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

Expert verified
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

01

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: \[A = ε × M × l\]
02

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: \[50 ppb = 50 \times 10^{-9} mol/L\]
03

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).
04

Calculate the absorbance

Now we can plug in our values into the Beer-Lambert law equation and compute the absorbance: \[A = ε × M × l\] \[A = (3.45 \times 10^3 M^{-1} cm^{-1}) \times (50 \times 10^{-9} mol/L) \times (1 cm)\] \[A = 3.45 \times 50 \times 10^{-6}\]
05

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.

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