Lower temperature gives more collisions?

[Erland]

In "Our Mathematical Universe" by Max Tegmark, in the footnote at p. 208, Tegmark writes "..the well known fact that things are more likely to bump into each other when you lower the temperature, just as slow neutrons are more likely than fast ones to strike targets in a nuclear reactor"

Is this really true? It seems completely wrong to me. Suppose that we lower the speeds of all particles in a container by 1/2. The effect will be that the particles move in slow motion compared to before. Hence, all the collisions between the particles which occurred in a time interval of length t before, now occur in a time interval of length 2t, so that fewer collisions occur in a time interval of length t than before, not more collisions.

What is the truth?

 

[voko]

That statement is definitely true for the fission reactions employed in conventional nuclear reactors. Part of that has to do with the "size" of the neutron, which becomes smaller as its velocity increases (that's a quantum mechanical effect). Whether it is universally true is much less clear. Even the verbiage "more likely" is ambiguous, because one can deal with different notions of likelihood. Perhaps it is true in the context of that passage, but you have not supplied the context.

 

[Erland]

The context is that Penrose claims that the brain is a quantum computer. Tegmark says no, decoherence will destroy the quantum effects in the brain. Tegmark is criticized by Penrose supporters. This footnote is his defence in the book.

The more I read this footnote, the more confused I become. Does the number of collisions increase or decrease with lower temoerature?

Photocopy of footnote attached.

 

[Khashishi]

Ug, that's a hair-brained passage. It's true that sometimes collision frequency goes up at lower temperature (as in the case of high temperature plasmas), but this sounds like a gross generalization. Normal uncharged gas molecules collide more as you raise the temperature.

Collision frequency goes down with temperature for some types of collisions with long range interactions, such as charged particles, since the faster two charged particles move by each other, the less time there is for their trajectories to be bent by each other. For short ranged collisions, like between bowling balls, frequency goes up with temperature.

 

[klimatos]

Normal uncharged gas molecules collide more as you raise the temperature.

Not necessarily. In the free atmosphere, at constant pressure, molecular collision frequencies increase with a drop in temperature. The increased molecular number density more than makes up for the slower molecular velocities.

 

[cjl]

Not necessarily. In the free atmosphere, at constant pressure, molecular collision frequencies increase with a drop in temperature. The increased molecular number density more than makes up for the slower molecular velocities.

Which makes sense if you think about it - if you hold pressure constant, the momentum flux against a surface must be constant from collisions with gas particles. If you drop the average speed of the particles, the average momentum transferred by a single collision also drops, so for the overall momentum flux to remain constant, the collision rate must increase.

 

[Khashishi]

Ah, but at constant pressure is a very artificial stipulation, since pressure increases with temperature.

 

[klimatos]

Ah, but at constant pressure is a very artificial stipulation, since pressure increases with temperature.

That depends upon what you mean by "constant" pressure. It is a very common thing for air temperatures to significantly warm as the day progresses without any corresponding significant increase in air pressures. Instead, molecular number density (n) often drops in proportion to the temperature increase and the equation of state (P=nkT) is very roughly maintained.

For global weather systems, high temperatures are most commonly associated with low pressure systems (tropical lows, for instance) and low temperatures most commonly associated with high pressure systems (polar highs, for instance). When it comes to atmospheric pressures, air densities are often more important than air temperatures.

I repeat that one should be very careful when applying equilibrium equations obtained in the laboratory to the non-equilibrium free atmosphere.

 

[Erland]

Many of the molecules in the brain Tegmark talks about are actually charged: sodium and potassium ions.