Determine the molar absorptivity of a substance in aqueous solution. A solution of \(p\)-nitrophenol containing \(8.8 \mu \mathrm{g} \mathrm{mL}^{-1}\) gave an absorbance of \(0.535\) at \(404 \mathrm{~nm}\) in a cuvette of path length \(1 \mathrm{~cm}\). What is the molar absorptivity of a \(p\)-nitrophenol at \(404 \mathrm{~nm}\), expressed to 3 significant figures? (Note: \(M_{r}\) of \(p\)-nitrophenol is \(291.27\).)

The Beer-Lambert Law, also known as Beer's Law, is a fundamental principle in spectrophotometry that relates the absorption of light by a substance to its chemical concentration. Beer's Law is mathematically expressed as \( A = \epsilon \cdot c \cdot l \), where \( A \) is the absorbance, \( \epsilon \) is the molar absorptivity, \( c \) represents the molar concentration of the solution, and \( l \) is the path length through which the light travels.

Essentially, Beer's Law posits that there's a direct proportionality between absorbance and the product of molar concentration and path length. The greater the concentration of the absorbing species, the more light gets absorbed. Conversely, a longer path length also increases the absorbance, assuming concentration remains constant. This law is utilized extensively in quantitative analysis to determine concentrations of solutes in a variety of chemical and biological applications.

Spectrophotometry is an analytical technique used to measure how much a chemical substance absorbs light by measuring the intensity of light as a beam of light passes through sample solution. The principle behind spectrophotometry is based on the fact that each compound absorbs or transmits light over a certain range of wavelength.

This technique can be used to measure the concentration of a known solute in a solution, by using the relationship derived from the Beer-Lambert Law. In practice, a spectrophotometer is used to measure the amount of light that passes through the solution, and the instrument then translates this into an absorbance reading. This reading, with the proper calibrations and considerations for the path length and molar absorptivity, can be used to calculate the concentration of the solute.

Chemical concentration refers to the amount of a substance present in a given volume of solution. It can be expressed in various ways including molarity, molality, mass percent, and so on.

To determine the concentration in a quantitative manner, analytical chemists often rely on the Beer-Lambert Law. Knowing the absorbance (which is derived from spectrophotometry) and the molar absorptivity of a substance, we can calculate the molar concentration. As we saw in the exercise, the use of this law makes it possible to ascertain the concentration of a substance in a solution by translating optical absorption data into a quantitative figure. Calculating chemical concentration is crucial in fields ranging from pharmaceuticals to environmental science, as it allows for the precise chemical analysis of solutions.

Molar concentration, typically measured in moles per liter (mol/L), is a measure of the concentration of a solute in a solution. To convert a given mass concentration to molar concentration, one needs the molar mass of the solute. The molar mass is the mass in grams of one mole of the substance.

In the provided exercise, the mass concentration given in \( \mu g/mL \) is converted to molar concentration using the molar mass of the substance, which is \( p \)-nitrophenol in this case. By dividing mass per unit volume by the molar mass, we obtain the molarity of the solution. This step is foundational for inputting into Beer's Law, which then allows us to solve for molar absorptivity or any other variable in the equation, assuming the others are known.