The **Beer-Lambert law (or Beer's law)** is
the linear relationship between absorbance and concentration of an absorbing
species. The general Beer-Lambert law is usually written as:

A = a() * b * c

where A is the measured absorbance, a() is a wavelength-dependent
absorptivity coefficient, **b** is the path
length, and **c **is the analyte concentration.
When working in concentration units of molarity, the *Beer-Lambert law*
is written as:

A = * b * c

where is the wavelength-dependent molar
absorptivity coefficient with units of M^{-1} cm^{-1}. Data
are frequently reported in percent transmission (I/I_{0} * 100) or in
absorbannce [A = log (I/I_{0})]. The latter is particularly convenient.
[common coefficients of near-ultraviolet absorption
bands of some amino acids and nucleotides]

Sometimes the extinction coefficient is given in other units; for example,

A = E

^{1%}* b * c

where the concentration C is in gram per 100 ml of solution. This useful when the molecular weight of the solute is unknown or uncertain.

**Q: Cytosine has a molar
extinction coefficient of 6*10 ^{3} at 270 nm at pH 7. Calculate the
absorbance and percent transmission of 1*10^{-4} and 1*10^{-3}
M cytosine solution in a 1-mm cell. [Solution....]**

**Q: A protein with extinction
coefficient E ^{1%} = 16 yields an absorbance of 0.73 when measured in
a 0.5-cm cell. Calculate the weight concentration. [Solution....]**

Experimental measurements are usually made in terms of **transmittance**
(T), which is defined as:

T = I / I

_{o}

where I is the** light intensity** after it passes through the sample and
I_{o} is the **initial light intensity**. The relation between A
and T is:

A = -log T = - log (I / I

_{o}).

*Absorption of light by a sample*

Modern absorption instruments can usually display the data as either transmittance,
%-transmittance, or absorbance. An unknown concentration of an analyte can be
determined by measuring the amount of light that a sample absorbs and applying
Beer's law. If the absorptivity coefficient is not known, the unknown concentration
can be determined using a working curve of absorbance versus concentration derived
from standards.

I_{o} is the intensity entering the sample at z=0, I_{z}
is the intensity entering the infinitesimal slab at z, dI is the intensity
absorbed in the slab, and I is the intensity of light leaving the sample.
Then, the total opaque area on the slab due to the absorbers is
* N * A * dz. Then, the fraction of photons absorbed will be
* N * A * dz / A so,

dI / I

_{z}= - * N * dz

Integrating this equation from z = 0 to z = b gives:

ln(I) - ln(I

_{o}) = - * N * bor - ln(I / I

_{o}) = * N * b.

Since N (molecules/cm^{3}) * (1 mole / 6.023x10^{23} molecules)
* 1000 cm^{3} / liter = c (moles/liter) and 2.303 * log(x) = ln(x) then

- log(I / I

_{o}) = * (6.023x10^{20}/ 2.303) * c * b- log(I / I

_{o}) = A = * b * c

where = *
(6.023x10^{20} / 2.303) = * 2.61x10^{20}

Typical cross-sections and molar absorptivities are:

(cm^{2}) (M^{-1}cm^{-1}) absorption - atoms 10^{-12}3x10^{8}molecules 10^{-16}3x10^{4}infrared 10^{-19}3x10 Raman scattering 10^{-29}3x10^{-9}

- deviations in absorptivity coefficients at
due to electrostatic interactions between molecules in close proximity**high concentrations (>0.01M)** -
**scattering of light due to particulates**in the sample -
**fluoresecence or phosphorescence**of the sample - changes in refractive index at high analyte concentration
- shifts in chemical equilibria as a function of concentration
- non-monochromatic radiation, deviations can be minimized by using a relatively flat part of the absorption spectrum such as the maximum of an absorption band
- stray light

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