As I shared previously (here), in 1857 Rudolf Clausius published [1] his derivation of the kinetic theory of gases in which he connected the world of moving and colliding atoms to the ideal gas law: PV = nRT.

Based on his work, Clausius hypothesized that absolute temperature is proportional to the average kinetic energy of the system molecules. He then calculated the mean speed of molecules at ambient conditions (298K).

• Oxygen = 461 m/s
• Nitrogen = 492 m/s
• Hydrogen = 1844 m/s

Of the many breakthroughs this work enabled, I wanted to share one that was stimulated by a seemingly simple question.

The Question

Upon reading these results, Dutch meteorologist C. H. D. Buys-Ballot [2] (1817-1890) asked a seemingly simple question (to paraphrase), if the speed of the molecules is so high, then why does it take on the order of minutes to smell the release of an odorous gas from across the room? 

What a great question to ask!  Stephen Brush [3] wrote about the importance of such science “gadflies” as Buys-Ballot to the progress of science. It was this group who asked the difficult penetrating questions that motivated others to excellence while unfortunately not being later acknowledged by historians for their critical role.  “Clausius gets the credit for introducing the mean free path, while Buys-Ballot is forgotten.” 

Buys-Ballot penetrating question forced Clausius to take a step back and do a reality check.  If the speed is so fast, why is the diffusion so slow?  Clausius rose to the occasion—“I rejoice at the discussion of this point by M. Buijs-Ballot”—and set about to add a layer of complexity to his physical model that incorporated collisions between molecules, proposing this phenomena as the mechanism impeding diffusion. 

The Answer

The question Clausius pondered was, “How far on average can the molecule move, before its center of gravity comes into the sphere of action of another molecule?” for which the “sphere of action” was the radius for which intermolecular forces are significant around a molecule.  Of significance is the fact that, at that time, Clausius—and really no one—had any real evidence for what a molecule looked like and what kind of forces it contained. So he assumed each molecule to have a specific radius and all molecules except one to be spread evenly throughout a given volume with the one exception moving at a mean velocity through this structure.  Based on his calculated probability of how far a moving sphere would pass through evenly-spaced obstacles before collision, Clausius determined that

and based further on his educated guess that the second ratio is about 1000:1—which is consistent with the approximate increase in volume that occurs when a liquid becomes a vapor—he concluded that a molecule is likely to travel about 1000 times its radius before colliding with another molecule.  This result has since held up.  Looking at the numbers, the radius of either nitrogen or oxygen is around 8 x 10^-9 cm and the mean free path is about 1000 times higher at around 8 x 10^-6 cm.

Interesting consequences of the kinetic theory of gases

Clausius’ push to quantify the atomic world helped open the door for others to consider the “If they exist, then…” startling implications of the theory.  Of this group, I share the work of three. 

• James Clerk-Maxwell determined a fundamental equation for gas viscosity based on Clausius’ mean free path theory.  He then used Stokes viscosity data in this equation to estimate that each gas particle makes about 8 billion collisions per second! [4] 
• Ludwig Boltzmann developed theories to quantify the rate at which a given population distribution moves towards the equilibrium Maxwell distribution.  He showed that because of such high collision frequencies, a distribution in which all gas molecules start with the same velocity would achieve a Maxwell distribution after only a hundred-millionth of a second. [5] 
• Josef Loschmidt (1821-1895) started looking at Clausius’ work in 1865 and those of others and noted two relationships.  Molecular volume depends on Nd³ while Clausius’ mean free path depends on Nd² as it’s the density of circumference-based surface area (and not volume) that hinders atomic motion.  Thus, he reasoned that if he could quantify both terms, he could solve for the combined variables in each.  He estimated Nd³ by considering the volume change of gas liquefaction: the total volume of molecules in a given volume of gas is equal to the total volume of condensate generated from this mixture.  He then estimated Nd² by considering the combined gas viscosity studies of Stokes, Maxwell and O.E. Meyer for which viscosity was mathematically linked to “mean free path” and thus to Nd².  With two equation and two unknowns, Loschmidt was able to estimate d of about 10 x 10^-8 cm.  While he didn’t further focus on this, one could use the same data to then calculate N and, using his data, this turns out to be 2 x 10¹⁸ molecules for 1 cc volume at 0^oC and 1 atm pressure.  This number was later revised to 2.687 x 10¹⁹ and became known as the Loschmidt number and also later evolved into the similar Avogadro’s number of 6.02 x 10²³ molecules per gram-mole based on the standard volume of a perfect gas being 22420.7 cc atm mole^-1. [6]

Such calculations as these and others, being based on the implications of moving and colliding atoms, were rather fascinating to consider and the fact that they started to make the invisible visible started to enhance the credibility of the atomic theory itself.

References

[1] Clausius, R., “The Nature of the Motion Which We Call Heat.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 111–34. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub, 2003.

[2] Clausius, R., “On the Mean Lengths of the Paths Described by the Separate Molecules of Gaseous Bodies.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 135–47. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub, 2003.

[3] Brush, Stephen G., “Gadflies and Geniuses in the History of Gas Theory.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 421–50. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub, 2003, pp. 421-450.

[4] Maxwell, James Clerk, “Illustrations of the Dynamical Theory of Gases.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 148–71. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub. 2003, p. 166.

[5] Boltzmann, Ludwig, “Reply to Zermelo’s Remarks on the Theory of Heat.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 392–402. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub., 2003, p. 402.

[6] Loschmidt, Josef in Stephen G. Brush’s The Kind of Motion We Call Heat:: A History of the Kinetic Theory of Gases in the 19th Century. Book 1: Physics and the Atomists. 3rd print. North-Holland Personal Library, 1986, pp. 75-76 and in Brush [3], pp. 429-430.

END