Problem 84 - Chemistry: The Central Science

Question

An impurity in water has an extinction coefficient of \(3.45 \times 10^{3} \mathrm{M}^{-1} \mathrm{~cm}^{-1}\) at \(280 \mathrm{~nm}\), its absorption maximum (A Closer Look, p. 582 ). Below \(50 \mathrm{ppb}\), the impurity is not a problem for human health. Given that most spectrometers cannot detect absorbances less than \(0.0001\) with good reliability, is measuring the absorbance of a water sample at \(280 \mathrm{~nm}\) a good way to detect concentrations of the impurity above the 50 -ppb threshold?

Step-by-Step Solution

Answer

Measuring the absorbance of a water sample at 280 nm is a good way to detect concentrations of the impurity above the 50 ppb threshold. This is because the spectrometer's minimum detectable concentration (2.90 x 10^(-8) M) is well below the 50 ppb threshold (2.76 x 10^(-6) M), ensuring reliable detection of impurity concentrations above the limit.

Step 1: Convert the threshold to molar concentration

Convert the given 50 ppb threshold to molar concentration. ppb stands for parts per billion, which means parts of impurity per billion parts of water. First, we need to convert ppb to a ratio: $$ 50 \, ppb = \frac{50}{10^9} $$ Then, we'll divide this ratio by the molar concentration of water. Since water has a molar mass of 18.015 g/mol and a density of 0.997 g/mL at room temperature, we can calculate the molar concentration of water as: $$ \frac{0.997 \, g/mL}{18.015 \, g/mol} = 0.0553 \, mol/mL $$ Now, we can find the threshold molar concentration by multiplying the ppb ratio by the molar concentration of water: $$ \frac{50}{10^9} \cdot 0.0553 \, mol/mL \approx 2.76 \times 10^{-6} \, M $$

Step 2: Calculate the minimum concentration detectable by spectrometer

To calculate the minimum concentration detectable by spectrometer, we will use the Beer-Lambert law, which states that absorbance (A) is equal to the product of the molar absorptivity (extinction coefficient, ε), concentration (c), and the path length (l): $$ A = \epsilon \cdot c \cdot l $$ In this exercise, we have the following values: - A = 0.0001 (minimum detectable absorbance) - ε = 3.45 x 10^3 M^(-1) cm^(-1) (extinction coefficient) - l = 1 cm (using the standard cuvette width) We can now solve for concentration (c) by rearranging the Beer-Lambert law: $$ c = \frac{A}{\epsilon \cdot l} = \frac{0.0001}{3.45 \times 10^{3} \, M^{-1} cm^{-1} \cdot 1 \, cm} \approx 2.90 \times 10^{-8} \, M $$

Step 3: Compare the calculated concentration with the threshold

We found that the minimum concentration detectable with a spectrometer is 2.90 x 10^(-8) M. Comparing this value to the 50 ppb threshold (2.76 x 10^(-6) M), we can see that the spectrometer's detection limit is well below the threshold, meaning it can reliably detect concentrations above the 50 ppb limit: $$ 2.90 \times 10^{-8} \, M < 2.76 \times 10^{-6} \, M $$ Since the calculated minimum concentration is less than the 50 ppb threshold, measuring the absorbance of a water sample at 280 nm is a good way to detect concentrations of the impurity above the 50 ppb threshold.

Key Concepts

Spectrophotometry

Spectrophotometry is a crucial analytical technique used to measure how much light a chemical substance absorbs at a given wavelength. By shining a specific wavelength of light through a sample and measuring the intensity of the transmitted light, spectrophotometers provide valuable information about the sample's composition. The key principle behind spectrophotometry is that each compound absorbs or transmits light over a certain range of wavelengths.
In the context of the exercise, spectrophotometry is used at 280 nm, a wavelength where the impurity in question has its peak absorption. This allows for a precise assessment of the impurity's concentration in the water sample. The accuracy of detection, however, depends on various factors including the instrument's sensitivity, the clarity of the sample, and the specific properties of the impurity.

Molar Absorptivity

Molar absorptivity, also known as the molar extinction coefficient (symbolized as \( \epsilon \)), is a measure of how well a chemical species absorbs light at a particular wavelength. It is an intrinsic property of the substance and indicates the amount of light that can be absorbed by a mole of the substance at a specific concentration and path length. The value \( 3.45 \times 10^{3} \mathrm{M}^{-1} \mathrm{~cm}^{-1} \) given in the problem signifies the impurity's ability to absorb light at 280 nm.
Calculating absorption using molar absorptivity is a fundamental application of the Beer-Lambert law, allowing students to understand how molecular characteristics influence light absorption. The higher the molar absorptivity, the more effectively the substance can be detected through spectrophotometry, making it a critical factor when determining whether a spectrophotometer can identify an impurity.

Molecular Concentration

Molecular concentration refers to the amount of a substance in a given volume of solution. It is usually expressed in molarity (M), which is the number of moles per liter (mol/L). In this exercise, the term 'ppb' or parts per billion, is a fractional way to express concentrations of trace substances—like the impurity in water. To relate ppb to molarity, one must consider the substance's molar mass and the solvent's density.
In the step-by-step solution, the conversion of 50 ppb to molarity involves using the molar concentration of water, illustrating a real-world application of molar concentration. This type of calculation is fundamental when working with concentration thresholds, as it allows one to evaluate whether specific levels exceed safety standards or detection limits.

Detection Limit

The detection limit of an analytical instrument is the lowest quantity or concentration of a substance that can be reliably distinguished from its absence. It is a crucial parameter that defines the sensitivity of the instrument. For spectrometers, the detection limit is often determined by the minimum absorbance value that can be confidently measured above the baseline noise.
In the provided exercise, the spectrometer's detection limit is an absorbance of 0.0001. By using the Beer-Lambert law, we translate this absorbance to a minimum detectable concentration. The comparison between this limit and the safety threshold (50 ppb) determines if the method is adequate for detecting impurity levels that could affect human health. Understanding detection limits ensures that the limits of the analytical techniques employed are recognized and adhered to for accurate assessment.