Flash back to the literal days of Sir Isaac Newton and horse drawn carriages. Natural Philosophy was the hobby of the wealthy elite. It is in these humble beginnings that we learn the story of thermodynamic properties and equations of state.

Robert Boyle (1627-1691) was one of the first “scientists”, actually at the time known as a natural chemist, to consider the behavior of gases while changing temperature and pressure. It’s widely considered that his laboratory assistant, Robert Hooke (from Hooke’s Law) built one of Boyle’s original apparatus on the assumption that air was a fluid of particles connected by small invisible springs. The apparatus was a small J-shaped glass tube with which mercury would be added to trap the air under the weight of the mercury column. What they found became know as Boyle’s Law, or that pressure*volume is equivalent at constant temperature, in other words pressure is inversely proportional to volume under constant conditions (temperature).

Approximately 100 years later Jacques Charles (1746-1823), a famous French inventor and balloonist, was experimenting with the relationship between gas temperature and volume at constant pressure. He found that with the pressure constant (like in a balloon for instance) the ratio of V/T is constant. In other words Charles Law states that V1/T1=V2/T2 at constant pressure. This relationship was later confirmed in 1801 by John Dalton who published his careful experimental results.

About the same time in 1811, Amedeo Avogadro (1776-1856) an Italian physicist published an essay titled, “”Essay on Determining the Relative Masses of the Elementary Molecules of Bodies and the Proportions by Which They Enter These Combinations”. Avogadro’s essay concluded that the number of molecules in a fixed volume of gas at constant temperature and pressure is constant, or V/n=k where k is the constant.

With these pieces in place in 1834, Emile Clapeyron, a French Engineer, was able to combine Boyle’s Law, Charles’ Law, and Avogadro’s Law into the Clapeyron Equation, or what we know today as the ideal-gas law: P * v = R * T. R was a constant for each gas that had to be determined empirically.

It wasn’t until the late 1800’s that Ludwig Boltzmann was able to derive the ideal-gas constant from purely statistical mechanics principles. Boltzmann and others knew that the ideal-gas “law” was only a law in the limiting behavior it provides. Far enough away from the critical region you could assume fluids to behave by this law. The problem started to become if P * v / R * T ≠ 1, then what is it equal to? In other words, how compressible (Z) are fluids that causes them to deviate from the ideal-gas law?

Heiki Kammerlingh-Onnes (1853-1926) was a dutch physicist famously racing James Dewar to be the first to liquefy helium. This work helped him to discover “superconductivity” when one of his mercury thermometers stopped displaying electrical resistance below 4.2 K. He was awarded the Nobel Prize of 1913 “for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium”. Kammerlingh-Onnes succeeded where Dewar failed due to two critical character traits: 1) he built and maintained positive relationships with folks internationally that allowed him to get supplies of helium which were very rare at the time. 2) he had an extensive knowledge of the physics of fluids.

Kammerlingh-Onnes utilized the Linde-Hampson cycle for liquefaction. The Linde-Hampson cycle requires very large pressure swings and extensive precooling with refrigerant gases to accomplish the throttling necessary for the final temperature drop. Without modern accepted equations of state for working in the supercritical fluid regions of these gases, Kammerlingh-Onnes needed an alternative for modeling when throttling of helium would lead to actual cooling and not heating. The problem with the ideal-gas law is that enthalpy = Cp * T and is independent of pressure. Which means that an ideal gas through a throttle always has T1 = T2 and no temperature drop occurs. So again, what does P * v / R * T = ? The root of the problem lies in the attraction of the atoms and molecules that causes the density to change.

Take, for example, the following measurements of the density of helium that Kammerlingh-Onnes could have taken during his experiments:

Temperature = 77 K (Liquid nitrogen bath for cooling)

Pressures = 5 bar, 10 bar, 15 bar, 20 bar

Densities = 3.1002 kg/m^3, 6.1488 kg/m^3, 9.1461 kg/m^3, 12.093 kg/m^3

When we calculate P * v / R * T = Z = we get 1.0083, 1.0168, 1.0254, 1.0340, definitely not 1! Moreover, the higher the density, the more we deviate from 1. If we plot these versus density we get the following:

A similar process to this is likely what Kammerlingh-Onnes used to realize that a power-law expression is convenient for representing this behavior of fluids and helped to develop his virial expansion version of an EOS:

P * v / R * T = Z = 1 + B / v + C / v^2 + D / v^3…..

Here the 1, B, C, D, etc. are “virial coefficients”. “Virial” was originally defined by Rudolf Clausius to describe the forces and energies between bodies and particles. Virial theorem is a direct result of Lagrange’s identity used in orbital mechanics. The first virial coefficient is 1. The second virial coefficient, B, describes the forces between two bodies interacting and is a function of temperature. The third virial coefficient, C, three bodies (still a function of temperature), ad infinitum. This rigorous connection to statistical mechanics gave the virial equation a foundational significance.

In the above plot I used the linear regression tool in EES with Z as the dependent variable and density as the independent variable and the default (power-law) form of Z = a_0+a_1*rho+a_2*rho^2. The calculated virial coefficients (A=0.9984, B=0.002693, and C=0.00001088) agree to a fair extent with the accepted values in the literature determined from both experimental measurements and theoretical calculations (A=1, B=0.00264, and C=0.00001414). As you can see though, the accuracy of the method for predicting the higher order virials becomes more and more inadequate. The fluid information we are using to determine these values is not a sensitive measure of 4 and 5 body particle interactions. Hence, the virial equation is only applicable for supercritical fluids away from the vapor dome. The complexities of the equations required to calculate the virial coefficients from first principles are also cumbersome, yet still used today.