Question

The table gives molar absorptivity values for caffeine (FM 194.19u) and benzoic acid (FM 122.12u) at several wavelengths.

ε (M⁻¹cm⁻¹)
230 nm 240 mn 260 nm 274 nm
Caffeine 5 061 3 106 6 351 9 741
Benzoic acid 11 052 6 961 711.3 1 043

A 2.000-mL aliquot of an aqueous sample containing both caffeine and benzoic acid was diluted to 50.00 mL. An absorbance spectrum was obtained in a 1.000-cm cuvette, yielding absorbance values of 0.798 4, 0.495 1, 0.566 9, and 0.866 2 at 230, 240, 260, and 274 nm, respectively. Find the concentrations of both substances in the original sample in units of mg/L.

Answer 1

To find the concentrations of caffeine and benzoic acid in an original sample, apply Beer's Law with the given absorptivity values and absorbance readings, considering the dilution factor and the formula mass of each compound to express concentrations in mg/L.

Step-by-step explanation:

Calculating Concentrations of Caffeine and Benzoic Acid

To calculate the concentrations of caffeine and benzoic acid in an aqueous sample after a dilution, we make use of Beer's Law, which is given by A = ε x c x l, where A is absorbance, ε is the molar absorptivity, c is the concentration, and l is the path length. Here, we use the given absorbance values at different wavelengths and the molar absorptivity values for both compounds to determine their concentrations.

First, we find the diluted concentrations of caffeine and benzoic acid by creating a system of linear equations corresponding to the given absorbance at each wavelength. Then, we use matrix operations or other algebraic methods to solve the simultaneous equations. After finding the diluted concentrations, we calculate the concentrations in the original 2.000 mL aliquot by taking into account the dilution factor, which is the result of diluting to 50.00 mL. Finally, we convert these concentrations to mg/L using the formula mass (FM) of each compound.

For example, at 274 nm, only caffeine absorbs significantly, so we can use this wavelength to directly calculate the caffeine concentration. We divide the observed absorbance for caffeine (0.8662) by the product of molar absorptivity (ε8073;= 9,741 M⁻¹cm⁻¹) and path length (1 cm), yielding the diluted caffeine concentration in moles per liter. To find the caffeine concentration in the original sample, we multiply the diluted concentration by the dilution factor (50.00 mL / 2.000 mL). We then use the formula mass of caffeine to convert the concentration from moles per liter to mg/L.

Calculation of benzoic acid concentration follows a similar process, combining the absorbance values at different wavelengths until both concentrations are found. Once the initial concentrations of caffeine and benzoic acid are determined, the results provide insight into the composition of the original sample in terms of mg/L, which is helpful for analytical and quality control purposes.