Problem 32 - Analytical Chemistry

Question

Suppose you need to prepare a set of calibration standards for the spectrophotometric analysis of an analyte that has a molar absorptivity of \(1138 \mathrm{M}^{-1} \mathrm{~cm}^{-1}\) at a wavelength of \(625 \mathrm{nm}\). To maintain an acceptable precision for the analysis, the \(\% \mathrm{~T}\) for the standards should be between \(15 \%\) and \(85 \%\) (a) What is the concentration for the most concentrated and for the least concentrated standard you should prepare, assuming a 1.00 \(\mathrm{cm}\) sample cell. (b) Explain how you will analyze samples with concentrations that are \(10 \mu \mathrm{M}, 0.1 \mathrm{mM}\), and \(1.0 \mathrm{mM}\) in the analyte.

Step-by-Step Solution

Answer

Concentration range: \(6.20 \times 10^{-5} \, \mathrm{M}\) to \(7.23 \times 10^{-4} \, \mathrm{M}\). Dilute 0.1 mM and 1.0 mM samples; 10 µM is fine.

Step 1: Understand the Beer-Lambert Law

The Beer-Lambert Law describes the linear relationship between absorbance and concentration in spectrophotometry. Given by the equation \(A = \varepsilon cl\), where \(A\) is the absorbance, \(\varepsilon\) is the molar absorptivity (\(1138 \, \mathrm{M}^{-1} \, \mathrm{cm}^{-1}\)), \(c\) is the concentration, and \(l\) is the path length (1.00 cm). Transmittance (\(T\)) relates to absorbance by \(A = -\log_{10}(T)\).\%

Step 2: Calculate the Absorbance Range

The percent transmittance (\(%T\)) is related to absorbance by \(%T = 10^{-A} \times 100\). Thus, to find the absorbance corresponding to \(15\%\) and \(85\%\) transmittance:- For \(15\%\) transmittance: \(A = -\log_{10}(0.15)\approx 0.823\)- For \(85\%\) transmittance: \(A = -\log_{10}(0.85)\approx 0.0706\).

Step 3: Determine Concentration Bounds

Using the Beer-Lambert Law \(A = \varepsilon cl\) and solving for \(c\),- For \(A = 0.823\): \[c = \frac{A}{\varepsilon l} = \frac{0.823}{1138 \times 1.00} \approx 7.23 \times 10^{-4} \, \mathrm{M}\] - For \(A = 0.0706\): \[c = \frac{0.0706}{1138 \times 1.00} \approx 6.20 \times 10^{-5} \, \mathrm{M}\]

Step 4: Solution for Part (b)

To analyze samples:- For the \(10 \, \mu\mathrm{M}\) concentration (\(10^{-5} \, \mathrm{M}\)), it falls within the absorbance range calculated and can be directly measured.- For \(0.1 \, \mathrm{mM}\) (\(10^{-4} \, \mathrm{M}\)), it's above the maximum calculated (\(7.23 \times 10^{-4} \, \mathrm{M}\)), so dilute to within range.- For \(1.0 \, \mathrm{mM}\) (\(10^{-3} \, \mathrm{M}\)), it's drastically above, needing significant dilution to fit the absorbance window.

Key Concepts

Spectrophotometric Analysis

Spectrophotometry is a method used to measure how much a chemical substance absorbs light by measuring the intensity of light as a beam of light passes through the sample solution. It is a valuable tool in several scientific fields, especially chemistry and biology. The core idea behind spectrophotometric analysis is that each compound absorbs or transmits light over a certain range of wavelengths.
In practice, spectrophotometry is used to determine the concentration of a solute in a solution. The device commonly used for this measurement, a spectrophotometer, quantifies the amount of light absorbed at a specific wavelength.
The principles of spectrophotometric analysis are fundamental in understanding how absorbance measurements relate to the sample concentration, which is often applied through the Beer-Lambert Law. This analysis is crucial in areas like biochemistry, environmental testing, and industrial applications.

Molar Absorptivity

Molar absorptivity, often denoted as \(\varepsilon\), is a measure of how well a chemical species absorbs light at a particular wavelength. It is a constant for a given substance under specific conditions and is expressed in L mol\(^{-1}\) cm\(^{-1}\). This constant is critical because it allows calculations that relate an analyte's concentration to the amount of light absorbed.
Utilizing molar absorptivity, we can evaluate solutions mathematically and quantitatively with the Beer-Lambert Law formula:

• \(A = \varepsilon c l\) where \(A\) is absorbance, \(c\) is concentration, and \(l\) is path length.

A high molar absorptivity indicates strong absorption of light, signifying that even low concentrations can cause significant absorbance. Understanding molar absorptivity is crucial in applications like designing sensors and determining the concentration of solutions in analytical labs.

Calibration Standards

Calibration standards are solutions with known concentrations that are used to create a calibration curve in spectrophotometry. This curve helps in determining the concentration of an unknown sample by comparing its absorbance to that of the standards.
These standards are crucial to ensure precision and accuracy in measurements. Each standard's concentration is known and measured at specific \(% T\) values, providing a relationship between absorbance and concentration.
When preparing calibration standards:

• Ensure the concentration range appropriately covers the expected concentration of unknowns.
• Verify that the standards' \(% T\) falls between 15% and 85% for optimal accuracy, as extreme values can lead to inaccuracies.

Calibration standards are fundamental to achieving reliable and reproducible spectrophotometric results.

Percent Transmittance

Percent transmittance (\(% T\)) is the percentage of light that passes through a solution, relative to the incident light that strikes it. It is a critical concept in spectrophotometric analysis, linked closely with absorbance.
The relationship between absorbance (\(A\)) and transmittance is described via the equation:

• \(A = -\log_{10}(T)\), where \(T\) is the decimal form of \(% T\).

When interpreting spectrophotometric data, remember:

• Low \(% T\) indicates high absorbance and thus a higher concentration of the absorbing species.
• High \(% T\) means low absorbance, suggesting a lower concentration.

Understanding \(% T\) helps in quickly assessing the concentration of analytes in various solutions.

Absorbance and Concentration Relationship

The relationship between absorbance and concentration is central to spectrophotometry, defined mathematically by the Beer-Lambert Law. This law states that absorbance (\(A\)) is directly proportional to the concentration (\(c\)) of the sample.
Key points of this relationship include:

• Absorbance increases as concentration increases, highlighting a linear relationship when plotted.
• This direct correlation permits precise calculations of unknown concentrations by measuring absorbance.

It is essential for students to understand that the path length \(l\) is also a factor in this relationship, alongside molar absorptivity \(\varepsilon\). Such understanding enhances lab accuracy and the interpretation of spectrophotometric data, crucial in scientific research and industrial applications.