## Band absorptance formulations

4.5.1 Doppler-broadened rotational limes

The absorptance of a molecular band can be calculated based in an analogous way to line absorptance. At low pressure the rotational lines of a band are Doppler broadened and essentially non-overlapping, as the full line width, ~ 2bD, typically 0.002 cm"1, becomes smaller than the line spacing, d, typically 1.0 cm"1. Ignoring any asymmetries between P and R branches, we can consider, to a first approximation, that the line strength decreases exponentially with increasing rotational quantum number from the band centre and assume that the lines are symmetrical about it. Edwards and Menard (1964) adopted a normalized line strength distribution of the form

2Ao where kj is the rotational line strength for a transition with lower level J, A0 is referred to as the bandwidth parameter, and k is the total line strength given by k = 2^ kj , (4.72)

and so is the sum of the line strengths of the individual rotational lines. The band absorptance, A, can then be computed from the individual line absorptances, AJ, given by

YDV n and, to a good approximation, yd, see eqn (4.11), is evaluated at the band centre wave number for all lines. If we now let u = kX/Aq and 5 = a/^Yd/d, we can then introduce the dimensionless parameter, w = u/2S and obtain yj = we-Jd/A°, (4.75)

where the line strength, k, is in units of cm-2. For most situations, the lines are close together since the line spacing is of the order of 1 cm-1 and the band width is of the order of 100 cm-1, we can replace the summation in the band absorptance expression by the integral

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