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I am learning in my heat transfer class that the thermal conductivity decreases from solids to liquids to gases i.e. $$K_{solid} > K_{liquid}>K_{gas}$$ The reason cited is that (based on my understanding) because the spacing between molecules increases from solids to gases, transfer of thermal energy becomes more inefficient, so the thermal conductivity decreases. This was fairly intuitive.

But we later learnt that, according to the kinetic theory of gases, the thermal conductivity is directly proportional to the mean free path. Mean free path is the average distance travelled by a molecule before experiencing a collision. So, $$\lambda_{gas} > \lambda_{liquid} > \lambda_{solid}$$

So, shouldn't $K_{gas}$ be the greatest? Both relations seem to go against each other. So are there other factors that compensate for this deviation?

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What you're missing is the bonds between constituents, which are very strong in a solid, less strong in a liquid, and weak to nonexistent in a gas. Kinetic theory assumes that there are no interactions besides collisions, and in a totally uncoupled situation like that, the only thing that matters is the likelihood of collisions, which can only hamper the transfer of energy across the gas.

But when you have significant coupling between constituents, then energy can be transferred as an excitation of the coupling, rather than through collisions (in fact, the notion of "collisions" in a solid is itself a little shaky, as constituents are essentially fixed in place in a lattice). If you look at these lattice excitations as solutions of a kind of approximate wave equation, then, based on what we know about the speed of waves in a medium, the stronger the coupling relative to the density of the constituents, the faster these waves will travel.

So, in a situation where energy is primarily carried in collective excitations rather than collisions, the stronger the collective behavior, the faster the energy will travel.

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