Chemistry Reference
In-Depth Information
2.3.1
Number Distribution,
M
n
The distribution we have just assumed to define the arithmetic mean is a number
distribution, since the record consists of numbers of molecules of specified sizes.
The sum of these numbers comprises the integral (cumulative) number distribu-
tion.
Figure 2.1
represents one such distribution. The scale along the abscissa is
the molecular weight while that on the ordinate could be the total number of
molecules with molecular weights less than or equal to the corresponding value
on the abscissa. However, it is easier to compare different distributions if the
cumulative figures along the ordinate are expressed as fractions of the total num-
ber of molecules in each sample, and
Fig. 2.1
is drawn in this way. The units of
the ordinate are therefore mole fractions and extend from 0 to 1; the integral
distribution is now said to be normalized.
In mathematical terms, the cumulative number (or mole) fraction X(M)is
defined as
X
M
X
ð
M
Þ 5
x
i
(2-5)
i
where x
i
is the fraction of molecules with molecular weight M
i
. The differential
number function is simply the mole fraction x
i
, and a plot of these values against
corresponding M
i
's yields a differential number distribution curve, as in
Fig. 2.2
.
If the distribution is normalized, the area under the x
i
M
i
curve in
Fig. 2.2
will
be unity. (See Section 2.4.2 for units.)
To compile the number distribution we have expressed the proportion
of species with molecular weight M
i
as the corresponding mole fraction x
i
.
1.0
0.8
0.6
0.4
0.2
0
Molecular weight,
M
i
FIGURE 2.1
A normalized integral distribution curve.
