VdW
P Fullsack
What is the size of the vacuum?
Let us aks the question bluntly: is there something or nothing in this world?
Gases are intangible. Air around us seems immaterial. But we can touch the water of the ocean.
Is there something in the ocean, but nothing in the air?
Our story starts with Amedeo Avogadro, who made the first decisive progress towards answering this question.
Avogadro made the following brilliant hypothesis:
If you fill a container at a certain pressure and temperature with a gas, any gas, there is a precise number of molecules of gas in the container.
This number is called the Avogadro number when you take 22.4 liters of gas in ambient conditions (30 degrees Celsius and standardized atmospheric pressure).
The universal law of Avogadro marks the birth of the ‘physics’ theory of atomism, who had started in chemistry with Dalton.
But this law has a flaw. Because atoms or molecules are only revealed by a count, they have no material extension. For Avogadro, gases are clouds of mathematical points.
And mathematical points have no diameter, nor size.
By the middle of the nineteenth century, all physicsts started to elaborate on the mechanical interactions between molecules and their link with heat.
Maxwell, Clausius and Boltzmann shaped this into the kinetic theory of gases.
But atoms and molecules were now in a strange position. They has a diameter in the KTG but still nobody had ever been able to measure this diameter.
A big conceptual change was needed.
This second revolution in the theory of gas came from another amazing (dutch) physicist, named Van der Waals.
In his doctoral dissertation, he closely followed recent ideas that held that there was a continuity between gases and liquids.
And he postulated th following new equation of state to replace the equation of state of ideal gases.
\[(P+ \frac{a n^2}{V^2})(V-nb) = nT\] \(P,V,T,n\) are the pressure, volume, absolute temperature (in Kelvin degrees) and number of moles of the gas filling the container.
In the VdW equation, the parameter \(a\) is due to attractive forces that each molecule exert on its neighbors. These forces are called Van der Waals forces. Their effect is cancelled statistically inside the gaz but not near the walls of the container, see the following pictures.
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The other Van der Waals parameter, \(b\), is the incompressible volume per mole for a gas. It is directly proportional to the size of gas molecules.
The main aim of this activity is to try to infer the size of atoms from experimental data collected on gases. Of course there will be some statistics (even though the part of stats is the least interesting part of the whole story).
Hunting rat number 1
Let us try to determine the values of the Van der Waals constant a and b from experimental data.
The value of these constants depends on the chemical element we choose (i.e. on the gas). Each molecule will elicit a value of this pair.
Let us take a common element, nitrogen, as an example.
It is very common: it surrounds us, but we cannot breathe it.
Now make the following experiment.
We fill a closed vessel of 1 cubic meter with pure nitrogen gas. We force the vessel to be at the temperature $\(T=300K\) which is about room temperature. We start by placing 20 moles of nitrogen in the vessel and let it equilibrate at the required temperature. Then we make a measure of the pressure, that is of the force that the gas exerts on the walls of the vessel.
We redo this by adding some gas in (20 more moles). Again, let it equilibrate and measure the pressure. We keep doing this, thus increasing the pressure. We ideally should continue until we reach the highest pressure that our vessel can sustain without exploding.
I have found online a table of experimental measurements thus obtained in an engineering handbook (see references).
I have placed this table on the course website.
Let us load it
dnp = read.csv("../Data/np.csv",header=T)
head(dnp)## n_mol P_Pa
## 1 20 49361
## 2 25 61538
## 3 30 73649
## 4 35 85694
## 5 40 97674
## 6 45 109588Let’s take a look at the data.
P <- dnp$P_Pa
n <- dnp$n_mol
plot(dnp$n,P)
Waoo, it sounds totally boring!
We will see later that we would be mistaken. There is something real in the data!
OK. Now let us fit the equation of state of Van der Waals.
We will use one appropriate R function to find the parametric Van der Waals curve that approches the best, in the least-square sense, the data.
We will admit that we have determined the constant of ideal gas:
\[R = 8.3143 JK^{−1}mol\]
For fitting, we will assume that pressure data is normal, and homoskedastic (same variance).
What formula are we going to fit?
We just rewrite the VdW equation of state as:
\(P=-a \frac{a n^2}{V^2}+\frac{nRT}{V-nb}\)
# we call the function nls to fit the formula
# we have to start with values that are not too far from
# the solution
# this is a limitation of the method
# I embed the formula of VdW and I use orders of magnitudes
# as starting values for the a and b parameters
nf <- nls(P ~ - a*n^2 + (n*300.*8.3143)/(1. - b*n) ,start=list(a = 1.,b = 0.0005))I have my rat! The values of \(a\) and \(b\) come out from the data AND from the chosen model.
Let us compare with tables available online:
Gas a ( (L2·atm)/mol2) b (L/mol)
He 0.03410 0.0238
Ne 0.205 0.0167
Ar 1.337 0.032
H2 0.2420 0.0265
N2 1.352 0.0387
O2 1.364 0.0319
Cl2 6.260 0.0542
NH3 4.170 0.0371
CH4 2.273 0.0430
CO2 3.610 0.0429
Not bad. We found \(a=1.439\) (instead of \(1.364\)) and \(b=0.00005136\) (instead of \(0.0000371\)).
Can we do better?
What about trying other temperatures. We chose to maintain the vessel at \(300K\) but maybe we should try other temperatures?
Ok, I found another data set, in which we now have the temperatures. Here again, we force the temperature onto the vessel, but the pressure is free to equilibrate. The pressure is how our system responds (the output of the experiment). So we observe the pressure, but impose \((n,T)\).
Let us load the data and recalculate a fit:
dnp = read.csv("../Data/Tnp.csv",header=T)
dtnp = read.csv("../Data/Tnp.csv",header=T)
P <- dtnp$P_Pa
n <- dtnp$n_mol
T <- dtnp$T_K
R <- 8.3143
nf <- nls(P ~ - a*n^2 + (n*T*R)/(1. - b*n) ,start=list(a = 1.,b = 0.0005))
nf## Nonlinear regression model
## model: P ~ -a * n^2 + (n * T * R)/(1 - b * n)
## data: parent.frame()
## a b
## 1.408e+00 3.899e-05
## residual sum-of-squares: 2.66
##
## Number of iterations to convergence: 3
## Achieved convergence tolerance: 3.086e-06The estimated value of \(a\) is now \(1.408\) and the estimated value of \(b\) is \(0.00003899\). Both values have improved compared to published tables.
Conclusion
How good did we do? Our main goal was to estimate the size of atoms and molecules via the equation of state of Van der Waals.
The VdW radius is one of several measures that we can use as a proxy for the size of an atom. It has more to do with the zone of close attraction of atomic nuclei than with the size of the atom proper. Other commonly used sizes are the atomic radius and the covalence radius.
To calculate the radius of a molecule/atom, you first find the volume:
\[v=b/N_A\]
then, assuming your atom is a sphere (which of course is not right, but good enough for an order of magnitude), we can match the radius to the volume:
\(v=\frac{4}{3}\pi r^3\)
Modern physics can see individual atoms and measure their radii. It turns out that the VdW radius of the atom of Nitrogen is 155 picometers while the radius itself is 65 picometers. Hence we got, if not the correct answer, at least a good order of magnitude.
But wait, we should be looking at the molecule of Nitrogen, not a single atom!
Then, the VdW does in fact really well!
Try to see why yourselves by browsing the Wikipedia article on VdW radius.
In this article, the values of the diameter \(d\) and VdW activation volume \(b\) are taken from Weast’s Hamdbook of chemical engineering (1981).
Hunting rat number 2
That is all good but let’s be honest. I downloaded the experimental table above from an engineering web site, but was the data real?
As no indication whatsoever was given on how the data was acquired, I decided to make an inquiry as to how I would carry the experiment myself.
Think about a way to actually do the experiment.
What do we need? We need to have known values of \((n,P)\) or \((n,T,P)\).
How can I put in an initially empty balloon or containerprecise number of moles of molecules of a gas? And in fact of nitrogen?
I have an idea! Would I find a chemical reaction which would generate a precise number of moles of gas, I could use it to fill my bottle/ballon/tank of gas!
Then I remembered Faraday!
This other brilliant physicist is represented in the following beautiful portrait, so typical of the Victorian Era.
More to be added here soon.
Consequences
Derived from VdW equation of state
The value of the VdW equation is that if finally provided a continuous link between gas and condensed states (liquids). It opened both the road for a proper understanding of phase transitions and for zooming in on individual atoms and molecules.
By giving atoms and molecules a spatial extension, atomism was finally becoming a reality. It would belong to Wolfgang Pauli to explain the reason why atoms spread in space, which is due to the principle of exclusion.
Although the determination of \(a\) and \(b\) cannot pretend to absolute rigor due to the phenomenological nature of these quantitities, they remain essential for an understanding of thermodynamics.
They also allow you to calculate loads of interesting quantities such as the work performed in a Joule dilatation, the temperature of a gas as a function of volume and pressure, the compressibility curves, Amagat diagrams and so on.
Physicists continued to fill their bottles and tanks with gas or liquids and pushed as far as they could towards extreme temperatures and extreme pressures.
At the low temperature end, Onnes discovered the supraconductivity and superfluidity of Helium, a phenomenon that later explained by Bose and Einstein.
In terms of high pressure, there is an interesting link with the HMS Challenger expedition, and the search for an equation to describe … water under high pressure.
Peter Guthrie Tait proposed an equation of state relating the density of fluids to the pressure (see the book: From Deep Seas to Laboratory 3, by Frederic Aitken and Jean-Numa Foulc).
Many variants of the PVT state equation have been proposed, and they are all strongly influenced by the VdW (1873) equation.
Oouch, a new relation !
We have seen how to obtain the constant a and b.
And of course, we know, because you and I have read the dissertation of Van der Waals (1873) that a and b are related to the critical pressure and critical temperature.
\[a = \frac{27}{64}\frac{R^2 P_c^2}{T_c} \]
and
\[b=\frac{1}{8}\frac{RT_c}{P_c}\]
So may be we can find a relation between \(a\), \(b\) and the boiling temperature of gases.
Here is another question. If I try many, many different gas, what will the distribution of \((a,b)\) pairs look like in the plane?
To test this, I downloaded a list of gas and their estimated Van der Waals constants \((a,b)\).
d = read.csv("../Data/vdw.csv",header=T)## Warning in if (!header) rlabp <- FALSE: the condition has length > 1 and only
## the first element will be used## Warning in if (header) {: the condition has length > 1 and only the first
## element will be usedhead(d)## name formula a b
## 1 Aluminum trichloride AlCl3 42.630 0.24500
## 2 Ammonia NH3 4.225 0.03713
## 3 Ammonium chloride NH4Cl 2.380 0.00734
## 4 Argon Ar 1.355 0.03201
## 5 Boron trichloride BCl3 15.600 0.12220
## 6 Boron trifluoride BF3 3.980 0.05443plot(d$a,d$b)
Waoo, amazing!
Is this a new law or nature or what?
I read online that \(a\) is not a related to molecular size. Do you think that this is correct?
What could possibly explain this approximate proportionality of \(b\) and \(a\)?
Notice that using the relations above:
\[ b/a = 27/8 R T_c \]
But the critical temperature vary widely among different gases!