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If you are making a calibration curve for $\ce{CuSO4}$ where you plot absorbance against concentration, does the wavelength have to be exactly at the maximum of the peak? Or can it be somewhere else along the curve of the peak as long as it is consistent among all the measurements and still get an accurate result?

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No it doesn’t. Measurements can in principle be made at any wavelength.

If there are two colored specimens in the same solution, it is actually better to measure at the wavelength where the difference between the spectra is as high as possible.

Measuring at a peak has some advantages: the absorbance is high, giving a good signal/noise ratio, and if the wavelength varies a little it has less effect on the absorbance when measured at a peak than at a slope of the curve.

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Yes, it does, and especially for calibration curves [1, p. 439, 455]:

For spectrophotometric analysis, we normally choose the wavelength of maximum absorbance for two reasons:

(1) The sensitivity of the analysis is greatest at maximum absorbance; that is, we get the maximum response for a given concentration of analyte.

Sensitivity $b$ is the slope of your linear calibration curve:

$$y = a + bx$$

where $y$ – signal (absorbance); $a$ – background noise or erroneous values (can be often taken 0 for a dual-beam spectrometer with a blank probe); $x$ – concentration. From differential analysis

$$b = \frac{\mathrm{d}y}{\mathrm{d}x}\approx \frac{Δy}{Δx},$$

so that you are interested in getting the largest signal possible (at the maxima) in order to have the best sensitivity with the set of given concentration, which you cannot increase of decrease infinitely due to limited solubility and device limitations – most spectrophotometers won't register $A > 2$ properly. Further [1, p. 439, 455]:

(2) The curve is relatively flat at the maximum, so there is little variation in absorbance if the monochromator drifts a little or if the width of the transmitted band changes slightly. Beer’s law is obeyed when absorbance is constant across the selected waveband.

...

Measurements should be made at a wavelength of maximum absorbance. Instrument errors tend to be minimized if the absorbance falls in the range $A < 0.3 - 2$.

References

  1. Harris, D. C. Quantitative Chemical Analysis, 9th ed.; W. H. Freeman: New York, 2015. ISBN 978-1-4641-3538-5.
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