### Introduction

Spectrophotometry is the science of measuring the light-absorbing and light-transmitting characteristics of a substance.

Image Source: AD Instruments

**Spectrophotometry** is the science of measuring the light-absorbing and light-transmitting characteristics of a substance. Many substances absorb light and transmit light of specific wavelengths within the ultraviolet (200 - 400 nm), visible (400 - 700 nm) and near-infrared (700 - 1000 nm) regions of the electromagnetic spectrum. These light-absorbing / light-transmitting characteristics of a substance are useful in determining the presence and concentration of that substance in a sample.

### The Spectrophotometer

Light path through a visible-light spectrophotometer.

Image Source: Biol 1440, University of Central Arkansas

The instrument used to measure the amount of light of a specific wavelength absorbed or transmitted by a substance is called a **spectrophotometer**. In a spectrophotometer, a sample of the substance is placed across the path of a light beam of a chosen specific wavelength. The spectrophotometer determines the intensity of the light entering the sample and the intensity of the light leaving the sample, then calculates the amount of light transmitted and absorbed by the substance.

A diagram of the light path through a visible-light spectrophotometer is shown to the right. A beam of light emerges from its source and passes through a prism, which dissects the light into a continuous spectrum of wavelengths. The user can select which single specific wavelength of light passes to the sample with the monochromator. Light of a single wavelength is called **monochromatic light**. The monochromatic light that passes to the sample is the known as **incident light**, its intensity is represented the value I_{o}. As the incident light passes through the sample, a certain amount of the light will be absorbed by the sample. The monochromatic light that is not absorbed emerges from the sample; it is called **transmitted light**, and its intensity is represented by the value I (or I_{1}). The intensity of the transmitted might is detected by a photodetector.

Once the transmitted light is detected, the instrument calculates the fraction of the incident light transmitted by the sample, a value known as **transmittance (T), T = I**_{1} / I_{0}. From the transmittance value, the instrument will calculate the amount of the monochromatic light absorbed by the sample, a value known as **absorbance (A)**, using the formula **A = - log10 T**. Both the transmittance and absorbance are displayed on the display screen of the instrument.

### Transmittance & Absorbance

Incident light (I_{0}) transmitted through a sample.

Image Source: Wikipedia

The amount of monochromatic light absorbed by a sample is determined by comparing the intensities of the incident light (I_{0}) and transmitted light (I_{1}). The ratio of the intensity of the transmitted light (I_{1}) to the intensity if the incident light (I_{0}) is called *transmittance (T)* .

**T = I**_{1} / I_{0}

Because the intensity of the transmitted light (I_{1}) is never greater than the intensity of the incident light (I_{0}), transmittance (T) is always less than 1.

In practice, one usually multiplies T by 100 to obtain the **percent transmittance (%T)**, which ranges from 0 to 100%.

**%T = T * 100**

If the T of a sample is 0.40, the %T of the sample is 40%. This means that 40% of the photons in the incident light emerge from the sample as transmitted light and reach the photodetector. If 40% of the photons are transmitted, 60% of the photons were absorbed by the sample.

From the transmittance or % transmittance, one can calculate the quantity known as *absorbance (A)*. Absorbance is the amount of light absorbed by a sample. It is calculated from T or %T using the following equations:

**A = - log**_{10} T or **A = log**_{10} (1/T)

**A = 2 - log**_{10} %T

These equations reveal that transmittance and absorbance are inversely related. That is, the more a particular wavelength of light is absorbed by a substance, the less it is transmitted. Moreover, the inverse relationship between A and T is not linear, it is **logarithmic**. Therefore, if 50% of the photons of monochromatic light are transmitted by a sample, and 50% of the photons are absorbed, T = 0.5, but A is not 0.5, A is 0.3, due to the inverse logarithmic relationship beween T and A. If 10% of the photons of monochromatic light are transmitted by a sample, and 90% of the photons are absorbed, T = 0.1, but A is not 0.9, A = 1.0. When A is 2.0, 99% of the photons of monochromatic light are absorbed, and when A is 3.0, 99.9% of the photons of monochromatic light are absorbed.

The inverse logarithmic relationship between absorbance and transmittance and between absorbance and %T are clearly shown in the graphs below. In these graphs, as transmittance (top graph) and %T (bottom graph) increase from 0 to 1.0 and 0% to 99%, respectively, absorbance decreases logarithmically from 2.0 to 0.

Image Source: Stephen Gallik, Ph. D.

### Sample Calculations

#### Three Sample Calculations of Absorbance from T and %T

__Calculation #1:__

if T = I_{1} / I_{0} = 0.999

then %T = T * 100 = 99.9

and A = 2 - log_{10} %T = 2 - log_{10} 99.9 = 2 - 1.9995 = 0.0005

__Calculation #2:__

if T = I_{1} / I_{0} = 0.50

then %T = T * 100 = 50

and A = 2 - log_{10} %T = 2 - log_{10} 50 = 2 - 1.69897 = 0.301

__Calculation #3:__

if T = I_{1} / I_{0} = 0.20

then %T = T * 100 = 20

and A = 2 - log_{10} %T = 2 - log_{10} 20 = 2 - 1.301 = 0.699

### Beer's Law

The amount of light absorbed by a sample is dependent on the concentration of the pigment in the sample (c), path length (l), and the extinction coefficient of the pigment (Ε).

Image Source: Wikipedia

Determining the amount of monochromtic light absorbed by a substance is most-commonly used to determine the concentration of that substance in a sample. The concentration (c) of a substance in a sample is one of three factors that affect the amount of light absorbed by a sample. The other two are path length (l), that is the distance the light travels through the sample, and the extinction coefficient of the absorbing substance (Ε). The extinction coefficient is simply a measure of how strongly a substance absorbs light of a given wavelength. The relationship between transmittance or absorbance and these three factors is expressed by **Beer's Law**, one of the fundamental laws of spectrophotometry. **Beer's Law states that the intensity of transmitted light **__decreases exponentially__ as each of these three factors increases. That is,

**T = I**_{1}/I_{0} = 10^{-Εlc}

Putting this in terms of absorbance, the absorbance of a given wavelength of absorbed light __increases linearly__ as each of these three factors increases. That is,

**A = log**_{10} (1/T) = Εlc

Since spectrophotometry is most-commonly used to determine the presence and concentration of a particular substance in a sample, we are most interested in understanding the relationship between absorbance or transmittance and concentration. The two graphs below show the relationship between %T and C (right graph) and the relationship between A and C (left graph). The graphs show that %T decreases exponentially as concentration rises while A increases linearly as concentration rises. Understanding these relationships, we can easily use %T and/or A to determine the concentration of light-absorbing substance in a sample. Since the relationship between A and C is a relatively simple linear one, scientists usually use absorbance measurements when determining the concentration of a particular substance rather than %T.

Graphs show the relationship between %T and C (right graph) and the relationship between A and C (left graph).

Image Source: unknown