Last time we introduce the Theory of Corresponding States through the historical context via Van der Waals’ equation of state. At the end of class we tried a simple exercise to estimate the density of nitrogen and hydrogen via classical corresponding states. Although nitrogen was accurate to 0.8%, hydrogen was off by 12%. Here’s a plot over our cryo fluids of interest this semester:
So clearly, something weird is happening with the very low temperature fluids neon, deuterium, hydrogen, and helium. The deviations are primarily due to the classical corresponding states curves being developed for hydrocarbon and refrigerant blends. The heavy gases like krypton, and the very light gases are clearly not well modeled by the technique. Why?
Early in the semester we introduced the Kammerlingh-Onnes’ virial equation:
If the corresponding states in compressibility (Z) space are not working then something must be happening to the virial coefficients of these light gases. Plotting the virials we see that is indeed the case where a classical curve exists and the quantum fluids depart from:
But how can this be if the particles are acting as billiard balls? As early as 1922 Byk suggested that all deviations from classical corresponding states should be due to Quantum Mechanics. That is indeed a very early suggestion. Although Wave-particle duality was first postulated by Einstein in 1905:
“It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do“
it wasn’t until 1924 when Louis de Broglie extended the theory to all matter. De Broglie’s premise was simple, if light has mass, which it does, a very, very small amount, then all mass should also have wave-particle duality. When he submitted his Ph.D. thesis the committee sent it to Einstein and inquired whether he should receive a diploma. Einstein famously replied something along the lines of: “The matter of the diploma is up to the committee. But if it’s up to me, I’d award him the Nobel Prize.” Which he did receive in 1929:
The Nobel Prize in Physics 1929 was awarded to Louis de Broglie “for his discovery of the wave nature of electrons“.
Mathematically speaking the reasoning went like this:
E = mc² where m is mass and c is the speed of light, has to equal E = hf where h is Plank’s constant and f is the frequency of a light wave.
The wavelength is defined as speed over frequency: λ = c/f and the momentum defined as energy over speed: i = E/c. By substituting these into above we arrive at the definition of a de Broglie wavelength:
λ = h/i the de Broglie wavelength is equal to Plank’s constant divided by the momentum of the particle, atom, or molecule in question. But what does this mean? As the mass of molecules decreases, so does the momentum, since Plank’s constant is a constant, the wavelength increases. Eventually the particles will behave more wave-like than billiard ball-like.
For reference a heavy baseball hit at 110 miles/hour has a de Broglie wavelength λ = 9.5 x 10^-35 meters. A hydrogen molecule moving at 1 x 10^6 m/s has a wavelength λ = 2 x 10^-13. The wavelength of hydrogen is 21 orders of magnitude larger than the baseball because the mass is so small!
Now the key challenge is relating relevant properties of these light gases to the de Broglie wavelength. Going all the way back to statistical mechanics we’ll find a direct relationship for calculating the 2nd virial coefficient:
B = 2πNa ∫[1-exp(-U/kbT)]r²dr
where Na is Avogadro’s number, U is the intermolecular potential, kb is Boltzmann’s constant and r is the distance between molecules. The intermolecular potential is the attraction and repulsive forces between molecules. The Lennard-Jones 12-6 potential is one of the simplest forms:
From this, σ is the radius (distance between molecules) at the maximum well depth (ε) which provides a measure of the molecule speed. Combining these with the definition of the de Broglie wavelength above allows estimation of an approximate wavelength for the molecules:
λ = Nah/(Mε)^0.5 where M, the molecular weight, times the well depth and taking the square root gives momentum.
In 1947 de Boer proprosed a dimensionless comparison between the de Broglie wavelength for a molecule above and the average radius of interaction σ to see if on a scale relevant to the molecules waves or particles dominated. Here’s a plot of the de Boer, aka Quantum Parameter calculated for the cryogenic fluids:
From this we can see that tritium is the balance point where wave and particle effects balance and as we reduce in mass, the molecules become more wavelike than particle like. For example, the hydrogen wavelength is nearly double the average radius of particle interaction! Moreover, we can now use this Quantum Parameter to scale (similar to Z) and map the fluids relative to one another. First we have to non-dimensionalize the temperature and pressure by the lennard-jones parameters (similar to how Van der Waals used the critical point as reducing parameters): T*=RT/ε and P*=Naσ³P/ε Now plot versus quantum parameter. First reduces temperature versus quantum parameter:
Now the pressure:
So if we know the difference in intermolecular potential between these quantum fluids, we can estimate properties using the slope of the fit lines above and the equations below. I then used the equations to transform parahydrogen vapor pressures into normal hydrogen vapor pressures for developing the equations of state we talked about previously.
Another application was during my Ph.D. dissertation when we were trying to correlate the shear strengths of solid cryogens. I had a hunch that the shear stress would reduce in a similar way as the pressure. When I tried it, I almost fell out of my chair when I made the plot:
Which then meant we were able to estimate shear strength data for tritium with a high degree of confidence: