Question

A spectrophotometer is used to measure the concentration of a molecule in a solution based on Beer's law using cuvette of path length 10mm and detectable absorbance of 0.01. The molar absorptivity of the solution is $10^4$ l/mol-m. The minimum concentration of the solution that can be detected is

• A. $1\mu\text{mol/l}$
• B. $24\mu\text{mol/l}$
• C. $5\mu\text{mol/l}$
• D. $10\mu\text{mol/l}$

Hint:

Always double-check unit dimensions when working with analytical instrumentation questions.
In almost all standard bio-instrumentation applications, the cuvette path length is normalized to \( 1\text{ cm} \) (which equals \( 10\text{ mm} \)).
Keeping this standard benchmark in mind makes calculations very straightforward.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to find the minimum detectable concentration of a solution using Beer-Lambert's Law, given the path length, detectable absorbance, and molar absorptivity of the solution.

Step 2: Key Formula or Approach:
Beer-Lambert's Law is expressed as:
\[ A = \epsilon \cdot c \cdot l \] where:
\( A \) is the absorbance (dimensionless).
\( \epsilon \) is the molar absorptivity.
\( c \) is the concentration of the solution.
\( l \) is the path length of the cuvette.

Step 3: Detailed Explanation:

• Identify the given parameters:
- Path length, \( l = 10\text{ mm} = 1\text{ cm} = 10^{-2}\text{ m} \).
- Absorbance, \( A = 0.01 = 10^{-2} \).
- Molar absorptivity, \( \epsilon = 10^4\text{ L}/(\text{mol}\cdot\text{cm}) \).
Note: Although the text states the units as "l/mol-m", spectrophotometer calculations in chemistry and bio-instrumentation typically express the path length in centimeters (\( \text{cm} \)) and the molar absorptivity in \( \text{L}/(\text{mol}\cdot\text{cm}) \). Using these standard units resolves any dimensional inconsistencies in the exam question.

• Rearrange the Beer-Lambert formula to solve for the concentration \( c \):
\[ c = \frac{A}{\epsilon \cdot l} \]
• Substitute the values using centimeter-based units (\( l = 1\text{ cm} \) and \( \epsilon = 10^4\text{ L}/(\text{mol}\cdot\text{cm}) \)):
\[ c = \frac{0.01}{10^4 \cdot 1} = \frac{10^{-2}}{10^4} = 10^{-6}\text{ mol/L} \]
• Convert the concentration from moles per liter (\( \text{mol/L} \)) to micromoles per liter (\( \mu\text{mol/L} \)):
Since \( 1\ \mu\text{mol} = 10^{-6}\text{ mol} \):
\[ c = 1 \times 10^{-6}\text{ mol/L} = 1\ \mu\text{mol/L} \]
• This corresponds to the minimum concentration that can be detected by the spectrophotometer under the given parameters.