Question

A knowledge of molar absorptivities is particularly useful in biochemistry, where UV spectroscopy can provide an extremely sensitive method of detection. Imagine, for instance, that you wanted to determine the concentration of vitamin \(A\) in a sample. If pure vitamin \(A\) has \(\lambda_{\max }=325(\epsilon=50,100)\), what is the vitamin A concentration in a sample whose absorbance at \(325 \mathrm{~nm}\) is \(A=0.735\) in a cell with a pathlength of \(1.00 \mathrm{~cm} ?\)

Short answer

The concentration of vitamin A is approximately \(1.466 \times 10^{-5} \mathrm{~mol/L}\).

Step-by-step solution

Identify the Given Values

First, list the given values from the problem: \( \lambda_{\max} = 325 \mathrm{~nm} \), molar absorptivity \( \epsilon = 50,100 \mathrm{~L/mol \, cm} \), absorbance \( A = 0.735 \), and path length \( l = 1.00 \mathrm{~cm} \).

Recall Beer's Law Formula

Beer's Law relates absorbance to concentration and is given by the equation \( A = \epsilon c l \), where \( A \) is the absorbance, \( \epsilon \) is the molar absorptivity, \( c \) is the concentration, and \( l \) is the path length.

Rearrange the Formula for Concentration

To find the concentration \( c \), rearrange the equation \( A = \epsilon c l \) to solve for \( c \), giving \( c = \frac{A}{\epsilon l} \).

Substitute the Values into the Equation

Substitute the known values into the equation: \( c = \frac{0.735}{50,100 \, \mathrm{L/mol \, cm} \times 1.00 \, \mathrm{cm}} \).

Calculate the Concentration

Calculate the concentration: \( c = \frac{0.735}{50,100} \approx 1.466 \times 10^{-5} \mathrm{~mol/L} \).

Key concepts

Molar Absorptivity

Molar absorptivity, also known as the molar extinction coefficient, is a measure of how strongly a chemical species absorbs light at a particular wavelength. This value is significant because it helps determine the concentration of a solution, given its absorbance under UV spectroscopy. Molar absorptivity is denoted by the symbol \( \epsilon \) and expressed in units of \( \text{L/mol} \cdot \text{cm} \).
A high molar absorptivity indicates that even a low concentration of the substance can absorb a significant amount of light. For instance, in the exercise discussing vitamin A, the molar absorptivity is given as \( 50,100 \text{ L/mol} \cdot \text{cm} \), implying strong absorption at the wavelength of \( 325 \text{ nm} \).

When working with UV spectroscopy, knowing the molar absorptivity allows scientists to accurately measure the concentration of solutions based on the absorbance data, making it a crucial parameter in analytical chemistry.

Beer's Law

Beer's Law, a key principle in UV spectroscopy, describes the linear relationship between absorbance and concentration. It is mathematically represented as \( A = \epsilon c l \), where:

• \( A \) is the absorbance of the solution,
• \( \epsilon \) is the molar absorptivity,
• \( c \) is the concentration of the solution,
• \( l \) is the path length of the sample cell, typically in centimeters.

This law allows for the calculation of unknown concentrations by measuring absorbance, assuming that \( \epsilon \) and \( l \) are constant and known. The direct proportionality emphasized by Beer's Law is why it is so powerful in quantitative analysis.

It is important to maintain proper calibration and accounts for any deviations or instrumental errors that may arise from overly concentrated solutions or other anomalies to ensure accurate results.

Concentration Determination

Determining the concentration of a substance in a solution using UV spectroscopy involves applying Beer's Law. From Beer's Law \( A = \epsilon c l \), we can rearrange the equation to isolate concentration \( c \):\[ c = \frac{A}{\epsilon l} \]Substituting the given values from the problem, \( A = 0.735 \), \( \epsilon = 50,100 \text{ L/mol} \cdot \text{cm} \), and \( l = 1.00 \text{ cm} \), gives:\[ c = \frac{0.735}{50,100 \times 1.00} \]Upon calculating, \( c \) is found to be approximately \( 1.466 \times 10^{-5} \text{ mol/L} \), expressing the concentration of vitamin A in the solution.

This method is powerful in biochemistry and allows for precise and efficient analysis of various compounds in solutions. By comparing the absorbance measurements with known values of molar absorptivity, one can readily find the concentration of unknown solutions.

Vitamin A Analysis

Analyzing vitamin A, particularly using UV spectroscopy, is crucial when testing for its concentration in various samples such as food or biological fluids. Vitamin A has a specific absorption maximum (\( \lambda_{\max} \)) at a wavelength of 325 nm, where its molar absorptivity is known to be quite high at 50,100 \( \text{L/mol} \cdot \text{cm} \).

In practical applications, you might use this information to ensure dietary products meet nutritional standards or to study the bioavailability of vitamin A in the body. Understanding the UV spectrum of vitamin A aids in its identification and quantification, contributing to various fields such as nutrition science and pharmacology.

By employing the UV spectroscopy method, researchers can track and analyze vitamin A levels efficiently, providing a non-invasive tool to monitor this vital nutrient in clinical and food science settings.