A satellite instrument capable of measuring the CO v=0 rightarrow v = 1 fundamen

Question

A satellite instrument capable of measuring the CO v=0 rightarrow v = 1 fundamental absorption band near 2143 cm -1 at a specific altitude in the atmosphere, reveals that the P(6) line is the strongest line in the P-branch of the band and that the P(6), and P(7) lines occur at 2120.2 cm -1 and 2116.4 cm -1, respectively. Estimate the atmospheric temperature at the point sampled by the instrument. What are the error limits associated with this temperature determination? (Assume that CO is a rigid molecule and that the strength of the lines depend only on the population in the absorbing rotational level.)

Explanation / Answer

Rotational constant of CO:

B = 0.5 x spacing between P(6) and P(7) lines

= 0.5 x (2120.2 - 2116.4) = 1.9 cm-1

P(6) line is strongest => maximum populated level Jmax = 6

Jmax = (kT/2hcB)0.5 - 0.5

where k is the Boltzmann constant, h is the Planck constant, B is the rotational constant in cm-1 and c is the speed of light in cm/s

6 = ((1.381 x 10-23 x T)/(2 x 6.626 x 10-34 x 2.998 x 1010 x 1.9))0.5 - 0.5

Temperature T = 231 K

Since the wavenunbers can only be measured to 0.1 cm-1 resolution,

Absolute error in measurement of B = 2 x 0.1 = 0.2 cm-1 (B is measured by subtracting 2 wavenumbers, so absolute errors add up).

%error in T = %error in B = 100*(0.2/1.9) = 10.5%

Absolute error in T = (10.5/100) x 231 = 24 K

So T = (231 ± 24) K

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