Beer-Lambert Law- Definition, Derivation, and Limitations

The Beer-Lambert law, also known as the Beer-Lambert-Bouguer law or simply Beer`s law, is a fundamental relation in spectroscopy that relates the absorption of light by a solution to the properties of the solution and the light. The law states that the logarithm of the ratio of the intensity of the incident light (Io) to the intensity of the transmitted light (I) is proportional to the product of the path length of the absorbing medium (x), the concentration of the absorbing solution (c), and a constant called the molar absorptivity or extinction coefficient (ε) of the solute at a given wavelength. Mathematically, the law can be expressed as:

$$\log \frac{I_0}{I} = \epsilon c x$$

The Beer-Lambert law is useful for determining the concentration of a solute in a solution from its absorbance, or vice versa, if the molar absorptivity and the path length are known. The law can also be used to compare the absorption spectra of different compounds and to identify unknown substances by their characteristic absorption peaks.

The Beer-Lambert law is based on two assumptions: that the light is monochromatic, meaning that it has a single wavelength, and that the solution is homogeneous, meaning that it has a uniform concentration and composition throughout. The law also assumes that there is no scattering or reflection of light by the solution or the container, and that the solute molecules do not interact with each other or change their chemical state due to light absorption.

The Beer-Lambert law was derived independently by several scientists in the 18th and 19th centuries, including Pierre Bouguer, Johann Heinrich Lambert, August Beer, and Jacques Babinet. The law is named after Beer and Lambert, who published their results in 1852 and 1760, respectively. The law is also sometimes attributed to Bouguer, who was the first to measure the attenuation of light by atmospheric air in 1729.

The Beer-Lambert law has many applications in various fields of science and engineering, such as analytical chemistry, biochemistry, physics, astronomy, meteorology, environmental science, and medicine. The law is commonly used to measure the concentration of solutes in solutions by using spectrophotometers, which are instruments that measure the intensity of light at different wavelengths. The law can also be used to study the structure and function of molecules by analyzing how they absorb light of different frequencies. For example, the Beer-Lambert law can be used to determine the amount of oxygen in blood by measuring its absorption of red and infrared light. The law can also be used to estimate the amount of pollutants in air or water by measuring their absorption of ultraviolet or visible light.

The Beer-Lambert law relates the intensity of light transmitted through a solution to the concentration and path length of the absorbing solute. To derive this law, we need to combine two empirical laws: Lambert`s law and Beer`s law.

Lambert`s law states that the decrease in intensity of light with thickness of the absorbing medium at any point is directly proportional to the intensity of light. Mathematically, it can be expressed as:

$$-\frac{dI}{dx} \propto I$$

Where $I$ is the intensity of light at a distance $x$ from the source, and $-\frac{dI}{dx}$ is the rate of decrease of intensity with thickness $dx$. The negative sign indicates that the intensity decreases as $x$ increases. The proportionality constant is called the absorption coefficient and is denoted by $a$. Therefore, we can write:

$$-\frac{dI}{dx} = aI$$

This is a differential equation that can be solved by integration. Rearranging and integrating both sides, we get:

$$\int_{I_0}^{I} \frac{dI}{I} = -a \int_{0}^{x} dx$$

Where $I_0$ is the initial intensity of light at $x=0$, and $I$ is the final intensity of light at $x=l$, where $l$ is the path length of the solution. Solving the integrals, we get:

$$\ln \frac{I}{I_0} = -al$$

Taking the exponential of both sides, we get:

$$\frac{I}{I_0} = e^{-al}$$

This equation shows that the ratio of transmitted intensity to incident intensity depends on the absorption coefficient and the path length of the medium.

Beer`s law states that the decrease in intensity of light with thickness of the absorbing medium at any point is directly proportional to the concentration of the absorbing solute. Mathematically, it can be expressed as:

$$-\frac{dI}{dx} \propto c$$

Where $c$ is the concentration of the solute in mol/L. The proportionality constant is called the molar absorption coefficient and is denoted by $b$. Therefore, we can write:

$$-\frac{dI}{dx} = bc$$

Combining Lambert`s law and Beer`s law, we get:

$$-\frac{dI}{dx} = abcI$$

This equation shows that the rate of decrease of intensity depends on the product of three factors: absorption coefficient, molar absorption coefficient, and concentration. We can define a new constant called the molar extinction coefficient and denote it by $\epsilon$, such that:

$$\epsilon = ab$$

Therefore, we can write:

$$-\frac{dI}{dx} = \epsilon c I$$

This is another differential equation that can be solved by integration. Rearranging and integrating both sides, we get:

$$\int_{I_0}^{I} \frac{dI}{I} = -\epsilon c \int_{0}^{x} dx$$

Solving the integrals, we get:

$$\ln \frac{I}{I_0} = -\epsilon c l$$

Taking the exponential of both sides, we get:

$$\frac{I}{I_0} = e^{-\epsilon c l}$$

This equation shows that the ratio of transmitted intensity to incident intensity depends on the molar extinction coefficient, concentration, and path length of the solution.

To make this equation more convenient to use, we can take the logarithm to base 10 of both sides and multiply by -1. This gives us:

$$-\log \frac{I}{I_0} = \epsilon c l \log e$$

Using the property that $\log e = 2.303$, we get:

$$-\log \frac{I}{I_0} = 2.303 \epsilon c l $$

This equation is known as the Beer-Lambert law. It relates the absorbance of a solution to its concentration and path length. The absorbance is defined as:

$$A = -\log \frac{I}{I_0} $$

The Beer-Lambert law can also be written as:

$$A = 2.303 \epsilon c l $$

This equation shows that absorbance is directly proportional to concentration and path length for a given wavelength of light and a given solute.

The Beer-Lambert law is useful for determining the concentration of a solute in a solution by measuring its absorbance at a known wavelength and path length, or vice versa. It is also useful for studying the interaction of light with matter and the properties of absorbing solutes.

The equation log I/Io = - ∈ × 𝑐 × 𝑥 is the mathematical expression of the Beer-Lambert law, which relates the absorbance of light by a solution to the concentration and path length of the solution. Let us break down each term in the equation and explain what they mean.

• log I/Io is the logarithm of the ratio of the intensity of transmitted light (I) to the intensity of incident light (Io). This ratio is also called the transmittance (T) of the solution. The logarithm is taken to convert the exponential relationship between I and Io into a linear one. The negative sign indicates that as the concentration or path length increases, the transmittance decreases.
• ∈ is the molar extinction coefficient, which is a constant that depends on the nature of the solute and the wavelength of the light. It measures how strongly the solute absorbs light at a given wavelength. It has units of L/mol/cm.
• 𝑐 is the concentration of the solute in mol/L. It represents how much solute is present in a given volume of solution. The higher the concentration, the more light is absorbed by the solution.
• 𝑥 is the path length of the solution in cm. It represents how far the light travels through the solution. The longer the path length, the more light is absorbed by the solution.

The equation log I/Io = - ∈ × 𝑐 × 𝑥 can be used to calculate any of these variables if the others are known. For example, if we know the molar extinction coefficient, the concentration and the path length of a solution, we can use the equation to find out how much light is transmitted through it. Alternatively, if we know how much light is transmitted through a solution of known path length and wavelength, we can use the equation to find out its concentration or molar extinction coefficient. This is how spectrophotometers work: they measure the transmittance or absorbance of a solution and use the Beer-Lambert law to determine its concentration or molar extinction coefficient.

The equation log I/Io = - ∈ × 𝑐 × 𝑥 assumes that there are no other factors that affect the absorption of light by the solution, such as scattering, fluorescence, chemical equilibrium or non-monochromatic radiation. These factors can cause deviations from linearity and limit the applicability of the Beer-Lambert law in some cases. We will discuss these limitations in point 4.

The Beer-Lambert law assumes that the light intensity decreases exponentially as it passes through a solution of constant concentration and path length. However, this assumption may not always be valid in real situations. There are several factors that can cause deviations from the linear relationship between absorbance and concentration. Some of these factors are:

• High concentration: When the concentration of the absorbing solution is too high (usually above 0.01 M), the molecules may interact with each other electrostatically and change their absorption properties. This can result in a change in the molar extinction coefficient or a non-uniform distribution of the molecules along the path length. To avoid this problem, the solution should be diluted to a low concentration before measuring its absorbance.
• Scattering: When the solution contains particles that are comparable in size to the wavelength of light, some of the light may be scattered instead of absorbed. This can reduce the intensity of the transmitted light and increase the apparent absorbance. Scattering can also occur due to bubbles, dust, or impurities in the solution. To avoid this problem, the solution should be filtered or centrifuged to remove any particulates before measuring its absorbance.
• Fluorescence or phosphorescence: When the solution contains molecules that can emit light after absorbing it, some of the absorbed light may be re-emitted in a different wavelength. This can interfere with the measurement of absorbance and cause errors. Fluorescence or phosphorescence can also occur due to impurities or contaminants in the solution. To avoid this problem, the solution should be purified or protected from light before measuring its absorbance.
• Refractive index: When the concentration of the absorbing solution is high, it can change the refractive index of the solution and affect the angle of incidence and refraction of light. This can alter the path length of light and cause deviations from the Beer-Lambert law. To avoid this problem, the refractive index of the solution should be measured and corrected for before measuring its absorbance.
• Chemical equilibrium: When the absorbing solution contains molecules that can undergo chemical reactions or changes in their molecular structure as a function of concentration, temperature, pH, or other factors, their absorption properties may change accordingly. This can result in a shift in the absorption spectrum or a change in the molar extinction coefficient. To avoid this problem, the chemical equilibrium of the solution should be maintained or controlled before measuring its absorbance.
• Non-monochromatic radiation: When the light source used for measuring absorbance is not monochromatic (i.e., it contains more than one wavelength), some wavelengths may be absorbed more than others by the solution. This can result in a non-linear relationship between absorbance and concentration. To avoid this problem, a monochromatic light source or a narrow-band filter should be used for measuring absorbance.

These are some of the common limitations of the Beer-Lambert law that should be considered when applying it to measure the concentration of an absorbing solution. By taking appropriate precautions and corrections, these limitations can be minimized or overcome to obtain accurate results.

In this article, we have learned about the Beer-Lambert law, which describes the relationship between the absorbance of light by a solution and the concentration and path length of the absorbing medium. We have also derived the equation log I/Io = - ∈ × 𝑐 × 𝑥 from the laws of Lambert and Beer, and explained the meaning of each term in the equation. Finally, we have discussed some of the limitations of the Beer-Lambert law, such as non-linearity at high concentrations, scattering of light, fluorescence or phosphorescence, changes in refractive index, shifts in chemical equilibria, and stray light. We have seen that the Beer-Lambert law is a useful tool for quantitative analysis of solutions, but it also has some assumptions and conditions that need to be met for its validity. We hope you have enjoyed reading this article and learned something new about the Beer-Lambert law. 😊

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