How to chose wavelength in spectrophotometry? How to use Beer-Lambert Law?

How is the wavelength chosen to measure the concentration of an ion determined? and What does the wavelength correspond to in an atom/ion?

When light is shined on a solution what happens to the ions in solution? Include how and why light is absorbed or emitted from a substance.

Would a high or low concentration absorb more light?

How does the beer-lambert law relate to concentration and information from the spectrophotometer?

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Working wavelength is chosen by analysing the spectrogram $A(\lambda)$. Such diagram is obtained by measuring absorbance $A$ in function of wavelength $\lambda$. Practically, you test a moderate concentration of your analyte and sweep the complete wavelength bandwidth of your analyser. Then you select wavelength where a maximum of absorbance occurs. If multiple maximum are present, you may prefer the highest one (best sensitivity) or a smaller one with less interferences (best selectivity). Both can happens simultaneously. The main criterion is:

$$\underset{\lambda}{\max}\left\{A(\lambda)\right\}$$

For information: the wavelength is linked to photon energy $E$ by the following relation:

$$E = h \nu = h \frac{c}{\lambda}$$

Where $\nu$ is the frequency of the EM wave, $c$ the speed of light and $h$ the Plank constant.

Absorbance is a property of matter, when an atom/ion/molecule absorbs a photon (visible or nearly visible IR/UV), it can promote electron to a highest energy level (UV, highest energy) or increase vibrational energy (IR, lowest energy) of the compound and the photon is destroyed.

A spectrophotometer assess the transmittance $T$ (ratio of transmitted $\phi$ and incident flux $\phi_0$ expressed as powers) of light beam through a sample cell.

$$T = \frac{\phi}{\phi_0}$$

And transmittance (dimensionless) is linked to absorbance (dimensionless) in the following way (functional scaling to achieve linearisation between signal and physical quantity):

$$A = -\log_{10}(T)$$

Beer-Lambert Law is a limit law conditioned by several hypothesis and it is only valid for low concentrations. When Beer-Lambert applies you can linearly link concentration $x$ with absorbance.

$$A(x) = \epsilon(\lambda)lx$$

Where $\epsilon$ is a proportional coefficient (with suitable units) called extinction coefficient which describes how matter absorbs light at wavelength $\lambda$. The quantity $l$ stands for the optical path length through the sample cell.

Numerous substances validate Beer-Lambert Law but some does not. Therefore you must assess linearity before quantifying unknown concentrations. This is usually done when performing calibration curve. Then you know your dynamic range and sensitivity of your analytical method and you can regress concentration from absorbance.

I suggest you to read more about those concepts in a reference book such as Skoog & West, Fundamentals of Analytical Chemistry or Vogels, Textbook of Quantitative Chemical Analysis. There is also a free electronic book available on internet called Analytical Chemistry 2.0.