Beer Lambert Law Calculator

Absorbance, Transmittance & Optical Concentration Solver

Input Parameters

M−¹ cm−¹
Spectrophotometer Live Model Active
Absorbance (A)
0.30000
Light transmitted through sample matches inputs.
Mathematical Working & Substitutions
A = ε × c × l

Understanding the Beer-Lambert Law

What is the Beer-Lambert Law?

The Beer-Lambert Law (also known as Beer's Law or the Beer-Lambert-Bouguer Law) is a fundamental relationship in spectroscopy and analytical chemistry. It states that the absorbance of a chemical species in solution is directly proportional to its concentration, the optical path length, and the molar absorptivity of the solute.

The Mathematical Formula

The primary representation of the Beer-Lambert law is:

A = ε × c × l

Where the parameters are defined as:

Transmittance vs. Absorbance

Transmittance (T) is the fraction of incident light that passes through the sample:

T = I / I0   and   T% = (I / I0) × 100%

Because light absorption decreases the transmitted light exponentially, absorbance is defined logarithmically:

A = −log10(T) = 2 − log10(T%)

Thus, an absorbance of 0.0 corresponds to 100% transmittance (no light absorbed). An absorbance of 1.0 means only 10% of light is transmitted (90% absorbed), and an absorbance of 2.0 means only 1% is transmitted (99% absorbed).

Analytical Use: Calibration Curves

In research and chemical laboratories, the Beer-Lambert Law is utilized to determine unknown concentrations. By measuring the absorbance of several standards (solutions of known concentration), chemists construct a **Calibration Curve** (linear plot of Absorbance vs. Concentration). Using linear regression, the line equation is found:

A = m × c + b

Where m represents the slope (proportional to ε and l) and b is the y-intercept (stray background absorbance). Measuring the absorbance of an unknown sample then allows its concentration to be determined directly by solving for c.

Solved Examples

Example 1: Calculating Absorbance

Problem: A solute has a molar absorptivity of 1.5 × 104 L mol−1 cm−1. Calculate the absorbance of a 2.0 × 10−5 M solution in a standard 1.0 cm cuvette.

Solution:

  1. Identify parameters: ε = 15,000 M−1 cm−1, c = 2.0 × 10−5 M, l = 1.0 cm.
  2. Apply formula: A = ε × c × l
  3. Substitute: A = 15000 × (2.0 × 10−5) × 1.0
  4. A = 0.300. (The solution has an absorbance of 0.300).
Example 2: Absorbance from Transmittance Percentage

Problem: A sample shows a light transmittance of 35.0% at a specific wavelength. Determine its absorbance.

Solution:

  1. Given transmittance percentage T% = 35.0%.
  2. Apply formula: A = 2 − log10(T%)
  3. Substitute: A = 2 − log10(35.0)
  4. Calculate: A = 2 − 1.544 = 0.456.
Example 3: Solving for Unknown Concentration

Problem: An organic dye has an absorptivity coefficient of 8,400 M−1 cm−1. Measured in a 0.5 cm cuvette, the sample has an absorbance of 0.420. Find its concentration.

Solution:

  1. Given: A = 0.420, ε = 8,400, l = 0.5 cm.
  2. Rearrange Beer-Lambert Law: c = A / (ε × l)
  3. Substitute: c = 0.420 / (8400 × 0.5)
  4. Calculate: c = 0.420 / 4200 = 1.0 × 10−4 mol/L (or 100 μM).

Frequently Asked Questions

What are the physical and chemical limitations of the Beer-Lambert Law?
The Beer-Lambert Law is highly linear only at low concentrations (typically ≤ 0.01 M). Deviances occur at higher concentrations due to:
  • Chemical interactions: Solute molecules associate, dissociate, or react with solvent, changing absorptivity.
  • Refractive index changes: High concentrations alter the solution's refractive index.
  • Stray light: Scattering of light by particles or inside the instrument.
Why is the standard path length of cuvettes exactly 1 cm?
A path length of 1.0 cm is standard because it provides a balance between having enough sample volume for an accurate reading without requiring excess quantities of reagents, and keeping calculations simple (since path length l = 1 simplifies the product ε × c × l to ε × c).
How does stray light affect absorbance measurements?
Stray light is radiation of unwanted wavelengths that reaches the detector. It causes a negative deviation from the Beer-Lambert law, capping the maximum absorbance the spectrophotometer can accurately read. At very high concentrations, stray light causes the calibration curve to bend and flatten out.
What wavelength is selected when measuring absorbance?
Absorbance is typically measured at the wavelength of maximum absorption (λmax). This wavelength is chosen because the rate of change of absorbance with concentration is highest at this point, providing maximum sensitivity and reducing errors from minor wavelength drifts.