Re: unit in size distribution

On Sep 13, 10:17 am, Bob <bbx107....@xxxxxxxxxxxxxx> wrote:
On Tue, 11 Sep 2007 22:08:01 -0700, Mike87 <Sulfate...@xxxxxxxxx>
wrote:





On Sep 12, 12:46 pm, Bob <bbx107....@xxxxxxxxxxxxxx> wrote:
On Tue, 11 Sep 2007 20:55:40 -0700, Mike87 <Sulfate...@xxxxxxxxx>
wrote:

Hi

Usually we express aerosol size distribution in ln(natural log) scale.
Since total no. concentration is in unit of cm-3.
So size distribution in log-scale will be in unit of micron-1.cm-3.

It was told that "we cannot take the logarithm of a dimensional
quantity".

correct

Therefore, ln(radius) is unitless.

Well...

It is true mathematically that a log does not have units (and must be
of a unitless number).

Why?

log (a*b) = log a + log b

log (3 meters) = log 3 + log (meters)

Now what?

100 cm = 1 m

Take log of both sides...

log (100 cm) = (1 m)

That is fine, but you can't go further without dealing with those
units. Obviously, just dropping them won't work. log (100) <> log (1).

For simply graphing, there is no particular problem. If you want, you
can think of it as graphing data that have been normalized, by
dividing by 1 meter (or 1 "whatever your unit is").

No valid meaningful equation will take the log (or exponent) of a
number with units. If it appears to, there is a hidden assumption, and
you really need to find out what it is.

For example... someone proposes that if two distances are equal, then
the logs of the distances are (numerically) equal. The little case
above with 1 m = 100 cm disproves that -- unless (hidden assumption)
both measurements have the same units.

and another question...

also in logscale

N/( sqrt(2*pi)*lnSigma ) * exp( -(lnr-lnr_mean)^2 / ( 2*ln^2 Sigma ) )

Let me rewrite this eq.:

dN N
( -(lnr-
lnr_mean)^2 )
---- = -------- * exp( -
-------------------------------------------------------------------)
dlnr sqrt(2*pi)*lnSigma
( 2*ln Sigma *ln Sigma )


I use three ( to represent big bracket.

N: no. cm-3
r, r_mean: micron
Sigma: standard deviation of r , also in micron

So all ln(parameters) are all unitless. right?
Then the unit of dN/dlnr is the same as that of N . right?

thank you for your answer

Mik

.

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