Chapter 19 derived how diffusion, viscosity, and thermal conductivity scale with temperature, mass, and molecular size. Here we compare those scalings with measured values for three gases at the same temperature and pressure.

Table 1. Transport coefficients of helium, argon, and nitrogen at 298.15 K and 1 atm

	Helium	Argon	Nitrogen
Molar mass M (g/mol)	4.003	39.95	28.01
Moles per m³ at 1 atm	40.9	40.9	40.9
Molar heat capacity Cv (J/(mol·K))	12.47	12.48	20.82
Mean speed ⟨v⟩ (m/s)	1256	398	475
Effective diameter d (Å), from viscosity	2.16	3.59	3.70
Diffusion coefficient D (self-diffusion, cm²/s)	1.759	0.182	0.209
Viscosity η (μPa·s)	19.8	22.6	17.8
Thermal conductivity k (W/(m·K))	0.155	0.0177	0.0258

Each gas holds the same 40.9 mol/m³ at this temperature and pressure, so the comparison is between equal numbers of molecules per volume. W/(m·K) is watts per meter per kelvin. [A]

Does diameter contribute?

Diameter enters every coefficient in the table as 1/d². A larger molecule is a bigger target, its free flights between collisions are shorter, and D, η, and k all fall. Diameter works alongside mass, and the table shows how the two compete. Nitrogen is seven times heavier than helium, which alone would make it more than two and a half times as viscous. But nitrogen’s effective diameter is 1.7 times larger, which cuts its collision cross-section by a factor of almost three, and the two effects nearly cancel, leaving nitrogen about 10 percent less viscous than helium.

The diameters in the table are not independent, because they were inferred from the viscosities. The effective diameter is not a measured size: it is the diameter of the rigid sphere that would reproduce the gas’s viscosity, and it absorbs the attraction, the softness of the repulsion, and everything else the rigid-sphere model leaves out. The real test, then, is whether these diameters predict the other coefficients. Using only mass and these diameters, the diffusion coefficients of argon and nitrogen relative to helium come out within about 10 percent of the data, and the monatomic conductivity ratio of argon to helium comes out exactly. For nitrogen, the predicted conductivity is about 30 percent too high. The cause is not diameter, since nitrogen also stores energy in rotation, which the elementary picture treats only roughly. The effective diameter is also not a fixed size: molecules that collide faster push closer together and behave as if they were smaller, so the diameter depends on temperature. [B]

References

[A] Self-diffusion coefficients are the values recommended by D. R. Burgess Jr., Self-Diffusion and Binary-Diffusion Coefficients in Gases, NIST Technical Note 2279 (2024) (https://doi.org/10.6028/NIST.TN.2279). Viscosity, thermal conductivity, and heat capacity were computed with CoolProp 8.0.0 at 298.15 K and 101.325 kPa, using the correlations of V. D. Arp (helium viscosity), B. A. Hands and V. D. Arp, Cryogenics 21, 697–703 (1981) (helium conductivity), and E. W. Lemmon and R. T. Jacobsen, International Journal of Thermophysics 25, 21–69 (2004) (argon and nitrogen).

[B] The viscosity of rigid elastic spheres, η = (5/16)√(πmk_BT)/(πd²), solved for d using the computed viscosity at 298.15 K. The self-diffusion data give 1.96, 3.43, and 3.50 Å through D = (3/(8(N/V)d²))√(k_BT/πm). The comparisons of predicted and measured ratios in the text are my own calculation.

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