Riddle me this: why does a gas deviate from ideal behavior?

Years ago, during on-campus interview season at college, a friend of mine majoring in electrical engineering told of how difficult one of his interviews was.  “The interviewer asked me how an oscilloscope worked, and I carefully explained how to plug in the different wires and then how to adjust the knobs and so on.  He then said, ‘No, that’s not what I meant.  I meant, how does the internal circuitry of the oscilloscope work?’”  Yikes!

I wrote Block by Block – The Historical and Theoretical Foundations of Thermodynamics to explain how the “internal circuitry” of thermodynamics works, seeking to link the microscopic world of atoms to the macroscopic world of thermodynamic phenomena.  While I explained much of what I wanted to, I was left with some unanswered questions, and these motivated me to create this website and continue my journey.

This post is my first in an anticipated “Riddle me this” series to start sharing my questions with you along with a crowd-sourcing invitation to help me answer them.  If you know the answers to any of my questions, please respond, either via my social media accounts (LinkedIn, Facebook, Twitter) or in the “Leave a Reply” section at the bottom of this post.  As I’m seeking documented evidence supporting the answers, the more detailed and referenced your response, the better.  Thank you in advance.  Now let’s get started.

Riddle me this: why exactly does a gas deviate from ideal behavior?

I’m very interested in understanding exactly why a gas deviates from ideal behavior.  Consider the ideal gas law:

                PV = nRT

where P = pressure, V = volume, n = number of moles, R = gas constant, and T = absolute temperature

Rudolf Clausius showed how this equation could be derived from first principles based on the kinetic theory of gases.  It is exact and contains no adjustable parameters.  Deviation is typically observed by comparing the actual measured pressure against that predicted by the equation, re-arranged as this:

                P (ideal) = nRT / V

In my own research on the cause of the deviation, the reason most often cited is the presence of attractive forces.  But this is like my friend’s initial answer about how an oscilloscope works.  It doesn’t really explain anything. 

Some of my contacts suggested to me that the answer lies in Van der Waals’ (VDW) equation.

                  P (VDW) = [nRT/(V – nb)] – a(n²/ V²)

                  where

a is a constant related to the attractive forces, and

b is a constant related to the volume of the atoms or molecules

Now I can study this equation, plug in numbers, and see indeed that for real gases the equation provides much greater accuracy than the ideal gas law.  I can also see how changes in both intermolecular attractive force (a) and atomic volume (b) from one gas system to another based on separate physical property data result in accurate predictions of corresponding changes in pressure.  I can sort of logically see how this equation works.  And yet, what I can’t see is an explanation of exactly what’s happening at the molecular level to cause the pressure to deviate from that predicted by the ideal gas law.  Why exactly does intermolecular attraction result in a change in pressure, all other things being equal?  The VDW equation doesn’t explain this.

As an aside, it’s interesting to read about the origin of the VDW equation.  As suggested by Kipnis et al. in their book Van der Waals and Molecular Science (Chapter 3), in the late 1800s Dutch scientist Johannes Diderik van der Waals first arrived at his equation, based on “elementary reasoning and some experimental information,” and then afterward rationalized its physical meaning.  In other words, the equation lacks a fundamental underpinning. While the empirical nature of the equation was recognized by scientists of the day, such as James Clerk Maxwell and Ludwig Boltzmann, who commented that Van der Waals found his equation “to some extent by inspiration,” this did not detract from their applause of Van der Waals’ work, for they made immediate use of it in their own research efforts.  Indeed, in 1905 Boltzmann and Josef Nabl described Van der Waals as “the Newton of the theory of the deviations of gases from ideality.”  In the end, the VDW equation played a critical role in the early days of thermodynamics.  And yet, its derivation did not tell us why the deviations occurred.

So where does this leave us?  This is the question.  I’ve not been able to find many who have sought links between non-ideal gas behavior and the physical phenomena as its cause.  I’ve read one explanation that said the deviation occurs because gas molecules near the wall boundary experience greater “pull back” attractive forces from the system and so don’t strike the wall as hard, but this never made sense to me because this would also cause lower temperature at the wall.  The deviation from the ideal gas law I’m referring to occurs when temperature, volume, and number of moles are held constant.

Does the formation of dimers result in the deviation?

Based on a suggestion from a colleague of mine, Tom Kinney, I found an explanation that made sense to me in David Oxtoby et al’s Principles of Modern Chemistry (2012, 7^th edition, p. 420).  Pressure can decrease from that predicted by the ideal gas law due to the temporary attraction of molecules to form dimers, “so reducing the rate of collisions with the walls.”  In their review (1976) of this topic, Blaney and Ewing defined these gas phase dimers, which became known as Van der Waals molecules, as “weakly bound complexes of small atoms or molecules held together, not by chemical bonds, but by intermolecular attractions.”  These dimers, whose existence has been validated by spectroscopy, form whenever two atoms or molecules have sufficiently low kinetic energies and sufficiently close proximity to capture each other in the very early stages of condensation.  Other researchers proposed the same cause-effect phenomena as Oxtoby et al., stating that low temperatures cause monomers to form dimers, which effectively lowers the pressure of the system.

While the mathematical models involved in these studies, which primarily focus on the virial equation of state and specifically on the second virial coefficient, support the hypothesis that the occurrence of dimers influences the observed deviation from the ideal gas law, such support falls short of definitely demonstrating the cause-effect link.

With this background, I conclude this post with a comment and then some questions.  First, the comment.  It’s interesting to me that I never read about this dimer hypothesis in my thermodynamics textbooks even though I spent many hours working through problems involving equations of state, such as the VDW equation and its successor, the Redlich-Kwong equation.  I wish that a textbook had been available to explain such micro-to-macro connections.  Perhaps it’s time to write it.

Regarding my questions, these are meant for you, the reader.  Simply put, I don’t know what causes the deviation.  But while I don’t know, I’ve still thought about it.  It seems to me that if the dimer hypothesis is true, then modeling efforts should focus on the “n” variable in the ideal gas law, which quantifies the number of moles in the system, since the formation of dimers directly reduces n and so reduces the predicted pressure.  Have you read of anyone considering this or otherwise studying this?  Conversely, have you read of anyone who has tried to refute the hypothesis by quantifying deviation in the absence of dimers?  I have not found such published research.  Finally, I’m working with one hypothesis here.  If you know of another, please let me know.  I welcome your thoughts on this riddle.