HandsOn 25 - Location of the Precipitation Point

The following derivation uses a full complement of mathematics. Its purpose is to clarify the usual derivations of the location of the precipitation point in a diffusion tube (found in many chemistry books, for example) are inexact in their procedure, though correct in the final outcome.

**Normal distribution.** The *normal* or *Gaussian*
distribution describes the results of a random walk. This is the
distribution observed in Chapter 2. The Gaussian distribution
describes
the probability *P*(*x*) that a walker who starts at *x*
= 0 will be at
position *x* after taking *N* steps randomly back and forth
along a
line, each step of average length
*L*_{step}

In our model of the diffusion chamber, molecules diffuse from the two
ends. We can choose *x* = 0 for the origin of one molecule, and
*x* =
for the origin of the other molecule, where is the length of the
diffusion chamber. Then the probability *P*_{1}(*x*)
of finding a molecule
of the first type at position *x* is

There is precipitation of solid (dust) when two molecules of different
kinds meet one another and react. The first precipitation occurs when
the *first* molecules to meet each other, namely in the extreme
tails of the two distributions.
Hence, the probability of ``meeting the tails'' of the two distributions
would be the product
*P*_{1}(*x*) *x* *P*_{2}(*x*),
which would be also the
probability
*P*_{pre}(*x*) of observing
precipitation
at position *x*

- [*x*^{2}/(2*L*_{step}^{2}*N*_{1}) + ( -
*x*)^{2}/(2*L*_{step}^{2}*N*_{2})]

is least negative. If you are familiar with calculus, find this by
taking the derivative of this precipitation probability
Here we note that the use of distributions (6.13)
and (6.14)
is correct only for the
*first* precipitation. When
the precipitation has depleted the molecules in the tails of the
distributions, then the distributions are no longer Gaussian. In fact,
it turns out that when a steady state has been reached (when
precipitation removes molecules at the same rate as new molecules leave
the two ends of the diffusion tube), the distributions become straight
lines descending from each end to zero at the point *x* given by
Eq. 6.19.

Previous: HandsOn 24 - Diffusion Chamber