Transmittance and Absorbance

Introduction

A spectrometer cannot measure "how much light a molecule absorbs" directly. What it actually measures is the brightness of the beam before it reaches the sample and after it has passed through. From these two readings we define transmittance , the fraction of light that gets through, and absorbance , a logarithmic quantity that grows in step with the amount of absorbing substance. Most UV-visible spectra and colorimeter readings are given as absorbance, so understanding how it is defined is essential before using the Beer–Lambert law.

Core explanation

The two intensities. Let I₀ be the intensity of monochromatic light entering the sample and I the intensity leaving it. In practice, I₀ is measured by passing light through a blank , a cuvette containing only the solvent, so that losses from reflection at the cuvette walls and absorption by the solvent are cancelled out.

Transmittance. The transmittance is the fraction transmitted:

T = I/I₀

It ranges from 1 (no absorption) to 0 (all light absorbed). It is often quoted as a percentage: %T = 100 × I/I₀.

Why a logarithm? Imagine a sample that transmits half the light. A second identical layer behind it transmits half of what remains, giving one quarter overall; a third layer gives one eighth. Each equal thickness removes the same fraction of the light, not the same amount. This is exponential decay, and the natural way to turn exponential behaviour into a straight-line relationship is to take a logarithm.

Absorbance. Absorbance is defined as:

A = log₁₀(I₀/I) = −log₁₀T

With this definition, each identical layer adds the same amount of absorbance. Absorbance has no units, although some instruments label it "AU" (absorbance units).

Useful reference values:

%T T A --- --- --- 100 1.00 0.00 50 0.50 0.30 10 0.10 1.00 1 0.010 2.00 0.1 0.0010 3.00

An absorbance of 1 means only 10% of the light gets through; an absorbance of 2 means only 1%.

Why absorbance is preferred. Absorbance is directly proportional to the concentration of absorbing species and to the path length (the Beer–Lambert law), whereas transmittance falls exponentially with both. A straight-line relationship is far easier to use for calibration and calculation, so spectra are usually plotted as absorbance against wavelength.

Practical working range. Measurements are most precise when A lies between about 0.1 and 1.0. At very low absorbance, I and I₀ are almost equal and small errors dominate; at very high absorbance, so little light reaches the detector that noise and stray light become significant.

Formulae

T = I/I₀; %T = 100T; A = log₁₀(I₀/I) = −log₁₀T; and conversely T = 10^(−A).

Step-by-step reasoning

To convert a percentage transmittance into absorbance:

1. Divide %T by 100 to obtain T. 2. Take log₁₀ of T. 3. Change the sign: A = −log₁₀T. 4. Check: a smaller %T must give a larger A.

Visual explanation

Sketch two graphs against concentration. Transmittance starts at 1 and curves steeply downwards, flattening towards zero. Absorbance starts at 0 and rises as a straight line through the origin. The same data, plotted two ways, shows why absorbance is the useful quantity.

Real-world analogy

Stacking sunglasses is a good picture. If one pair lets through 20% of the light, two pairs let through 20% of 20%, which is 4%. Each pair multiplies the brightness by the same fraction; in absorbance terms, each pair simply adds 0.70.

Real-world example

Photographic neutral-density filters are labelled by optical density, which is the same quantity as absorbance. An ND 0.3 filter transmits 50% of the light and an ND 0.9 filter transmits about 12.5%. Photographers stack filters by adding their densities.

Why?

Why do we need a blank? The cuvette walls reflect some light and the solvent may absorb slightly. Measuring I₀ through a blank means that these losses are included in both readings and cancel, so the calculated absorbance belongs only to the solute.

Common misconception

"Absorbance is the percentage of light absorbed." It is not a percentage. An absorbance of 0.50 means that 10^(−0.5), about 32%, of the light is transmitted and about 68% is absorbed, not 50%.

Worked example

Question: A solution transmits 25.0% of the incident light. Calculate its absorbance.

Reasoning: T = 25.0 ÷ 100 = 0.250. A = −log₁₀(0.250) = 0.602. Because transmission is below one, the positive absorbance is physically sensible: more light was removed from the transmitted beam.

Answer: A = 0.602 (no units).

Quick check

1. What is the transmittance of a sample with an absorbance of 2.00, expressed as a percentage? Answer: T = 10^(−2.00) = 0.0100, so 1.00% of the light is transmitted.

Exam focus

Learn A = log₁₀(I₀/I) and T = I/I₀, and practise converting both ways, remembering to convert percentages to fractions first. State that absorbance has no units. Explain that absorbance is used because it is proportional to concentration.

Advanced insight

Some older texts and physics courses use the natural logarithm, defining absorbance as ln(I₀/I), which is 2.303 times larger than the decadic absorbance. Chemistry uses the decadic form, and the molar absorption coefficient is defined to match it; mixing the two conventions gives errors by a factor of 2.303.

Summary

Transmittance T = I/I₀ is the fraction of light passing through a sample, and absorbance A = log₁₀(I₀/I) = −log₁₀T. Because each layer removes a constant fraction of the light, transmittance falls exponentially while absorbance rises linearly with concentration and path length. Absorbance has no units and is most precise between about 0.1 and 1.0. I₀ is measured through a solvent blank.

Practice questions

1. Define absorbance in terms of I₀ and I. Answer: A = log₁₀(I₀/I), where I₀ is the incident intensity and I is the transmitted intensity. 2. Convert an absorbance of 0.40 into percentage transmittance. Answer: T = 10^(−0.40) = 0.398, so about 39.8% is transmitted. 3. A sample transmits 80.0% of the light. Calculate its absorbance. Answer: A = −log₁₀(0.800) = 0.097. 4. Explain why absorbance is plotted against concentration rather than transmittance. Answer: Absorbance is directly proportional to concentration, giving a straight line, whereas transmittance decreases exponentially and gives a curve.