of the absorbing substance, this integrates to In (///,,) = -kCl.
The most commonly used form of this Beer-Lambert law involves logarithms to the base 10:
C is in units of mol L ', / is in cm, and the constant of proportionality e (L mol~' cm~') is known as the molar absorptivity or molar extinction coefficient. The dimensionless quantity log(/(,/7) is known as the ab-sorbance, A, which is related to the transmittance by A = log(/()//) = -log T.
Most commercial spectrometers report absorbance, as defined in Eq. (Q), versus wavelength. This is very important to recognize, since as we will see later, calculations of the rate of light absorption in the atmosphere require the use of absorption coefficients to the base e rather than to the base 10. While the recent atmospheric chemistry literature reports absorption cross sections to the base e, most measurements of absorption coefficients reported in the general chemical literature are to the base 10. If these are to be used in calculating photolysis rates in the atmosphere, the factor of 2.303 must be taken into account.
In gas-phase tropospheric chemistry, the most common units for concentration, N, are molecules cm~3 and for path length, I, units of cm. The form of the Beer-Lambert law is then or
FIGURE 3.1 f Schematic diagram of experimental approach to the Beer-Lambert law.
where it must again be emphasized that a, known as the absorption cross section, must have been measured with the appropriate form of the Beer-Lambert law to the base e. The dimensionless exponent aNl is often referred to as the "optical depth."
In the past, some absorption coefficients for gases have been reported with concentrations in units of atmospheres, so that the absorption coefficient is in units of atm"1 cm"1. Since the pressure depends on temperature, the latter (usually 273 or 298 K) were also reported.
For most tropospheric situations involving gaseous species, the Beer-Lambert law is an accurate method for treating light absorption; similar considerations apply to nonassociated molecules in dilute solution. However, under laboratory conditions with relatively high concentrations of the absorbing species, deviations may arise from a variety of factors, including concentration- and temperature-dependent association or dissociation reactions, deviations from the ideal gas law, and saturation of very narrow lines with increasing
TABLE 3.4 Conversion Factors for Changing Absorption Coefficients from One Set of Units to Another
Both units in either logarithmic base e or base 10 X 2.69 X 1019 = (atm at 273 K)- 1 (an"1)
, 2 , , _,sX 2.46 X 1019 = (atmat298 K)-1 (cirT1) (cm molecule ; ... ,„,
t . , n„„ „i-i , -u x 4.06 x 10-20 = cm2 molecule- 1 (atm at 298 K) (cm > x 1-09 = (atmat 273 r)-. (cm-.)
m°l cm ) x 53g x 10_5 = (To].r at 29g K)_ , (cm_ 1}
Change of both logarithmic base and units
X 1.17 X 1019 = (atm at 273 K)- 1 (cm"1), base 10
, 2 , , -u u x 1.07 x 1019 = (atm at 298 K)-1 (cm-1), base 10 (cm molecule >• base L41 x 1Ql6 = ^ a, , fcm_ ^ ^ ^
X 3.82 X 10-21 = cm2 moleculebase e (L mol-1 cm-1), base 10 x 0.103 = (atm at 273 K)-1 (cm-1), base e
(atm at 273 K)-' (cm-1), base 10 X X T10"2<! = =m2 molecule-base e
(Torr at 298 K)-> (cm- >), base 10 * ™ * i^'7 = ^^olecule-base e
(atm at 298 K) (cm ), base 10 x ^ = (atm at ^ R)_, (cm_ base g concentrations, i.e., increasing pressures. Particularly important is the situation in which a "monochromatic" analyzer beam actually has a bandwidth that is broad relative to very narrow lines of an absorbing species. In this case, which is often encountered in the infrared, for example, the Beer-Lambert law is nonlinear. Clearly, to be on the safe side, it is good practice to verify the linearity of ln(/(,/7) plots as a function of absorber concentration when experimentally determining absorption coefficients.
Table 3.4 gives conversion factors for converting absorption coefficients from one set of units to another and for changing between logarithms to the base 10 and base e.
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