UV-visible spectroscopy and the Beer-Lambert law

The cooling water at Kingfisher Bioenergy runs through miles of carbon steel pipe and heat exchangers. If corrosion speeds up, dissolved iron in the water rises before any leak appears, so the operators want an iron result every day. Phosphorus in the plant's effluent is limited by its permit, and the treatment operators need daily numbers to adjust chemical dosing. Both measurements use the same tool: a spectrophotometer that measures how much light a colored solution absorbs. Spectrophotometry is quick, cheap and sensitive, and it is the gateway to all the spectroscopic methods in the following lessons.

Light and absorption

Light is electromagnetic radiation, described by its wavelength \(\lambda\), frequency \(\nu\) and the energy of each photon, \(E = h\nu = hc/\lambda\), where h is Planck's constant and c the speed of light. Short wavelengths carry more energy. Ultraviolet light from about 190 to 400 nm and visible light from 400 to about 750 nm carry enough energy to promote electrons in molecules from lower to higher energy levels. Infrared light, at longer wavelengths, excites molecular vibrations instead, which Lesson 7 covers.

A molecule absorbs only photons whose energy matches the gap between two of its energy levels. Molecules with conjugated double bonds, aromatic rings or metal ions bound to suitable ligands absorb in the ultraviolet or visible region. The color we see is the complement of what is absorbed: a solution that absorbs blue-green light near 510 nm, like the red-orange iron complex used at the plant, looks orange-red. Most analytes are not colored, so many methods first convert the analyte into a colored product with a reagent that reacts selectively.

Transmittance and absorbance

When light of power \(P_0\) passes through a sample, some is absorbed and the transmitted power P is lower. The transmittance is \(T = P/P_0\), often quoted as a percentage, and the absorbance is:

An absorbance of 1 means 10 percent of the light gets through, an absorbance of 2 means 1 percent. Absorbance is preferred to transmittance because, under the right conditions, it is directly proportional to concentration.

The Beer-Lambert law

The Beer-Lambert law states that absorbance is proportional to the path length b through the sample and the concentration c of the absorbing species:

The proportionality constant \(\varepsilon\) is the molar absorptivity, in L/(mol cm) when b is in centimeters and c in mol/L. It depends on the substance, the wavelength and to some extent the solvent and temperature. A large \(\varepsilon\) means a sensitive method. The iron complex with 1,10-phenanthroline has \(\varepsilon\) of about 11,000 L/(mol cm) at 510 nm, which allows iron to be measured at a fraction of a milligram per liter in a standard 1 cm cell.

The law can be pictured simply. Each thin slice of solution absorbs the same fraction of the light reaching it, so the power falls exponentially with distance, and the logarithm of the ratio, the absorbance, grows linearly with path length. Doubling the concentration doubles the number of absorbing molecules in each slice, which also doubles the absorbance.

Instruments

A spectrophotometer has five main parts. A source provides broad-band light: a deuterium lamp for the ultraviolet and a tungsten-halogen lamp for the visible. A monochromator, built around a diffraction grating and slits, selects a narrow band of wavelengths. The sample sits in a cuvette of known path length, usually 1.00 cm. A detector, a photodiode or photomultiplier, converts the transmitted light into a current. The electronics compare it with the blank to give absorbance.

The blank, a cuvette filled with the solvent and reagents but no analyte, is measured to set \(P_0\). That cancels absorption by the solvent, the reagents and the cuvette walls, and reflection losses at the windows. Glass and plastic cuvettes suit the visible region but absorb ultraviolet light below about 320 nm, so ultraviolet work needs quartz cuvettes. Cuvettes are handled by their frosted sides and wiped clean, because a fingerprint on the light path adds absorbance.

Single-beam instruments measure the blank and sample one after the other. Double-beam instruments split the light and measure both at once, which cancels drift in the lamp. Diode-array instruments pass all wavelengths through the sample and spread them onto an array of detectors, recording a full spectrum in a fraction of a second. Simple filter photometers, which use a colored filter or light-emitting diode instead of a monochromator, are common in field kits and online analyzers for water treatment.

Choosing conditions and sources of error

Measurements are usually made at the wavelength of maximum absorbance, \(\lambda_{max}\), where sensitivity is highest and small errors in wavelength setting matter least, because the absorbance curve is flat at its peak. Absorbances between about 0.1 and 1.0 give the best precision. Below that, small differences between large signals are hard to measure. Above it, very little light reaches the detector and noise and stray light dominate.

The Beer-Lambert law is a limiting law and fails under some conditions. Chemical deviations occur when the absorbing species takes part in an equilibrium that shifts with concentration, such as an indicator whose two forms absorb differently or a complex that dissociates on dilution. Instrumental deviations arise from polychromatic light, because the law holds exactly only for a single wavelength, and from stray light, light of other wavelengths that reaches the detector without passing through the sample. Stray light makes high absorbances read low, bending the calibration curve toward the concentration axis. At high concentrations, above roughly 0.01 mol/L, interactions between absorbing molecules also change \(\varepsilon\).

Colorimetric methods at the plant

Each method reacts the analyte with a reagent to form a strongly absorbing product, using conditions that favor the reaction and suppress interferences. For iron, hydroxylamine first reduces any iron(III) to iron(II), which forms a stable complex with three phenanthroline molecules. For phosphate, the molybdenum blue method is so sensitive that it reaches micrograms per liter, which is why it is the standard for effluent phosphorus. Enzyme activity, important when the plant evaluates new amylase products, is often measured by following the change in absorbance as a reaction proceeds, a kinetic method that applies the rate laws from CHE-250.

Interferences are the main weakness of color-forming methods. Silicate forms a molybdenum blue of its own and can raise phosphate results; arsenate does the same. Strongly colored or turbid samples absorb or scatter light whatever their analyte content. Standard methods deal with these by controlling acidity and reaction time, by filtering or by measuring a sample blank without the color reagent and subtracting it. When a new kind of sample comes into the lab, such as water from a new well or a process stream after a change of chemistry, the analyst checks for interferences with a spike recovery before trusting routine results.

The program below calibrates the iron method from Theo's transmittance readings, calculates the molar absorptivity, reads a diluted cooling-water sample, and shows how half a percent of stray light distorts high absorbances.

# Beer-Lambert calibration for iron in cooling water (phenanthroline complex, 510 nm)
import math

std = [0.0, 0.5, 1.0, 2.0, 3.0, 4.0]                   # Fe, mg/L
T_pct = [100.0, 79.5, 63.3, 40.1, 25.3, 16.1]          # percent transmittance, 1.00 cm cell
A = [-math.log10(t / 100) + 0.0 for t in T_pct]
for c, a in zip(std, A):
    print(f"{c:4.1f} mg/L  A = {a:.3f}")
n = len(std); xm = sum(std) / n; ym = sum(A) / n
b = sum((x - xm) * (y - ym) for x, y in zip(std, A)) / sum((x - xm) ** 2 for x in std)
a0 = ym - b * xm
print(f"slope = {b:.4f} L/mg (path 1.00 cm), intercept = {a0:.4f}")
eps = b * 55.845 * 1000              # molar absorptivity, L/(mol cm)
print(f"molar absorptivity = {eps:,.0f} L/(mol cm)")

# Unknown cooling-water sample, diluted 10.00 mL to 25.00 mL before the complex is formed
A_unk = 0.316
c_cell = (A_unk - a0) / b
print(f"in the cell: {c_cell:.2f} mg/L; in the sample: {c_cell * 25 / 10:.2f} mg/L")

# Stray light makes high absorbances read low: 0.5 percent stray light
s = 0.005
for true_A in (0.5, 1.0, 2.0, 3.0):
    T = 10 ** -true_A
    meas = -math.log10((T + s) / (1 + s))
    print(f"true A {true_A:.1f} reads as {meas:.2f}")

 0.0 mg/L  A = 0.000
 0.5 mg/L  A = 0.100
 1.0 mg/L  A = 0.199
 2.0 mg/L  A = 0.397
 3.0 mg/L  A = 0.597
 4.0 mg/L  A = 0.793
slope = 0.1984 L/mg (path 1.00 cm), intercept = 0.0003
molar absorptivity = 11,082 L/(mol cm)
in the cell: 1.59 mg/L; in the sample: 3.98 mg/L
true A 0.5 reads as 0.50
true A 1.0 reads as 0.98
true A 2.0 reads as 1.83
true A 3.0 reads as 2.22

The diluted sample reads 1.59 mg/L, so the cooling water itself contains about 4.0 mg/L iron after allowing for the 10 to 25 mL dilution. That is high for this system, and the operators check the corrosion inhibitor dosing. The stray-light model shows why the lab dilutes samples that read above about 1.0: a true absorbance of 2.0 would read as 1.83, an error of 8 percent that no calibration within the normal range would reveal.

Beyond simple absorption

If two absorbing species are present, their absorbances add. Measuring at two wavelengths where their absorptivities differ gives two equations that can be solved for both concentrations, and modern software extends this to many components using the full spectrum. Fluorescence spectroscopy measures light emitted when excited molecules return to the ground state. Because it measures a small signal against a dark background, it can be a thousand times more sensitive than absorption, and it is used for trace organics and in many biological assays. Turbidity, the cloudiness of water caused by suspended particles, is measured by the light they scatter, and it is one of the simplest online measurements in water treatment.