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Helps Chem and Hints for Homework Assignment #9
Making a baseline correction is easy to picture but harder to program. The baseline can
be established by picking a couple of regions on either side of a peak (or group of peaks) and
performing a least squares fit through them to get an equation describing the baseline. To do
this, however, you must copy the data points for the two regions to a section of a spreadsheet
where they can be listed sequentially as a single array. If you use the slope and intercept
functions instead of linest, you won=t need to worry about blank spaces, but you will still need to
input both regions as a single array. Then it is a simple matter in a spread sheet to subtract a
calculated baseline value from each data point across the peak (ycorr,i = yi ! [mbase xi + bbase]).
What about picking the region to integrate over? Take as an example the two
overlapping peaks in the figure below. Even though these two peaks strongly overlap, we can
see that picking the mid point between the two peaks (the bottom of the trough) as one limit of
integration would not lead to large errors in determination of peak area because the extra area
from one of the peaks would compensate for the missing area from the other peak of interest. In
fact, in this example of two
symmetrical, equal intensity
0.050659
peaks, the areas exactly
compensate each (If
f( other. )
you are interested, you can
g( )
play around with creating
h( )
overlapping peaks of
different intensity and see
. 13
6.83462610
what the error would be in
3
3
1.1.10
1 .10
the peak area determination
by this approach.)
The other limit of integration (for example, the left point for the left peak in the figure)
can be picked according to your best judgment of where the peak seems to come down close to
the baseline. Even though, mathematically, the peak continues on to infinity, the area under the
curve rapidly converges; that is, extending the integration to larger regions increases the area by
negligible amounts (and, indeed, the additional area becomes dominated by noise).
The trapezoidal rule for integration involves treating each pair of points as the corners of
a trapezoid, the area of which is x (yi + yi+1), where x = xi+1 xi. Simpson=s method
involves approximating every three consecutive points on a curve as defining a parabola. The
area under a segment of a parabola can be shown to be x (yi + 4 yi+1 + yi+2), where yi and yi+2
are the end points, yi+1 is the midpoint, and again x = xi+1 xi. [Note that this formula assumes
equally spaced points; an alternative formula is (1/6)(xi+2 xi)(yi + 4 yi+1 + yi+2).] A final
caution: when you set up your Simpson routines, make sure that each segment doesnt get
integrated twice.