oxygen in water. Today, we know that for “regular solutions”, this is a true statement.

Autograph letter signed, Princeton, September 24, 1936, with equations and a rare diagram in his hand, to Cassel. “Dear Sir, I cannot understand your statement for the following reasons. There are two phases present (1 and 2). 2= Vapor-Phase as well as a dissolved substance which creates an equilibrium between both. It should also be V1 respectively V2 as the specific volume for Unit Mass for the dissolved substance. P1 and p2 are the particle pressure. If one now makes a circle progress between the two phases with the Unit Mass from the dissolved substance, then as a whole the resulting work is 0. This gives the Formula: V1dp1 = V2dp2.

Now you set p2 = p vapor and p1 = p vapor + 2* gamma/R [I]. And you get the equation: V vapor dp vapor = V fluid (p vapor + 2* gamma/R). But now [I] counts for the total pressure from both phases but not for the particle pressure of the dissolved substance, like you predicted.

In truth, p1 will be independent of the excess pressure 2G/R, if the Unit Mass is dissolved in the ”Liquid” (1) and that does not require a volume change. Equation 1 seems totally unfounded and therefore your initial formula as well.  I am writing this to you to be fair to you, and not in an effort to tell you what to do in the effort. I am asking for patience in case I misunderstood you.”

Einstein also worked in 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.

This letter originated with Cassel’s family. Filled with explanations and calculations and relating to his own work , it is one of the most significant scientific letters of Einstein to reach the market in years. As for Cassel, he went on to receive a patent for the “Method and Apparatus for Measuring Surface Tension”.