When data is plotted on logarithmic
scales, the numbers are typically represented with one of two formats.
- The raw numbers themselves are plotted on a
logarithmic axis.
- The raw numbers are first converted to their
log values (exponents of 10) and these are plotted on a
linear axis.
The two scales above cover an identical range of numbers but differ in their utility as
illustrated by two representations
of the titration curve of acetic acid.
By the first format, the raw numbers themselves are plotted
on a log axis with the major
intervals of the X-axis (the darker vertical lines in this plot)
corresponding to a power of 10, e.g., 10^{-7}, 10^{-6}, etc.,
as read from
left to right!
By the second format, the raw numbers are first converted to their
logarithmic values and these are plotted on a linear
axis
as illustrated below.
Both graphs represent exactly the same data.
In other words, they are there quantitatively equivalent. However, the
different formats provide easy access to different types of information about the
titration. For example:
- By using the top plot, [H^{+}]
can be read
directly from the X-axis; for example, at 45% saturation, [H^{+}]
= 1.0 x 10^{-5}.
- By using the bottom plot, [H^{+}]
must be calculated by taking the antilog of the negative
value of the log value read from the X-axis; for example, at 45% saturation, [H^{+}] =
antilog (-5.0) = 10^{-5}.
- In contrast, the bottom plot is easier
use for determining the pK_{d} of acetic acid, defined as the negative
log of the dissociation constant, or -log ([H^{+}]_{50%})
with [H^{+}] at Ya = 50% saturation. As read directly
(right to left!) from the X-axis of the bottom plot, pKdn = 4.90
where Ya =
50% saturation.
- With the top plot, the value read for [H^{+}]
at 50% saturation from the X-axis -- i.e., 1.25 x 10^{-5} --
must be convert to the negative of its log
value;
i.e., - log (1.25 x 10^{-5}) = -
log (1.25) - log (10^{-5})
= - log (1.25) + 5
= - log (5/4) + 5 =
- log (5) + log
(4) +
5
= - 0.7 + 2 x 0.3 +
5 = - 0.7 + 0.6
+ 5 = - 0.1 +
5 = 4.9 = pKdn