# jee paper setting committee is fond of this cocnept though not mentioned in syllabus...example jee 2000,jee 2004 conatined problems based on these for

though not mentioned in syllabus..jee 2000,jee 2004 conatined problems based on these formulas

n optics, the Beerâ€“Lambert law, also known as Beer's law or the Lambertâ€“Beer law or the Beerâ€“Lambertâ€“Bouguer law relates the absorption of light to the properties of the material through which the light is traveling.
Contents

1 Equations
2 Derivation
3 Prerequisites
4 Chemical analysis
5 Beerâ€“Lambert law in the atmosphere
6 History
8 References

 Equations
Diagram of Beerâ€“Lambert absorption of a beam of light as it travels through a cuvette of width â„“.

The law states that there is a logarithmic dependence between the transmission (or transmissivity), T, of light through a substance and the product of the absorption coefficient of the substance, Î±, and the distance the light travels through the material (i.e. the path length), â„“. The absorption coefficient can, in turn, be written as a product of either a molar absorptivity (extinction coefficient) of the absorber, Îµ, and the concentration c of absorbing species in the material, or an absorption cross section, Ïƒ, and the (number) density N' of absorbers.

For liquids, these relations are usually written as:

T = {I\over I_{0}} = 10^{-\alpha\, \ell} = 10^{-\varepsilon\ell c}

whereas for gases, and in particular among physicists and for spectroscopy and spectrophotometry, they are normally written

T = {I\over I_{0}} = e^{-\alpha'\, l} = e^{-\sigma \ell N}

where I0 and I are the intensity (or power) of the incident light and the transmitted light, respectively; Ïƒ is cross section of light absorption by a single particle and N is the density (number per unit volume) of absorbing particles.

The base 10 and base e conventions must not be confused because they give different values for the absorption coefficient: \alpha\neq\alpha'. However, it is easy to convert one to the other, using \alpha' = \alpha \ln(10)\approx 2.303\alpha.

The transmission (or transmissivity) is expressed in terms of an absorbance which, for liquids, is defined as

A = -\log_{10} \left( \frac{I}{I_0} \right)

whereas, for gases, it is usually defined as

A' = -\ln \left( \frac{I}{I_0} \right).

This implies that the absorbance becomes linear with the concentration (or number density of absorbers) according to

A = \varepsilon \ell c = \alpha\ell \,

and

A' = \sigma \ell N = \alpha' \ell \,

for the two cases, respectively.

Thus, if the path length and the molar absorptivity (or the absorption cross section) are known and the absorbance is measured, the concentration of the substance (or the number density of absorbers) can be deduced.

Although several of the expressions above often are used as Beerâ€“Lambert law, the name should strictly speaking only be associated with the latter two. The reason is that historically, the Lambert law states that absorption is proportional to the light path length, whereas the Beer law states that absorption is proportional to the concentration of absorbing species in the material.[1]

If the concentration is expressed as a mole fraction i.e. a dimensionless fraction, the molar absorptivity (Îµ) takes the same dimension as the absorption coefficient, i.e. reciprocal length (e.g. mâˆ’1). However, if the concentration is expressed in moles per unit volume, the molar absorptivity (Îµ) is used in LÂ·molâˆ’1Â·cmâˆ’1, or sometimes in converted SI units of m2Â·molâˆ’1.

The absorption coefficient Î±' is one of many ways to describe the absorption of electromagnetic waves. For the others, and their interrelationships, see the article: Mathematical descriptions of opacity. For example, Î±' can be expressed in terms of the imaginary part of the refractive index, Îº, and the wavelength of the light (in free space), Î»0, according to

\alpha' = \frac{4 \pi \kappa}{\lambda_{0}}.

In molecular absorption spectrometry, the absorption cross section Ïƒ is expressed in terms of a linestrength, S, and an (area-normalized) lineshape function, Î¦. The frequency scale in molecular spectroscopy is often in cmâˆ’1, wherefore the lineshape function is expressed in units of 1/cmâˆ’1, which can look funny but is strictly correct. Since N is given as a number density in units of 1/cm3, the linestrength is often given in units of cm2cmâˆ’1/molecule. A typical linestrength in one of the vibrational overtone bands of smaller molecules, e.g. around 1.5 Î¼m in CO or CO2, is around 10âˆ’23 cm2cmâˆ’1, although it can be larger for species with strong transitions, e.g. C2H2. The linestrengths of various transitions can be found in large databases, e.g. HITRAN. The lineshape function often takes a value around a few 1/cmâˆ’1, up to around 10/cmâˆ’1 under low pressure conditions, when the transition is Doppler broadened, and below this under atmospheric pressure conditions, when the transition is collision broadened. It has also become commonplace to express the linestrength in units of cmâˆ’2/atm since then the concentration is given in terms of a pressure in units of atm. A typical linestrength is then often in the order of 10âˆ’3 cmâˆ’2/atm. Under these conditions, the detectability of a given technique is often quoted in terms of ppmâ€¢m.

The fact that there are two commensurate definitions of absorbance (in base 10 or e) implies that the absorbance and the absorption coefficient for the cases with gases, A' and Î±', are ln 10 (approximately 2.3) times as large as the corresponding values for liquids, i.e. A and Î±, respectively. Therefore, care must be taken when interpreting data that the correct form of the law is used.

The law tends to break down at very high concentrations, especially if the material is highly scattering. If the light is especially intense, nonlinear optical processes can also cause variances.
 Derivation

The derivation is quite simple in concept. There are many details, so think of this first paragraph as a conceptual overview. Divide the absorbing sample into thin slices that are perpendicular to the beam of light. The light that emerges from a slice is slightly less intense than the light that entered because some of the photons have run into molecules in the sample and did not make it to the other side. For most cases where measurements of absorption are needed, a vast majority of the light entering the slice leaves without being absorbed. Because the physical description of the problem is in terms of differences---intensity before and after light passes through the slice---we can easily write an ordinary differential equation model for absorption. The difference in intensity due to the slice of absorbing material dI is reduced; leaving the slice, it is a fraction Î² of the light entering the slice I. The thickness of the slice is dz, which scales the amount of absorption (thin slice does not absorb much light but a thick slice absorbs a lot). In symbols, dI = Î²Idz, or dI / dz = Î²I. This conceptual overview uses Î² to describe how much light is absorbed. All we can say about the value of this constant is that it will be different for each material. Also, its values should be constrained between -1 and 0. The following paragraphs cover the meaning of this constant and the whole derivation in much greater detail.

Assume that particles may be described as having an absorption cross section (i.e. area), Ïƒ, perpendicular to the path of light through a solution, such that a photon of light is absorbed if it strikes the particle, and is transmitted if it does not.

Define z as an axis parallel to the direction that photons of light are moving, and A and dz as the area and thickness (along the z axis) of a 3-dimensional slab of space through which light is passing. We assume that dz is sufficiently small that one particle in the slab cannot obscure another particle in the slab when viewed along the z direction. The concentration of particles in the slab is represented by N.

It follows that the fraction of photons absorbed when passing through this slab is equal to the total opaque area of the particles in the slab, ÏƒAN dz, divided by the area of the slab A, which yields ÏƒN dz. Expressing the number of photons absorbed by the slab as dIz, and the total number of photons incident on the slab as Iz, the fraction of photons absorbed by the slab is given by

dI_z = - \sigma N\,I_z\,dz .

Note that because there are fewer photons which pass through the slab than are incident on it, dIz is actually negative (It is proportional in magnitude to the number of photons absorbed).

The solution to this simple differential equation is obtained by integrating both sides to obtain Iz as a function of z

\ln(I_z) = - \sigma N z + C . \,

The difference of intensity for a slab of real thickness â„“ is I0 at z = 0, and Il at z = â„“. Using the previous equation, the difference in intensity can be written as,

\ln(I_l) - \ln(I_0) = (- \sigma \ell N + C) - ( - \sigma 0 N + C) = - \sigma \ell N \,

rearranging and exponentiating yields,

\ T = \frac{I_l}{I_0} = e ^ {- \sigma \ell N} = e ^ {- \alpha'\ell} .

This implies that

A' = - \ln\left( \frac{I_l}{I_0} \right) = \alpha' \ell = \sigma\ell N \,

and

A = - \log_{10}\left( \frac{I_l}{I_0} \right) = \frac{\alpha'\ell}{2.303} = \alpha \ell = \varepsilon \ell c. \,

The derivation assumes that every absorbing particle behaves independently with respect to the light and is not affected by other particles. Error is introduced when particles are lying along the same optical path such that some particles are in the shadow of others. This occurs in highly concentrated solutions. In practice, when large absorption values are measured, dilution is required to achieve accurate results. Measurements of absorption in the range of I1 / I0 = 0.1 to 1 are less affected by shadowing than other sources of random error. In this range, the ODE model developed above is a good approximation; measurements of absorption in this range are linearly related to concentration. At higher absorbances, concentrations will be underestimated due to this shadow effect unless one employs a more sophisticated model that describes the non-linear relationship between absorption and concentration.
 Prerequisites

There are at least six conditions that need to be fulfilled in order for Beerâ€™s law to be valid. These are:

The absorbers must act independently of each other;
The absorbing medium must be homogeneous in the interaction volume
The absorbing medium must not scatter the radiation - no turbidity;
The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium;
The incident radiation should preferably be monochromatic, or have at least a width that is more narrow than the absorbing transition; and
The incident flux must not influence the atoms or molecules; it should only act as a non-invasive probe of the species under study. In particular, this implies that the light should not cause optical saturation or optical pumping, since such effects will deplete the lower level and possibly give rise to stimulated emission.

If any of these conditions are not fulfilled, there will be deviations from Beerâ€™s law.
 Chemical analysis

Beer's law can be applied to the analysis of a mixture by spectrophotometry, without the need for extensive pre-processing of the sample. An example is the determination of bilirubin in blood plasma samples. The spectrum of pure bilirubin is known, so the molar absorbance is known. Measurements are made at one wavelength that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences.The concentration is given by c = Acorrected / Îµ.

For a more complicated example, consider a mixture in solution containing two components at concentrations c1 and c2. The absorbance at any wavelength, Î» is, for unit path length, given by

A(\lambda)=c_1\ \varepsilon_1(\lambda)+c_2\ \varepsilon_2(\lambda).

Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the concentrations c1 and c2 as long as the molar absorbances of the two components, Îµ1 and Îµ1 are known at both wavelengths. This two system equation can be solved using Cramer's rule. In practice it is better to use linear least squares to determine the two concentrations from measurements made at more than two wavelengths. Mixtures containing more than two components can be analysed in the same way, using a minimum of n wavelengths for a mixture containing n components.

The law is used widely in infra-red spectroscopy for analysis of polymer degradation and oxidation. The carbonyl group absorption at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated.
 Beerâ€“Lambert law in the atmosphere

This law is also applied to describe the attenuation of solar or stellar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The Beerâ€“Lambert law for the atmosphere is usually written

I = I_0\,\exp(-m(\tau_a+\tau_g+\tau_{\rm NO_2}+\tau_w+\tau_{\rm O_3}+\tau_r)),

where each Ï„x is the optical depth whose subscript identifies the source of the absorption or scattering it describes:

a refers to aerosols (that absorb and scatter)
g are uniformly mixed gases (mainly carbon dioxide (CO2) and molecular oxygen (O2) which only absorb)
NO2 is nitrogen dioxide, mainly due to urban pollution (absorption only)
w is water vapour absorption
O3 is ozone (absorption only)
r is Rayleigh scattering from molecular oxygen (O2) and nitrogen (N2) (responsible for the blue color of the sky).

m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of Î¸) to 1 / cos(Î¸), where Î¸ is the observed object's zenith angle (the angle measured from the direction perpendicular to the Earth's surface at the observation site).

This equation can be used to retrieve Ï„a, the aerosol optical thickness, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.

When the path taken by the light is through the atmosphere, the density of the absorbing gas is not constant, so the original equation must be modified as follows:

T = {I_{1}\over I_{0}} = e^{-\int\alpha'\, dz} = e^{-\sigma\int N dz}

where z is the distance along the path through the atmosphere, all other symbols are as defined above.[2] This is taken into account in each Ï„x in the atmospheric equation above.
 History

The law was discovered by Pierre Bouguer before 1729. It is often mis-attributed to Johann Heinrich Lambert, who cited Bouguer's Essai d'Optique sur la Gradation de la Lumiere (Claude Jombert, Paris, 1729) â€” and even quoted from it â€” in his Photometria in 1760. Much later, August Beer extended the exponential absorption law in 1852 to include the concentration of solutions in the absorption coefficient.

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Absorbance

"Optical density" redirects here. "Optical density" can also refer to index of refraction.[1]

In spectroscopy, the absorbance A (also called optical density) is defined as:[2]

A_\lambda = \log_{10}(I_0/I)\,,

where I is the intensity of light at a specified wavelength Î» that has passed through a sample (transmitted light intensity) and I0 is the intensity of the light before it enters the sample or incident light intensity (or power). Absorbance measurements are often carried out in analytical chemistry, since the absorbance of a sample is proportional to the thickness of the sample and the concentration of the absorbing species in the sample, in contrast to the transmittance I / I0 of a sample, which varies logarithmically with thickness and concentration.
Measuring absorbance

Absorptance[3] (not absorbance) is defined as: The ratio of the radiant flux absorbed by a body to that incident upon it. Also called [absorption] factor. Compare absorptivity. Total absorptance refers to absorptance measured over all wavelengths.Spectral absorptance refers to absorptance measured at a specified wavelength.

Absorptance is explained, as it relates to absorbance, on the Color and Vision Research Laboratories, Institute of Ophthalmology, UCL, in this way:

Absorbance spectra are typically used to define photopigment spectra because their shape, when normalized (i.e., plotted as a fraction of the maximum absorbance), is independent of pigment optical density (pigment concentration). In contrast, the absorptance spectra, like the spectral sensitivity of the human subject, broadens as the optical density increases.[4]

Outside the field of analytical chemistry, e.g. when used with the Tunable Diode Laser Absorption Spectroscopy (TDLAS) technique, the absorbance is often defined using the natural logarithm instead of the common logarithm, i.e. as

A_\lambda = \ln(I_0/I)\,,

See the Beer-Lambert law for a more complete discussion.
Contents

1 Explanation
2 Absorbance vs transmittance
3 Instrument measurement range
6 References

 Explanation

The term absorption refers to the physical process of absorbing light, while absorbance refers to the mathematical quantity. Also, absorbance does not always measure absorption: if a given sample is, for example, a dispersion, part of the incident light will in fact be scattered by the dispersed particles, and not really absorbed. However, in such cases, it is recommended that the term "attenuance" (formerly called "extinction") be used, which accounts for losses due to scattering and luminescence.[5]

Although absorbance does not have true units, it is quite often reported in "Absorbance Units" or AU (not to be confused with the Astronomical unit).
 Absorbance vs transmittance
Absorbanceâ†“ Transmittance (I / I0)â†“ Percent transmittance (100 * I / I0)â†“
0 1 100
0.1 0.79 79
0.25 0.56 56
0.5 0.32 32
0.75 0.18 18
0.9 0.13 13
1 0.1 10
2 0.01 1
3 0.001 0.1
 Instrument measurement range

Any real measuring instrument has a limited range over which it can accurately measure absorbance. An instrument must be calibrated and checked against known standards if the readings are to be trusted. Many instruments will become non-linear (fail to follow the Beer-Lambert law) starting at approximately 2 AU (~1% Transmission). It is also difficult to accurately measure very small absorbance values (below 10âˆ’4) with commercially available instruments for chemical analysis. In such cases, laser-based absorption techniques can be used, since they have demonstrated detection limits that supersede those obtained by conventional non-laser-based instruments by many orders of magnitude (detections have been demonstrated all the way down to 5 10âˆ’13). The theoretical best accuracy for most commercially available non-laser-based instruments is in the range near 1 AU. The path length or concentration should then, when possible, be adjusted to achieve readings near this range.

Some filters, notably welding glass, are rated by shade number, which is 7/3 times the absorbance plus one:[6]

shade number = \frac{7\left(- \log_{10} T\right)}{3} + 1
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49
Subhomoy Bakshi ·

why did u hide it dude???

un hide it!

And beer lambert law se jo question aaya tha..usko ekbar post karega plz??

1
swordfish ·

Unhide it!...only spoilers are hidden.