(a) Assuming moderately dilute helium gas so that binary...

(a) Assuming moderately dilute helium gas so that binary collisions of helium atoms determine the transport coefficients, derive an expression for the thermal conductivity of the gas. (b) Estimate the ratio of the thermal conductivity of gaseous ${ }^3 \mathrm{He}$ to that of gaseous ${ }^4 \mathrm{He}$ at room temperature. (c) Will this ratio become different at a temperature near 2 K ? Why?

(a) Assuming moderately dilute helium gas so that binary collisions of helium atoms determine the transport coefficients, derive an expression for the thermal conductivity of the gas.
(b) Estimate the ratio of the thermal conductivity of gaseous ${ }^3 \mathrm{He}$ to that of gaseous ${ }^4 \mathrm{He}$ at room temperature.
(c) Will this ratio become different at a temperature near 2 K ? Why?

Key Concepts

Isotope Mass Dependence

The mass of gas molecules plays an important role in determining their average thermal speed, since v ~ ?(kT/m), where m is the molecular mass. This is particularly relevant when comparing isotopes such as ³He and ?He, as the difference in mass leads to different average speeds and, consequently, affects the thermal conductivity. The ratio of the thermal conductivities for different isotopes at the same temperature is influenced by this mass difference.

Kinetic Theory of Gases

This concept explains how molecules in a gas carry energy and momentum, and how their random motions lead to macroscopic transport properties such as thermal conductivity. In a dilute gas, the thermal conductivity can be derived by considering the energy transported by molecules moving with an average thermal speed and colliding over a characteristic mean free path. The derivation typically uses the relation ? ~ (1/3) C_v * v * ?, where C_v is the specific heat per unit volume, v is the average molecular speed, and ? is the mean free path.

Mean Free Path and Binary Collisions

In moderately dilute gases, transport properties are primarily determined by binary collisions between molecules. The mean free path is the average distance a molecule travels between collisions and is related inversely to the number density and the collision cross-section. The concept of binary collisions is crucial in deriving transport coefficients because it quantifies how frequently molecules interact and thus how effectively energy or momentum is transported through the gas.

Quantum Statistical Effects at Low Temperatures

At temperatures near 2 K, quantum statistics become significant, as classical assumptions break down. For helium isotopes, ³He behaves as a Fermi liquid while ?He exhibits Bose-Einstein condensation effects. These quantum effects modify the collision dynamics and transport properties, leading to deviations from classical predictions for thermal conductivity. Thus, the ratio of thermal conductivities between the isotopes can change at very low temperatures due to the influence of quantum statistical behavior.

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