The Beer–Lambert law relates the absorption of light by a solution to the properties of the solution according to the following equation: A = εbc, where ε is the molar absorptivity of the absorbing species, b is the path length, and c is the concentration of the absorbing species.

Contents

- 1 What does B represent in Beer’s law?
- 2 What is the meaning of each term in the Beer-Lambert law?
- 3 What are the units of ε in beer’s law?
- 4 What is ε in chemistry?
- 5 Is Lambert’s law and Beer’s law the same?
- 6 What does Lambert law depend on?
- 7 Is Beer-Lambert law always linear?
- 8 What is the unit of absorbance?
- 9 What is the meaning of the y-intercept in beer’s law plot?

## What does B represent in Beer’s law?

What factors influence the absorbance that you would measure for a sample? Is each factor directly or inversely proportional to the absorbance? – One factor that influences the absorbance of a sample is the concentration (c). The expectation would be that, as the concentration goes up, more radiation is absorbed and the absorbance goes up.

- Therefore, the absorbance is directly proportional to the concentration.
- A second factor is the path length (b).
- The longer the path length, the more molecules there are in the path of the beam of radiation, therefore the absorbance goes up.
- Therefore, the path length is directly proportional to the concentration.

When the concentration is reported in moles/liter and the path length is reported in centimeters, the third factor is known as the molar absorptivity (\(\varepsilon\)). In some fields of work, it is more common to refer to this as the extinction coefficient.

When we use a spectroscopic method to measure the concentration of a sample, we select out a specific wavelength of radiation to shine on the sample. As you likely know from other experiences, a particular chemical species absorbs some wavelengths of radiation and not others. The molar absorptivity is a measure of how well the species absorbs the particular wavelength of radiation that is being shined on it.

The process of absorbance of electromagnetic radiation involves the excitation of a species from the ground state to a higher energy excited state. This process is described as an excitation transition, and excitation transitions have probabilities of occurrences.

- It is appropriate to talk about the degree to which possible energy transitions within a chemical species are allowed.
- Some transitions are more allowed, or more favorable, than others.
- Transitions that are highly favorable or highly allowed have high molar absorptivities.
- Transitions that are only slightly favorable or slightly allowed have low molar absorptivities.

The higher the molar absorptivity, the higher the absorbance. Therefore, the molar absorptivity is directly proportional to the absorbance. If we return to the experiment in which a spectrum (recording the absorbance as a function of wavelength) is recorded for a compound for the purpose of identification, the concentration and path length are constant at every wavelength of the spectrum.

The only difference is the molar absorptivities at the different wavelengths, so a spectrum represents a plot of the relative molar absorptivity of a species as a function of wavelength. Since the concentration, path length and molar absorptivity are all directly proportional to the absorbance, we can write the following equation, which is known as the Beer-Lambert law (often referred to as Beer’s Law), to show this relationship.

### Derivation of Beer Lambert Law

\ Note that Beer’s Law is the equation for a straight line with a y-intercept of zero.

### What are the symbols in the Beer-Lambert law equation?

Beer’s law, also called Lambert-Beer law or Beer-Lambert law, in spectroscopy, a relation concerning the absorption of radiant energy by an absorbing medium. Formulated by German mathematician and chemist August Beer in 1852, it states that the absorptive capacity of a dissolved substance is directly proportional to its concentration in a solution,

The relationship can be expressed as A = ε l c where A is absorbance, ε is the molar extinction coefficient (which depends on the nature of the chemical and the wavelength of the light used), l is the length of the path light must travel in the solution in centimetres, and c is the concentration of a given solution.

John P. Rafferty

## What is the meaning of each term in the Beer-Lambert law?

The amount of light absorbed by a solution is related to the analyte concentration by the Beer–Lambert law, which is expressed as follows: A = εbc, where ε is the molar absorptivity of the analyte, b is the path length (the distance the light travels through the solution), and c is the concentration of the analyte.

### What quantity is represented by ε in beer’s law?

A is the amount of light absorbed for a particular wavelength by the sample. ε is the molar extinction coefficient.

## What are the units of ε in beer’s law?

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Page ID 281609 When light passes through a solution that absorbs light, it enters the solution with an initial intensity (\(I_o\)) at a given wavelength, and it emerges with an intensity, \(I\). Figure \(\PageIndex \): Initial intensity (\(I_o\)) of light passes through a solution and emerges as intensity, \(I\). The path length is \(l\). (CC-BY; Kathryn Haas) The Beer-Lambert Law defines the relationship between absorbance at a given wavelength and the concentration of the solution.

\ The absorbance (A) is a unitless number because \(\frac } \) is unitless. The absorbance depends on the concentration (\(c\)) and the path length (\(l\)). The concentration of the sample solution is measured in molarity (M) and the length of the light path in centimeters (cm). The Greek letter epsilon (\(\varepsilon\)) in these equations is called the molar absorptivity (also called the molar absorption coefficient ).

The units of \(\varepsilon\) are \(\frac \) or \(L \times mol^ \times cm^ \). Chemists most often measure and report absorbed light in terms of wavelength (\(\lambda\)) in units of nanometers (nm). But the wavelength scale is inconvenient for measuring energy because it is inversely proportional to both frequency and energy.

### What is Lambert law mathematically?

7.10.2.1 Absorption coefficient – Lambert’s law of absorption states that equal parts in the same absorbing medium absorb equal fractions of the light that enters them. If in traversing a path of length dx the intensity is reduced from I to I – dI then Lambert’s law states that dI/I is the same for all elementary paths of length dx.

Thus Equation 7.8 may be obtained, where K is a constant known as the absorption coefficient. (7.8) d I I = − K d x Therefore Equation 7.9 follows, where C is a constant. (7.9) L o g I = − K x + C If I = I o at x = 0 then Equations 7.10 and 7.11 follow. (7.10) C = log I o (7.11) I = I o e − K x Note that in considering a medium of thickness x, I o is not the intensity of incident light due to there being some reflection at the first surface.

Similarly I is not the emergent intensity owing to reflection at the second surface. By measuring the emergent intensity for two different thicknesses the losses due to reflection may be eliminated. Read full chapter URL: https://www.sciencedirect.com/science/article/pii/B9780750611626500137

### How do you know if Beer-Lambert law is obeyed?

Laboratory Activity: Teacher Notes – Activity 1: Introduction to the Spectrophotometer Major Chemical Concept The quantity of light absorbed by a solution is commonly expressed either in terms of percent transmittance (%T), as in Part I of this activity, or in terms of absorbance (A).

Absorbance is defined as: A = log 10 I 0 I = log 10 1 T Transmittance (T) is usually expressed as %T = Tx100%, although it actually is defined as: T = I I 0 I 0 is the incident light and I is the transmitted light. Absorbance may be obtained from %T by the equation: A = log 10 (100) – log 10 (%T) = 2 – log 10 (%T) This last equation is used to convert %T to A.

The absorbance scale is a logarithmic scale. As absorbance increases, scale markings get closer together, and the meter becomes more difficult to read. Thus it is frequent practice to read %T and convert to A. For convenience a table showing %T and the corresponding absorbance (A) is included in the Appendix.

For quantitative work the absorbance is more useful than the transmittance because it is directly proportional to concentration. The relationship is known as the Beer-Lambert Law and is: A = abc where a (molar absorptivity or extinction coefficient) is a constant for a given absorbing chemical species, b is the radiation path length through the sample (measured in cm), and c is the stoichiometric molar concentration (mol/L) of the absorbing species.

Thus for a given absorbing species at a specific wavelength in a given sample holder of fixed path length (b), a plot of A versus concentration is a straight line if the Beer-Lambert Law is obeyed. To determine if the Beer-Lambert Law is obeyed over a given concentration range by a given species, measure absorbance as a function of concentration, using the same test-tube for all of the measurements.

- Plot absorbance vs.
- Concentration; check the linear nature of the curve.
- If the curve is linear, then the slope of the line may be calculated and used to determine the concentration by dividing the absorbance by the slope.
- An alternative method of determining concentration from the calibration curve is to mark off the measured absorbance on the ordinate, draw a line perpendicular to the ordinate until it intersects the curve, then drop a perpendicular line to the abscissa from the intersection point.

The intersection of the latter line and the abscissa gives the concentration. In practice, only the absorbance and concentration are variables and are in direct proportion, so A µ c, and A = k´C where k´ is the slope of the curve A vs.c. Level Part I of this activity can be used for all levels of students.

For basic students, it might be best for you to collect the data with the assistance of several students. For general students, groups of two or three should be able to collect the data for Part I, but Part II might be better done as a class activity. Honor students should be able to do the entire activity.

Expected Student Background This activity will probably be the first exposure students have with instrumentation. The experimental procedure is rather straightforward and should not pose great difficulty to students. By this point in the course, students should have the experience with routine laboratory procedures involving the use of glassware, how to clean glassware, and so forth.

- Time Teacher preparation time to mix the 0.020MCr(NO 3 ) 3 solution and set out the necessary equipment is about one hour.
- The spectrophotometers should be checked prior to use.
- The estimated time for students to perform the activity is50 min to complete Part I,
- Additional time up to 50 min may be required for Part II, particularly if students are required to prepare their own calibration curves.

Safety Read the Safety Considerations in the Student Version, Safety goggles should be worn during the activity. The chromium(III) nitrate solution is moderately toxic and should not be poured into the drain. Students are instructed to return the solution actually used in the measurements back into the original container.

- If there is some possibility that the solution supply will be contaminated, then an alternate disposal procedure should be devised.
- If it has not been contaminated, the solution may be saved for later use.
- There are several sources referred to in SourceBook that give recommended disposal procedures ( see Safety section ).

Local and state regulations must be followed. In view of the disposal problems, only the actual volume of solution needed with a modest excess should be prepared. Caution students to wash their hands with soap and water at the end of the activity Materials (For 24 students working in pairs) Nonconsumables 6 Spectronic 20 ” spectrophotometers (per class) 24 Small test-tubes (13-mmx100-mm) or 24 glass cuvettes 12 Plastic wash bottles Tissue paper (Kimwipes ” or equivalent) Buret, 50-mL Volumetric flasks, 50-mL and 100-mL (5 of each) or graduated cylinders, 50-mL and 100-mL Consumables 0.020M Chromium(III) nitrate solution, 250 mL (2.00gCr(NO 3 ) 3,9H 2 O per 250 mL solution) Distilled water Graph paper Advance Preparation You should perform the activity beforehand to be familiar with the procedure.

- Check the spectrophotometers to be certain they are operating correctly.
- Allow a 20 min warm-up period prior to use.
- To prepare the solution, use 2.00 g Cr(NO 3 ) 3 per 250 mL of solution.
- Weigh the solute carefully.
- The unknowns are prepared by successively diluting the 0.020M solution using 100-mL volumetric flasks to bring the volume Cr(NO 3 ) 3 in Column 3 to final volume (Column 6).

Use these solutions to prepare a calibration curve and student unknowns. Use a buret to measure all volumes. Have the 0.020M Cr(NO 3 ) 3 solution available in several plastic bottles, at least one per spectrophotometer. Make the dilutions according to the following table. Pre-Laboratory Discussion Demonstrate for students the techniques involved with using a spectrophotometer. In particular show them how to clean, fill, and place the test-tubes (or cuvettes) in the instrument. Show them how to set the percent transmittance to 0 and 100%.

A transparency master showing a front view of the Spectronic 20 ” with the controls identified is in the Appendix. Show students how to use the calibration curve to determine an unknown concentration. Students should have little difficulty in preparing a data table. For Part I a simple three column table is needed.

Part II needs a simple table to record the wavelength used, the percent transmittance recorded and the calculated absorbance. If necessary, help students design a data table. A useful activity for the pre-laboratory portion of the activity is described in Demonstration 5 in the Atomic Structure module.

- A beveled piece of white chalk is placed in a sample tube in the cell holder of the Spectronic 20.
- Looking “down the tube” allows one to see the color of the selected light radiation.
- It is suggested that students view the color every 25-50nm in the visible range.
- The monochromator that is part of the Welch ChemAnal ” System is another good way to show the colors of light used.

Mount the monochromator separately on the optical bench, and use a white piece of paper as a screen to view the colors corresponding with light selected every 25-50nm wavelength. One advantage of the ChemAnal ” monochromator is that the lid swings out of the way so that the “works” inside can be viewed directly while the instrument is being used, removing some of the “black box” aspects of the instrument.

Teacher-Student Interaction Once students begin the laboratory activity, circulate among them and watch that correct techniques are being used. Pertinent questions may be asked at this time. In particular students need to be aware that the test-tube/cuvette needs proper alignment in the sample holder, that 0 and 100% transmittance needs to be checked during usage at each new wavelength, and that care should be exercised in reading the instrument’s meter.

Anticipated Student Results The following table shows typical data obtained in Part I of the activity. The data show one transmittance maximum at 500nm, thus the graph of the data also show one peak at 500nm wavelength corresponding to an absorbance minimum. Answers to Implications and Applications

Two absorption maxima appear at the transmittance minima of about 415nm and 580nm. The absorption maxima correspond to violet and orange light (400-450nm and 600-650nm, respectively). On the basis of the answer to Question 2, a red colored solution would absorb a complementary color. In the case of Cr(NO 3 ) 3 the green color of the solution results because it absorbs its complementary magenta color and transmits green. Thus we expect that a red colored solution would transmit red light.Two uses of an absorption spectrum are to identify an unknown substance and to determine the concentration of a solution of a known substance. A sample substance in solution when placed in the path of light between the monochromator and detector of a spectrophotometer may transmit some, all, or none of light. Light absorption occurs at wavelengths at which the energy of light photons corresponds to the energy needed to excite electrons in atoms, ions, or molecules in the sample. Although the unaided human eye can detect subtle differences in the intensity of colored light, it is difficult to quantitatively measure the differences. Thus, the spectro-photometer is an extension of the sense of sight because it enables the measurement of color intensity to be done routinely and accurately.

Post-Laboratory Discussion During the post laboratory discussion, help students to graph A vs. l (nm). To measure the concentration of unknowns, absorbance maxima must be known in order to set the correct wavelength for the procedure. It would be useful to students to point out that the concentration of a substance is directly proportional to the area under the peaks in an absorbance spectrum.

For Part II, help students use the A vs. concentration graph. Have several copies and a transparency of the graph available. Use a hypothetical absorbance and the transparency to show how to read the concentration from the graph. Find the absorbance on the vertical axis and draw a line parallel to the concentration axis beginning at this point until it intersects the graph.

Then draw a line from the intersection parallel to the absorbance axis until it intersects the concentration axis. The intersection gives the concentration of the unknown.Figure 6. Calibration curve of absorbance vs. concentration for Cr(NO 3 ) 3 at 415 nm with example of unknown determination. Extensions

Students can be instructed in how to prepare their own calibration curve for this activity. This preparation would give them practice in quantitative subdilutions and calculation of dilution concentration. Each group of students could be assigned one subdilution, and data could be pooled. Other solutions lend themselves well to this kind of study. The absorbing species in the chromium(III) nitrate solution is Cr(H 2 O)6 3+ ion. When iron(III) chloride is dissolved in 0.1M HCl to form a 0.10M solution, the ion FeCl 4 -is formed. A 0.30M Cu(NO 3 ) 2 solution that is also 2.0M in NH 4 NO 3 forms the Cu(NH 3 ) 4 2+ ion. These solutions can be used for a study similar to the one in this activity.

Assessing Laboratory Learning

Laboratory technique may be assessed by noting how the experimentally determined concentration compares with the accepted value of the unknown concentration. Good technique would give a result within ±1%. A result within ±5% is quite acceptable and might be given a grade of “A.” A result within ±10% would be a “B,” and a result within ± 15% would be a “C.” Use a laboratory practical test to assess instruction. One possible test is to give students a solution of unknown concentration along with a calibration curve and the wavelength to use. Grade on the basis of how closely the student’s obtained value compares to the known concentration ofthe solution.

#### What are the three factors of Beer’s law?

From the formula for the absorbance, you can see that it is affected by three factors: the molar absorptivity of the solution, the path length which is the distance travelled by the light in the sample cell, and the concentration of the solution.

#### What is the unit of absorbance?

Absorbance is measured in absorbance units (Au), which relate to transmittance as seen in figure 1. For example, ~1.0Au is equal to 10% transmittance, ~2.0Au is equal to 1% transmittance, and so on in a logarithmic trend.

#### What is the symbol for absorbance?

The absorbance (symbol: A), usually the y axis of a uv spectrum, is defined as follows. If the sample absorbs no light, If the sample absorbs light, Thus, the greater the amount of the light absorbed by the sample, the larger the absorbance. See also Beer-Lambert Law, OCHEMPAL IS NOW IN THE FORM OF A BOOK Title: The Elements of Organic Chemistry; Subtitle: A Compendium of Terminology, Definitions, and Concepts for the Beginner.

## What is ε in chemistry?

In chemistry, the molar absorption coefficient or molar attenuation coefficient (ε) is a measurement of how strongly a chemical species absorbs, and thereby attenuates, light at a given wavelength. It is an intrinsic property of the species.

### What unit is ε in spectrophotometry?

As a result, ϵ has the units: L·mol – 1 ·cm – 1. The path length is measured in centimeters.

#### How do you calculate ε in chemistry?

Molar absorptivity Decadic absorbance divided by the path-length l and mole concentration c, of the absorbing material. ε = A 10 / cl. The molar absorptivity is a Beer-Lambert absorption coefficient. SI unit: m 2 mol – 1.’

#### What is the linear equation for Beer’s law?

(Image courtesy of Vernier Software & Technology). The equation for Beer’s law is a straight line with the general form of y = mx +b. where the slope, m, is equal to εl.

## Is Lambert’s law and Beer’s law the same?

Difference Between Beer’s Law and Lambert’s Law

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Beer’s law states that the amount of absorbed light is proportional to the solution concentration, whereas Lambert’s law states that the absorbance and path length are directly linked. Beer’s law and Lambert’s law are frequently used in combination with the Beer-Lambert law because they can show the connection between absorbance and both the path length of light inside the sample and the sample concentration.

## What does Lambert law depend on?

3.1 ABSORPTION, REFLECTION, AND REFRACTION – Section 2.2 includes discussion of two fundamental laws Grotthus-Draper’s principle and the second law of photochemistry, and also gives the details of Beer-Lambert’s law. These are fundamental principles for this discussion.

According the Beer-Lambert law (2.5), absorption of radiation depends on: • intensity of the incident beam • path length • concentration of absorbing species (chromophores) • extinction coefficient Figures 2.3, and 2.4 show, in addition, that the Beer-Lambert law is designed for monochromatic light and its absorption increases with decrease in radiation wavelength.

Finally, equation 2.6 gives the method of calculation of combined intensity of radiation of polychromic radiation, which is the usual case of exposure of real samples. This is certainly a good starting point, which can be further developed to answer pertinent questions.

This part of the mechanism can be described simply, as follows: “Potential stabilizing materials are expected to reflect, absorb, or refract UV radiation without emission of radiation wavelengths, which may be harmful to the protected materials.” This raises many practical issues, as follows: • effect of material mixtures • cross-section of absorption • effect of dispersion • action of organic absorbers • action of inorganic particulates • attenuation of radiation throughout cross-section of sample • surface ablation • effect of particle size • conditions of reflection • conditions of refraction • effect of refracted and absorbed radiation (see the next section) The above topics describe practical aspects of mechanisms of absorption, reflection, and refraction, and they are discussed below in the above order.

UV absorber or screener does not exist in polymeric material alone but it is dispersed within the matrix of the material to be protected. It is therefore pertinent that there is a competition for incoming radiation between UV absorber and other components of the mixture under study.

A | absorbance at particular wavelength |

a m | absorptivity of matrix |

a a | absorptivity of absorber |

c m | concentration of matrix |

c a | concentration of absorber |

b | thickness of measured sample |

This equation helps us to realize that both matrix and absorber compete for absorption of radiation. If we assume that A and b are unities and that the absorptivity of the UV absorber is 100 times higher than that of the matrix, then both matrix and absorber will absorb almost the same amounts of radiation if absorber concentration in the matrix is 1%.

This shows that we need absorbers having much higher absorptivities than that of matrix, but there will always be some residual radiation which will be absorbed by the matrix. This is the reason that matrix cannot be completely protected by UV absorbers or UV screeners added to the matrix. Figure 3.1 shows that absorbance increases with increased addition of carbon nanotubes, because they play a role of radiation screener.

It should be noted that the absorbance of nanotubes at concentration of 0.08 wt% (the highest concentration on the graph) was only about 10 times larger than absorbance of polymer. Comparing data from Figure 3.1 with the above example of absorptivities of matrix and absorber, SWNT in this example had absorptivity 12,500 times larger than polyimide, but polyimide containing 1 wt% of SWNT (it would be very large concentration of screener) still absorbs about 1% of incoming radiation. Figure 3.1, Absorbance of polyimide film containing different concentrations of single-walled carbon nanotubes. Copyright © 2005 This analysis also shows that the use of UV absorbers and screeners gives limited protection to polymers, especially those having strong chromophoric groups (strong absorption in the UV range).

a | absorbance |

λ | wavelength |

i | index for wavelength |

j | index for the number of components |

k | index for molar extinction coefficient |

p | index for number of components |

ɛ | molar extinction coefficient |

l | sample thickness (or pathway length) |

c | concentration of components |

This equation produces a sequence of equations for various wavelengths. These are fundamental equations used in so-called chemometrics, which is a subdiscipline of chemistry involved in the application of statistical and mathematical methods to problem solving in chemistry (in this case, helping to collect maximum information from optical data in application to photophysics).2 Absorption cross-section is a useful term because it helps to relate radiation intensity and absorption to the concentration of molecules: 3 σ ( λ ) = ln 1 C where:

σ(λ) | absorption cross-section in cm 2 per molecule at a given wavelength, λ |

I 0 | incoming radiation |

I | transmitted radiation |

l | optical path |

C | concentration in molecule cm −3 |

Absorption cross-section is very useful in comparison of data from different experiments because it helps to normalize conditions of experiments. Data give a very good understanding of the effects of different wavelengths on a particular compound. One of the expectations of photochemical studies is that the light intensity corresponds to the photochemical change.

Figure 3.2 shows results of an experiment in which hydrogen peroxide concentration and UV radiation intensity varied and their effect on kinetics of degradation of methyl tert-butyl ether was studied. In this simple experiment, a good linear relationship was obtained between supplied energy and rate of reaction, even though that concentration of hydroperoxide also varied.

In more complex studies of materials containing a mixture of various products (especially polymers), there is always a danger that increased intensity (above solar radiation) may change reaction kinetics and mechanisms. It is therefore always important to use a similar experiment to the one presented in Figure 3.2 to check the validity of the experiment. Figure 3.2, Rate constant of reaction between radicals formed from peroxide and methyl tert-butyl ether at different average light intensities. Copyright © 2005 Figure 3.4, Absorbance vs. relative dispersion index of SWNT in DMF. Carbon nanotubes are good UV radiation screeners, it was thus surprising to find that an increase in their concentration caused reduction of absorbance ( Figure 3.3 ). Further analysis of the phenomenon indicated that this reduction was caused by problems with their dispersion ( Figure 3.3 ). Figure 3.3, Absorbance vs. SWNT concentration in DMF. Copyright © 2007 This indicates that in stabilization processes, good distribution of stabilizer has a very strong influence on performance. In the case of organic absorbers, dispersion primarily depends on compatibility between stabilizer and other components of formulation, but it also depends on technological processes of dispersion.

Considering that stabilizers are used in small quantities, predispersion is always advisable. Good dispersion of inorganic stabilizers is even more difficult to achieve because it is complicated by properties of inorganic stabilizer (agglomerate formation, crystallinity, hardness, particle size, etc.), compatibility issues (e.g., acid/base interaction, polarity, etc.), and process conditions (intensity of mixing, mixing schedule, etc.).6 In both cases, a sound process has to be developed to maximize the effect of stabilizer addition.

Transmission of UV radiation through the sample is affected by absorption. Several quantities can be determined to evaluate optical density of material with and without stabilizer. These include: The mean free path represents the average distance between two successive interactions of photons in which the intensity of the incident photon beam is reduced by the factor of 1/e.

μ | linear attenuation coefficient |

The following relation represents the half-value thickness in which the intensity of the primary photon beam is reduced by half: 7 HVT = ln 2 μ During the processes of radiation passing through the material, stabilizer may be partially rendered inactive (see more on this subject in Chapter 5 ) and matrix laden with degradation products, which frequently change matrix absorption and vulnerabilities to UV exposure.

α | effective absorption/extinction coefficient |

F | flux of incoming radiation |

F T | threshold fluence |

Such processes are frequently observed when radiation fluence is too extensive for material to be able to prevent extensive damage (e.g., laser ablation). The effect of organic absorber can be predicted from equation 3.1 but the effects of screener (inorganic particles) are more difficult to predict because they depend not only on particle size and other physical properties of screener but also on the ability to disperse agglomerates.

σ | specific attenuation cross-section |

r | particle radius |

m | complex refractive index, which is function of wavelength, λ |

X | size parameter, X = 2πr/λ |

n(r) | size distribution |

ρ | particle density |

If we take titanium dioxide as an example of the effect of particle size on wavelength absorption, we will observe the following: 6 The particle size has an important influence on the performance of titanium dioxide, both as a pigment and as a UV screener (absorber).

For the pigment to have maximum opacity, the particle diameter must be equal to half of the wavelength (for a blue/green light to which the eye is most sensitive, the average wavelength is 460 nm, thus a particle diameter of 230 nm gives the maximum opacity). The color of the matrix (binder) has an influence here as well, and titanium dioxide must compensate.

For this reason, some grades of titanium dioxide are tailored to specific conditions and some are used to eliminate a yellow undertone. This is done by the choice of particle size. For this reason, commercial grades have particle sizes in a range from 200 to 300 nm.

- The amount of titanium dioxide is also crucial.
- If too little titanium dioxide is added, the distance between particles is too large and there is not enough opacity.
- If the amount is too great, it results in lower efficiency due to a particle crowding effect which causes particles to interfere with each other scattering efficiency.

Because the optimum light scattering of titanium pigments occurs when particle diameter is 0.23 μm, most pigments are manufactured to have the majority of particles closest to that in a range from 0.15 to 0.3 μm, depending on the application and the undertone required.6 Ultrafine grades are the exception.

They typically have particle sizes in a range from 0.015 to 0.035 μm and, because of their small particle size, they are transparent to visible light but absorb in the UV range.6 The best grades for sunscreens have particle diameter of 10 nm. At this particle size, they produce transparent looking sunscreens with excellent UV absorption qualities.

Figure 3.5 shows that the UV absorbance of ZnO particles increases with increasing size in the size range of 15–40 nm.10 The particles greater than 70 nm become opaque to UV radiation, whereas for particles greater than 70 nm the absorbance decreases with increasing size because of the decrease in particle density with increasing particle size.10 The reduction of particle size to less than 40 nm has a detrimental effect on the UVA/UVB absorbance ratio.10 Figure 3.5, Absorbance of UV radiation at 290 nm vs. particle size of ZnO. Copyright © 2014 Both organic and inorganic absorbers are able to absorb energy. The fate of this energy is discussed in the next section. Inorganic particles may also reflect and refract incoming radiation.

- Reflection of radiation which occurs on the material surface is the most desired outcome because energy is reflected into the surrounding space and therefore it does not affect material.
- If energy is reflected internally from the surface of an inorganic particle into, for example, a polymeric matrix, then this energy can be utilized for photochemical processes because light reflection does not affect its energy.

Refraction occurs when a light wave travels from a medium having a given refractive index to a medium with another refractive index at an angle. At the boundary between the media, the wave’s phase velocity is altered, usually causing a change in direction.

θ 1, θ 2 | angles of incidence and refraction |

n 1, n 2 | indices of refraction |

Refracted radiation retains some energy but the energy and wavelength of refracted radiation is different than that of incident radiation and inversely proportional to the ratio of refraction indices: λ 1 λ 2 = n 2 n 1 where:

λ 1, λ 2 | incoming and outgoing wavelength of radiation |

Read full chapter URL: https://www.sciencedirect.com/science/article/pii/B978189519886750005X

## Is Beer-Lambert law always linear?

Abstract – Beer’s law assumes a strictly linear dependence of the absorbance from concentration. Usually, chemical interactions and instrumental imperfection are made responsible for experimental deviations from this linearity. In this contribution we show that even in the absence of such interactions and instrumental errors, absorbance should be only approximately proportional to concentration.

- This can be derived from the quadratic dependence of the complex refractive index, and, by that, of the molar attenuation coefficient, from the dielectric constant and its frequency dispersion.
- Following dispersion theory, it is the variation of the real and the imaginary part of the dielectric function that depends linearly on concentration in the absence of interactions between the oscillators.

We show that this linear correlation translates into a linear dependence of the absorbance for low concentrations or molar oscillator strengths based on an approximation provided by Lorentz in 1906. Accordingly, Beer’s law can be derived from dispersion theory.

## What is the unit of absorbance?

Absorbance is measured in absorbance units (Au), which relate to transmittance as seen in figure 1. For example, ~1.0Au is equal to 10% transmittance, ~2.0Au is equal to 1% transmittance, and so on in a logarithmic trend.

## What is the meaning of the y-intercept in beer’s law plot?

Answer and Explanation: The calibration curve obtained from standard samples is used determine the concentration of unknown solutions. The y-intercept of the Beer’s law must always pass through the origin (0,0), i.e. a blank sample will have zero absorbance.

### How do you find unknown concentration using beer’s law?

The equation for Beer’s law is a straight line with the general form of y = mx +b. where the slope, m, is equal to εl. In this case, use the absorbance found for your unknown, along with the slope of your best fit line, to determine c, the concentration of the unknown solution.

#### What is a positive deviation in beer’s law?

What factors contribute to deviations from the Beer- Lambert’s Absorbance law? Deviations from Beer- Lambert’s law Beer–Lambert’s Absorbance law is a universally accepted relationship which helps calculation of concentration of an absorbing species from measured absorbance values. Under ideal conditions absorbance versus concentration plot is a straight line passing through the origin.