## Thursday, December 31, 2015

### Beer–Lambert law

An example of Beer–Lambert law: green laser light in a solution of Rhodamine 6B. The beam radiant power becomes weaker as it passes through solution
The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is traveling. The law is commonly applied to chemical analysis measurements and used in understanding attenuation in physical optics, for photons, neutrons or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

## History

The law was discovered by Pierre Bouguer before 1729.[1] It is often attributed to Johann Heinrich Lambert, who cited Bouguer's Essai d'optique sur la gradation de la lumière (Claude Jombert, Paris, 1729)—and even quoted from it—in his Photometria in 1760.[2] Lambert's law stated that absorbance of a material sample is directly proportional to its thickness (path length). Much later, August Beer discovered another attenuation relation in 1852. Beer's law stated that absorbance is proportional to the concentrations of the attenuating species in the material sample in 1852.[3] The modern derivation of the Beer–Lambert law combines the two laws and correlates the absorbance to both the concentrations of the attenuating species as well as the thickness of the material sample.[4]

## Beer–Lambert law

By definition, the transmittance of material sample is related to its optical depth τ and to its absorbance A as
$T={\frac {\Phi _{\mathrm {e} }^{\mathrm {t} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}}=e^{-\tau }=10^{-A},$
where
• Φet is the radiant flux transmitted by that material sample;
The Beer–Lambert law states that, for N attenuating species in the material sample,
$T=e^{-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}=10^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z},$
or equivalently that
$\tau =\sum _{i=1}^{N}\tau _{i}=\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\,\mathrm {d} z,$
$A=\sum _{i=1}^{N}A_{i}=\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\,\mathrm {d} z,$
where
Attenuation cross section and molar attenuation coefficient are related by
$\varepsilon _{i}={\frac {\mathrm {N_{A}} }{\ln {10}}}\,\sigma _{i},$
and number density and amount concentration by
$c_{i}={\frac {n_{i}}{\mathrm {N_{A}} }},$
where NA is the Avogadro constant.
In case of uniform attenuation, these relations become[5]
$T=e^{-\sum _{i=1}^{N}\sigma _{i}n_{i}\ell }=10^{-\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell },$
or equivalently
$\tau =\sum _{i=1}^{N}\sigma _{i}n_{i}\ell ,$
$A=\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell .$
Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.

The law tends to break down at very high concentrations, especially if the material is highly scattering. If the radiation is especially intense, nonlinear optical processes can also cause variances. The main reason, however, is the following. At high concentrations, the molecules are closer to each other and begin to interact with each other. This interaction will change several properties of the molecule, and thus will change the attenuation.

### Expression with attenuation coefficient

The Beer–Lambert law can be expressed in terms of attenuation coefficient, but in this case is better called Lambert's law since amount concentration, from Beer's law, is hidden inside the attenuation coefficient. The (Napierian) attenuation coefficient μ and the decadic attenuation coefficient μ10 = μ/ln 10 of a material sample are related to its number densities and amount concentrations as
$\mu (z)=\sum _{i=1}^{N}\mu _{i}(z)=\sum _{i=1}^{N}\sigma _{i}n_{i}(z),$
$\mu _{10}(z)=\sum _{i=1}^{N}\mu _{10,i}(z)=\sum _{i=1}^{N}\varepsilon _{i}c_{i}(z)$
respectively, by definition of attenuation cross section and molar attenuation coefficient. Then the Beer–Lambert law becomes
$T=e^{-\int _{0}^{\ell }\mu (z)\mathrm {d} z}=10^{-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z},$
and
$\tau =\int _{0}^{\ell }\mu (z)\,\mathrm {d} z,$
$A=\int _{0}^{\ell }\mu _{10}(z)\,\mathrm {d} z.$
In case of uniform attenuation, these relations become
$T=e^{-\mu \ell }=10^{-\mu _{10}\ell },$
or equivalently
$\tau =\mu \ell ,$
$A=\mu _{10}\ell .$

## Derivation

In concept, the derivation of the Beer–Lambert law is straightforward. Assume that a beam of light enters a material sample. Define z as an axis parallel to the direction of the beam. Divide the material sample into thin slices, perpendicular to the beam of light, with thickness dz sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the z direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by e(z) = −μ(ze(z) dz, where μ is the (Napierian) attenuation coefficient, which yields the following first-order linear ODE:
${\frac {\mathrm {d} \Phi _{\mathrm {e} }}{\mathrm {d} z}}(z)=-\mu (z)\Phi _{\mathrm {e} }(z).$
The attenuation is caused by the photons that did not make it to the other side of the slice because of scattering or absorption. The solution to this differential equation is obtained by multiplying the integrating factor
$e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}$
throughout to obtain
${\frac {\mathrm {d} \Phi _{\mathrm {e} }}{\mathrm {d} z}}(z)\,e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}+\mu (z)\Phi _{\mathrm {e} }(z)\,e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}=0,$
which simplifies due to the product rule (applied backwards) to
${\frac {\mathrm {d} }{\mathrm {d} z}}{\bigl (}\Phi _{\mathrm {e} }(z)\,e^{\int _{0}^{z}\mu (z')\mathrm {d} z'}{\bigr )}=0.$
Integrating both sides and solving for Φe for a material of real thickness , with the incident radiant flux upon the slice Φei = Φe(0) and the transmitted radiant flux Φet = Φe( ) gives
$\Phi _{\mathrm {e} }^{\mathrm {t} }=\Phi _{\mathrm {e} }^{\mathrm {i} }\,e^{-\int _{0}^{\ell }\mu (z)\mathrm {d} z},$
and finally
$T={\frac {\Phi _{\mathrm {e} }^{\mathrm {t} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}}=e^{-\int _{0}^{\ell }\mu (z)\mathrm {d} z}.$
Since the decadic attenuation coefficient μ10 is related to the (Napierian) attenuation coefficient by μ10 = μ/ln 10, one also have
$T=e^{-\int _{0}^{\ell }\ln {10}\,\mu _{10}(z)\mathrm {d} z}={\bigl (}e^{-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z}{\bigr )}^{\ln {10}}=10^{-\int _{0}^{\ell }\mu _{10}(z)\mathrm {d} z}.$
To describe the attenuation coefficient in a way independent of the number densities ni of the N attenuating species of the material sample, one introduces the attenuation cross section σi = μi(z)/ni(z). σi has the dimension of an area; it expresses the likelihood of interaction between the particles of the beam and the particles of the specie i in the material sample:
$T=e^{-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}.$
One can also use the molar attenuation coefficients εi = (NA/ln 10)σi, where NA is the Avogadro constant, to describe the attenuation coefficient in a way independent of the amount concentrations ci(z) = ni(z)/NA of the attenuating species of the material sample:
$T=e^{-\sum _{i=1}^{N}{\frac {\ln {10}}{\mathrm {N_{A}} }}\varepsilon _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}={\Bigl (}e^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }{\frac {n_{i}(z)}{\mathrm {N_{A}} }}\mathrm {d} z}{\Bigr )}^{\ln {10}}=10^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z}.$

## Validity

Under certain conditions Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte.[6] These deviations are classified into three categories:
1. Real—fundamental deviations due to the limitations of the law itself.
2. Chemical—deviations observed due to specific chemical species of the sample which is being analyzed.
3. Instrument—deviations which occur due to how the attenuation measurements are made.
There are at least six conditions that need to be fulfilled in order for Beer–Lambert law to be valid. These are:
1. The attenuators must act independently of each other.
2. The attenuating medium must be homogeneous in the interaction volume.
3. The attenuating medium must not scatter the radiation—no turbidity—unless this is accounted for as in DOAS.
4. The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium.
5. The incident radiation should preferably be monochromatic, or have at least a width that is narrower than that of the attenuating transition. Otherwise a spectrometer as detector for the power is needed instead of a photodiode which has not a selective wavelength dependence.
6. 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 Beer–Lambert law.

## Chemical analysis by spectrophotometry

Beer–Lambert 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 attenuation coefficient ε is known. Measurements of decadic attenuation coefficient μ10 are made at one wavelength λ that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration c is then given by
$c={\frac {\mu _{10}(\lambda )}{\varepsilon (\lambda )}}.$
For a more complicated example, consider a mixture in solution containing two species at amount concentrations c1 and c2. The decadic attenuation coefficient at any wavelength λ is, given by
$\mu _{10}(\lambda )=\varepsilon _{1}(\lambda )c_{1}+\varepsilon _{2}(\lambda )c_{2}.$
Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations c1 and c2 as long as the molar attenuation coefficient of the two components, ε1 and ε2 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 amount concentrations from measurements made at more than two wavelengths. Mixtures containing more than two components can be analyzed 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 and near-infrared spectroscopy for analysis of polymer degradation and oxidation (also in biological tissue). The carbonyl group attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated.

## Beer–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 optical depth for a slant path is τ′ = , where τ refers to a vertical path, m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = sec θ where θ is the zenith angle corresponding to the given path. The Beer–Lambert law for the atmosphere is usually written
$T=e^{-m(\tau _{\mathrm {a} }+\tau _{\mathrm {g} }+\tau _{\mathrm {RS} }+\tau _{\mathrm {NO_{2}} }+\tau _{\mathrm {w} }+\tau _{\mathrm {O_{3}} }+\tau _{\mathrm {r} }+\ldots )},$
where each τx is the optical depth whose subscript identifies the source of the absorption or scattering it describes:
m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of θ) to 1/cos θ, where θ 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 τa, 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.

## Saturday, December 26, 2015

### Lapse rate

The lapse rate is defined as the rate at which atmospheric temperature decreases with an increase in altitude.[1][2] The terminology arises from the word lapse in the sense of a decrease or decline. While most often applied to Earth's troposphere, the concept can be extended to any gravitationally supported parcel of gas.

## Definition

A formal definition from the Glossary of Meteorology[3] is:
The decrease of an atmospheric variable with height, the variable being temperature unless otherwise specified.
In the lower regions of the atmosphere (up to altitudes of approximately 12,000 metres (39,000 ft), temperature decreases with altitude at a fairly uniform rate. Because the atmosphere is warmed by convection from Earth's surface, this lapse or reduction in temperature is normal with increasing distance from the conductive source.

Although the actual atmospheric lapse rate varies, under normal atmospheric conditions the average atmospheric lapse rate results in a temperature decrease of 6.4 °C/km (3.5 °F or 1.95 °C/1,000 ft) of altitude above ground level.

The measurable lapse rate is affected by the moisture content of the air (humidity). A dry lapse rate of 10 °C/km (5.5 °F or 3.05 °C/1,000 ft) is often used to calculate temperature changes in air not at 100% relative humidity. A wet lapse rate of 5.5 °C/km (3 °F or 1.68 °C/1,000 ft) is used to calculate the temperature changes in air that is saturated (i.e., air at 100% relative humidity). Although actual lapse rates do not strictly follow these guidelines, they present a model sufficiently accurate to predict temperate changes associated with updrafts and downdrafts. This differential lapse rate (dependent upon both difference in conductive heating and adiabatic expansion or compression) results in the formation of warm downslope winds (e.g., Chinook winds, Santa Ana winds, etc.). The atmospheric lapse rate, combined with adiabatic cooling and heating of air related to the expansion and compression of atmospheric gases, present a unified model explaining the cooling of air as it moves aloft and the heating of air as it descends downslope.

Atmospheric stability can be measured in terms of lapse rates (i.e., the temperature differences associated with vertical movement of air). The atmosphere is considered conditionally unstable where the environmental lapse rate causes a slower decrease in temperature with altitude than the dry adiabatic lapse rate, as long as no latent heat is released (i.e. the saturated adiabatic lapse rate applies). Unconditional instability results when the dry adiabatic lapse rate causes air to cool slower than the environmental lapse rate, so air will continue to rise until it reaches the same temperature as its surroundings. Where the saturated adiabatic lapse rate is greater than the environmental lapse rate, the air cools faster than its environment and thus returns to its original position, irrespective of its moisture content.

Although the atmospheric lapse rate (also known as the environmental lapse rate) is most often used to characterize temperature changes, many properties (e.g. atmospheric pressure) can also be profiled by lapse rates...

## Mathematical definition

In general, a lapse rate is the negative of the rate of temperature change with altitude change, thus:
$\gamma = -\frac{dT}{dz}$
where $\gamma$ is the lapse rate given in units of temperature divided by units of altitude, T = temperature, and z = altitude.

Note: In some cases, $\Gamma$ or $\alpha$ can be used to represent the adiabatic lapse rate in order to avoid confusion with other terms symbolized by $\gamma$, such as the specific heat ratio[4] or the psychrometric constant.[5]

## Types of lapse rates

There are two types of lapse rate:
• Environmental lapse rate (ELR) – which refers to the actual change of temperature with altitude for the stationary atmosphere (i.e. the temperature gradient)
• The adiabatic lapse rates – which refer to the change in temperature of a parcel of air as it moves upwards (or downwards) without exchanging heat with its surroundings. There are two adiabatic rates:[6]
• Dry adiabatic lapse rate (DALR)
• Moist (or saturated) adiabatic lapse rate (SALR)

### Environmental lapse rate

The environmental lapse rate (ELR), is the rate of decrease of temperature with altitude in the stationary atmosphere at a given time and location. As an average, the International Civil Aviation Organization (ICAO) defines an international standard atmosphere (ISA) with a temperature lapse rate of 6.49 K/km[citation needed] (3.56 °F or 1.98 K/1,000 ft) from sea level to 11 km (36,090 ft or 6.8 mi). From 11 km up to 20 km (65,620 ft or 12.4 mi), the constant temperature is −56.5 °C (−69.7 °F), which is the lowest assumed temperature in the ISA. The standard atmosphere contains no moisture. Unlike the idealized ISA, the temperature of the actual atmosphere does not always fall at a uniform rate with height. For example, there can be an inversion layer in which the temperature increases with altitude.

Emagram diagram showing variation of dry adiabats (bold lines) and moist adiabats (dash lines) according to pressure and temperature

The dry adiabatic lapse rate (DALR) is the rate of temperature decrease with altitude for a parcel of dry or unsaturated air rising under adiabatic conditions. Unsaturated air has less than 100% relative humidity; i.e. its actual temperature is higher than its dew point. The term adiabatic means that no heat transfer occurs into or out of the parcel. Air has low thermal conductivity, and the bodies of air involved are very large, so transfer of heat by conduction is negligibly small.

Under these conditions when the air rises (for instance, by convection) it expands, because the pressure is lower at higher altitudes. As the air parcel expands, it pushes on the air around it, doing work (thermodynamics). Since the parcel does work but gains no heat, it loses internal energy so that its temperature decreases. The rate of temperature decrease is 9.8 °C/km (5.38 °F per 1,000 ft) (3.0 °C/1,000 ft). The reverse occurs for a sinking parcel of air.[7]

$P dV = -V dP / \gamma$
the first law of thermodynamics can be written as
$m c_v dT - V dp/ \gamma = 0$
Also since :$\alpha = V/m$ and :$\gamma = c_p/c_v$ we can show that:
$c_p dT - \alpha dP = 0$
where $c_p$ is the specific heat at constant pressure and $\alpha$ is the specific volume.

Assuming an atmosphere in hydrostatic equilibrium:[8]
$dP = - \rho g dz$
where g is the standard gravity and $\rho$ is the density. Combining these two equations to eliminate the pressure, one arrives at the result for the DALR,[9]
$\Gamma_d = -\frac{dT}{dz}= \frac{g}{c_p} = 9.8 \ ^{\circ}\mathrm{C}/\mathrm{km}$

When the air is saturated with water vapour (at its dew point), the moist adiabatic lapse rate (MALR) or saturated adiabatic lapse rate (SALR) applies. This lapse rate varies strongly with temperature. A typical value is around 5 °C/km (2.7 °F/1,000 ft) (1.5 °C/1,000 ft).[citation needed]

The reason for the difference between the dry and moist adiabatic lapse rate values is that latent heat is released when water condenses, thus decreasing the rate of temperature drop as altitude increases. This heat release process is an important source of energy in the development of thunderstorms. An unsaturated parcel of air of given temperature, altitude and moisture content below that of the corresponding dew point cools at the dry adiabatic lapse rate as altitude increases until the dew point line for the given moisture content is intersected. As the water vapour then starts condensing the air parcel subsequently cools at the slower moist adiabatic lapse rate if the altitude increases further.

The saturated adiabatic lapse rate is given approximately by this equation from the glossary of the American Meteorology Society:[10]
$\Gamma_w = g\, \frac{1 + \dfrac{H_v\, r}{R_{sd}\, T}}{c_{p d} + \dfrac{H_v^2\, r}{R_{sw}\, T^2}}= g\, \frac{1 + \dfrac{H_v\, r}{R_{sd}\, T}}{c_{p d} + \dfrac{H_v^2\, r\, \epsilon}{R_{sd}\, T^2}}$
 where: $\Gamma_w$ = Wet adiabatic lapse rate, K/m $g$ = Earth's gravitational acceleration = 9.8076 m/s2 $H_v$ = Heat of vaporization of water, = 2501000 J/kg $R_{sd}$ = Specific gas constant of dry air = 287 J kg−1 K−1 $R_{sw}$ = Specific gas constant of water vapour = 461.5 J kg−1 K−1 $\epsilon=\frac{R_{sd}}{R_{sw}}$ =The dimensionless ratio of the specific gas constant of dry air to the specific gas constant for water vapour = 0.622 $e$ = The water vapour pressure of the saturated air $p$ = The pressure of the saturated air $r=\epsilon e/(p-e)$ = The mixing ratio of the mass of water vapour to the mass of dry air[11] $T$ = Temperature of the saturated air, K $c_{pd}$ = The specific heat of dry air at constant pressure, = 1003.5 J kg−1 K−1

### Thermodynamic-based lapse rate

Robert Essenhigh developed a comprehensive thermodynamic model of the lapse rate based on the Schuster–Schwarzschild (S–S) integral equations of transfer that govern radiation through the atmosphere including absorption and radiation by greenhouse gases.[12] His solution "predicts, in agreement with the Standard Atmosphere experimental data, a linear decline of the fourth power of the temperature, T4, with pressure, P, and, as a first approximation, a linear decline of T with altitude, h, up to the tropopause at about 10 km (the lower atmosphere)."[12] The predicted normalized density ratio and pressure ratio differ and fit the experimental data well.[citation needed] Sreekanth Kolan extended Essenhigh's model to include the energy balance for the lower and upper atmospheres.[13][self-published source?][third-party source needed]

## Significance in meteorology

The varying environmental lapse rates throughout the Earth's atmosphere are of critical importance in meteorology, particularly within the troposphere. They are used to determine if the parcel of rising air will rise high enough for its water to condense to form clouds, and, having formed clouds, whether the air will continue to rise and form bigger shower clouds, and whether these clouds will get even bigger and form cumulonimbus clouds (thunder clouds).

As unsaturated air rises, its temperature drops at the dry adiabatic rate. The dew point also drops (as a result of decreasing air pressure) but much more slowly, typically about −2 °C per 1,000 m. If unsaturated air rises far enough, eventually its temperature will reach its dew point, and condensation will begin to form. This altitude is known as the lifting condensation level (LCL) when mechanical lift is present and the convective condensation level (CCL) when mechanical lift is absent, in which case, the parcel must be heated from below to its convective temperature. The cloud base will be somewhere within the layer bounded by these parameters.

The difference between the dry adiabatic lapse rate and the rate at which the dew point drops is around 8 °C per 1,000 ft. Given a difference in temperature and dew point readings on the ground, one can easily find the LCL by multiplying the difference by 125 m/°C.

If the environmental lapse rate is less than the moist adiabatic lapse rate, the air is absolutely stable — rising air will cool faster than the surrounding air and lose buoyancy. This often happens in the early morning, when the air near the ground has cooled overnight. Cloud formation in stable air is unlikely.

If the environmental lapse rate is between the moist and dry adiabatic lapse rates, the air is conditionally unstable — an unsaturated parcel of air does not have sufficient buoyancy to rise to the LCL or CCL, and it is stable to weak vertical displacements in either direction. If the parcel is saturated it is unstable and will rise to the LCL or CCL, and either be halted due to an inversion layer of convective inhibition, or if lifting continues, deep, moist convection (DMC) may ensue, as a parcel rises to the level of free convection (LFC), after which it enters the free convective layer (FCL) and usually rises to the equilibrium level (EL).

If the environmental lapse rate is larger than the dry adiabatic lapse rate, it has a superadiabatic lapse rate, the air is absolutely unstable — a parcel of air will gain buoyancy as it rises both below and above the lifting condensation level or convective condensation level. This often happens in the afternoon mainly over land masses. In these conditions, the likelihood of cumulus clouds, showers or even thunderstorms is increased.

Meteorologists use radiosondes to measure the environmental lapse rate and compare it to the predicted adiabatic lapse rate to forecast the likelihood that air will rise. Charts of the environmental lapse rate are known as thermodynamic diagrams, examples of which include Skew-T log-P diagrams and tephigrams.

The difference in moist adiabatic lapse rate and the dry rate is the cause of foehn wind phenomenon (also known as "Chinook winds" in parts of North America).