Question

An impurity in water has an extinction coefficient of \(3.45 \times 10^{3} M^{-1} \mathrm{cm}^{-1}\) at \(280 \mathrm{nm},\) its absorption maximum (A Closer Look, p. 576\() .\) 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?

Short answer

By using the Beer-Lambert Law equation, the absorbance at a threshold of 50 ppb and a pathlength of 1 cm is calculated as \(A = 172.5 \times 10^{-6}\). Since this value is greater than the minimum reliable absorbance of 0.0001 that a spectrometer can detect, measuring the absorbance at 280 nm is a good way to detect concentrations of the impurity above the 50 ppb threshold.

Step-by-step solution

Understand the Beer-Lambert Law equation

The Beer-Lambert Law relates the absorbance (A) of a sample to its extinction coefficient (ε), molarity (M), and pathlength (l) through the following equation: \[A = ε × M × l\]

Convert the concentration

In order to use the Beer-Lambert Law, we need to convert the given concentration threshold from ppb (parts per billion) to molarity (M). Since \(1 ppb = 1 \times 10^{-9}\) mol/L, we can convert 50 ppb to mol/L: \[50 ppb = 50 \times 10^{-9} mol/L\]

Decide on the pathlength

In order to decide if measuring the absorbance at 280 nm is a good way to detect concentrations of the impurity above the 50 ppb threshold, we need to consider the typical pathlength used in experiments - which is usually 1 cm. We will use that as our standard pathlength (l = 1 cm).

Calculate the absorbance

Now we can plug in our values into the Beer-Lambert law equation and compute the absorbance: \[A = ε × M × l\] \[A = (3.45 \times 10^3 M^{-1} cm^{-1}) \times (50 \times 10^{-9} mol/L) \times (1 cm)\] \[A = 3.45 \times 50 \times 10^{-6}\]

Compare the calculated absorbance with spectrometer's capabilities

Now we need to compare the calculated absorbance, A, to the minimum reliable absorbance that a spectrometer can detect (0.0001) to know if the measurement at 280 nm is a good way to detect concentrations of the impurity above the 50 ppb threshold. Calculate the absorbance value: \[A = 3.45 \times 50 \times 10^{-6}\] \[A = 172.5 \times 10^{-6}\] Since \(172.5 \times 10^{-6} > 0.0001\), we can conclude that measuring the absorbance at 280 nm is a good way to detect concentrations of the impurity above the 50 ppb threshold.

Key concepts

Extinction Coefficient

The extinction coefficient is an essential factor in the Beer-Lambert Law. It symbolizes how strongly a substance absorbs light at a particular wavelength. The larger the extinction coefficient, the more an element will absorb at that wavelength.
For example, in the exercise, the impurity in the water has an extinction coefficient of \(3.45 \times 10^3 \ M^{-1} \mathrm{cm}^{-1}\) at 280 nm.
This means that the impurity absorbs light significantly at this wavelength, making it detectable when measuring absorbance.

• This value is specific to each substance and varies with the wavelength of light used.
• It is expressed in units of \(M^{-1}\mathrm{cm}^{-1}\), which allows for easy calculation of absorbance based on concentration and path length.

Understanding the extinction coefficient helps in selecting the appropriate wavelength for spectrophotometry, ensuring accurate detection.

Absorbance

Absorbance is a measure of the amount of light absorbed by a solution. It's crucial for understanding how much of a substance is present in a sample.
According to the Beer-Lambert Law, absorbance (A) is directly proportional to the concentration of the sample. The equation is: \[ A = ε \times M \times l \]

• \(A\) represents absorbance.
• \(ε\) is the extinction coefficient.
• \(M\) is the molarity or concentration.
• \(l\) is the path length, generally 1 cm in many spectrometers.

The goal in spectrometry is to measure the absorbance accurately to infer the concentration of substances. For instance, in the problem, we find that if the absorbance is above 0.0001, the concentration of the impurity is detectable.

Spectrometry

Spectrometry, particularly UV-Vis spectrometry, is a powerful analytical technique for quantifying the concentration of compounds. It uses the principle that substances absorb light of specific wavelengths.
In the given exercise, spectrometry is used to measure absorbance at a wavelength of 280 nm to detect water impurities. This method relies on Beer's Law to calculate concentrations based on absorbance values.

• This technique is precise and can detect even low concentrations if the extinction coefficient and the path length are known.
• It involves passing light through the sample, and measuring changes in the light intensity to determine absorbance.

By comparing the measured absorbance with known values, the concentration of a substance can be calculated - providing insights into sample purity and content.

Molarity Conversion

Molarity refers to the concentration of a solution expressed as moles of solute per liter of solution. Converting different units of concentration like parts per billion (ppb) to molarity is essential in many chemical calculations.
For instance, in the exercise, the conversion from 50 ppb to molarity helped determine if the absorbance of the impurity could be measured reliably.

• Typically, \(1 \mathrm{ppb} = 1 \times 10^{-9}\) mol/L.
• This conversion is necessary to use the Beer-Lambert equation, which requires molarity.
• Understanding molarity and its conversions aids in accurately calculating concentrations and thus, precise absorbance measures.

Converting units to molarity is crucial for consistent and understandable results in spectrometric analyses.