By Christopher J. Foot
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4 Spread in meters of a localized pulse of thermal energy due (only) to molecular thermal conductivity in air at STP after 2 h and after 12 h. √ The functional form is the normal distribution with standard deviation σ = 2Dt, where D = κH /ρcp ≈ 2 × 10−5 m 2 s −1 and t is time in seconds. 024 J m−1 s−1 K−1 ). 4. The standard deviation of the spread of the elevated temperatures is only about 3 m after 12 hours. This means that the concept of parcel integrity for objects of the order of several hundred meters is safe for days if the only stirring mechanism is molecular diffusion.
3. 2 Distribution of velocities 29 and v can be taken to be roughly the vrms of the previous example. Hence, there are about 1027 collisions per second with the 1 m2 wall at STP. So where did the divisor 6 come from? It comes from a careful integration over all the angles, etc. A simple minded (but correct) way of obtaining it is to note that 13 of the molecules are going in the x direction, but only half of these are going in the +x direction. With or without the 6, this is a very large collision rate.
4). Imagine a molecule going left and right across the box and bouncing back at the walls. A round trip takes a time 2L/v0 where v0 is the speed of the molecule. At a reﬂection it experiences a change of momentum (m0 v0 ) = 2m0 v0 . Each reﬂection imposes an impulsive force to the wall (see the physics refresher below). The frequency of such reﬂections (by the entire box of molecules) is so large that the force is effectively steady (we shall see later that typical molecular speeds are hundreds of m s−1 ).