Beer–Lambert law in analytical spectrophotometry: which statement correctly expresses how absorbance depends on optical path length and analyte concentration?

Introduction / Context:
The Beer–Lambert law is the backbone of quantitative UV–Vis spectrophotometry. It connects the measured absorbance to the analyte concentration and the optical path length, enabling concentration determinations in chemistry, biology, and environmental testing.

Given Data / Assumptions:

• Dilute solutions with minimal scattering and no significant chemical interactions.
• Monochromatic light, or a sufficiently narrow bandwidth compared to the absorption band.
• Homogeneous sample and a defined optical path length l (typically 1 cm cuvettes).

Concept / Approach:
The Beer–Lambert relationship is A = ε * l * c, where A is absorbance, ε is molar absorptivity, l is path length, and c is concentration. Absorbance is additive and directly proportional to both l and c under ideal conditions.

Step-by-Step Solution:
Start with intensity relationship: A = -log10(T) = -log10(P / P0).Relate to concentration: A = ε * l * c.Thus, doubling l or c doubles A (holding other parameters constant).Conclude proportionality to both path length and concentration.

Verification / Alternative check:
Calibration curves (A vs c) yield a straight line through the origin for fixed l, confirming proportionality. Path length changes (e.g., 0.1 cm microcuvettes vs 1 cm) scale the slope by l.

Why Other Options Are Wrong:
Option b omits path length and misstates dependence. Option c confuses absorbance with the intensity ratio; A = log10(P0 / P), not equal to it. Option e is false; path length directly affects absorbance even at low concentration. Therefore a is correct.

Common Pitfalls:
Ignoring chemical equilibria, stray light, or instrumental bandwidth effects that can cause deviations from linearity.

Final Answer:
Absorbance is proportional to both the path length and the concentration of the absorbing species.