Reading and Interpreting Numbers
on Logarithmic Axes |

When data is plotted on logarithmic
scales, the numbers are typically represented with one of two formats.

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:

- 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,

By the second format, the raw numbers are first

- 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;., - log (1.25 x 10

i.e^{-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**

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© |
Duane W.
Sears Revised: September 30, 2010 |