# Tyndall showed us that dust is repelled by heat and attracted by cold

In 1870 John Tyndall was a prominent Irish scientist; he is now perhaps less known, possibly because he studied unglamorous things like dust. In a presentation that year he noted that a heated wire repelled dust particles. Note that this is separate from the convection of the air itself that the heated wire also causes. In addition to the convection the particles move relative to the air. Shortly afterwards his contemporary James Clerk Maxwell gave the following explanation, which is probably mostly right*.

Consider the surface of a dust (aerosol) particle suspended in air – which although small is much much larger than air molecules. Air molecules move at a speed that depends on temperature: vTH ~ (kT/m)1/2 , where T is the temperature (in Kelvin), m is the mass of the air molecule, and k is Boltzmann’s constant.

If the surface of the particle is in air in which the temperature is uniform then at any point on the surface particles have the same speed if they arrive on the surface from the left moving right, as they do if they come from the right and collide while moving to the left. But if there is, say a heated wire to the left, then the molecules coming from the left and moving right are hotter, and so faster, than those arriving from the right and moving left. In other words, when there is a temperature gradient, along say the x axis, dT/dx, then there is an imbalance. This pushes the particle along.

Particles come from about the gas’s mean free path away λ, so the net velocity is roughly

net velocity ~ − λ(dvTH/dx)

i.e., the net velocity of molecules colliding with the particle’s surface is approximately the gradient in the molecular speed times the distance they come from (the gas’s mean free path). The minus sign means that particles move down temperature gradients, i.e., away from heat and towards cold, just because the particles coming from the hot side are moving faster and so push harder than the ones coming from the cold side.

We now just write the gradient in thermal speed in terms of the gradient in temperature

net velocity ~ − λ(dT/dx)(k/mT)1/2= − (λvTH /T)(dT/dx)

and conveniently the product of the mean free path and thermal velocity is an easily measurable property of the gas, it is its kinematic viscosity ν ~ λvTH, so then

net velocity ~ − (ν /T)(dT/dx)

The kinematic viscosity of air ν ~ 10-5 m2/s, so near a hot wire where large temperature gradients of (dT/dx)/T ~ 100 / m, are possible, speeds of up to around 1 mm/s should be achievable. This explains Tyndall’s observation in the nineteenth century that there was almost no dust very near the hot wire, this motion down a temperature gradient had moved the dust particles away from the wire.

* The phenomenon of particles moving due to temperature gradients is now called called thermophoresis and the details are pretty complex and subtle.