High Pressure Limit. . . .
This is a continuation of playing with Eli's new chewy carrot colored Spectral Calculator toy the mice left in the burrow. A couple of days ago, the Rabetts looked at the effect of pressure, and before that temperature on CO2 bending mode absorption spectrum. Eli remarked on the fact that the peak of the pressure broadened absorption stays at about the same level for a constant volume mixing ratio above 50 mbar total pressure. This means that the peak of the absorption stays the same while the line gets wider, for example at 100 mbar and 1000 mbar total pressure and 380 ppm volume mixing ratio
The mice had a nice chatter about this. Eli wants to use this as a jumping off point.
There are three relevant line widths. The natural line width, which is a measure of the vibrationally excited state radiative lifetime and the associated uncertainty spread in the energy level. For vibrational lines these are sub MHz.
Since spectroscopy goes back and forth between MHz (very high resolution, microwave spectroscopy) and cm-1 (IR, visible, UV spectroscopy) we need an equivalence. Simply divide 1 MHz by the speed of light in cm/sec, 3 x 10^10 finding that 1 MHz is equivalent to 3.3 x 10^-5 cm-1.
The shape of the natural (isolated/no collisions) line is Lorentzian (from Wolfrum Mathworld).
where the Gamma (the thing like the hangman uses) is the full line width at half maximum (FWHM).
The second linewidth that we have to worry about is the Doppler width. Doppler broadening is a shift in frequency when something (the molecule) is moving towards you (increase) or away from you (decrease). For emission or absorption of light the shift will be wo(1+v/c) where wo is the frequency of absorption/emission in the rest frame, v the speed along the direction the photon moves in and c the speed of light. When you average over all possible directions of molecular motion, this turns out to be Gaussian
We can estimate the Doppler linewidth. The translational energy of the molecule is Et= 3/2 kT, where k is Boltzmann's constant, 1.38 x 10^-23 kg-m^2 /K-s^2. This yields Et= 6.2 x 10^-21 J, but we also know that Et= 1/2 m v^2, so v = sqrt(2Et/m) where m is in kg. The mass of one CO2 (C= 12 g/mole, O=16 g/mole. If you want to do this to four significant figures you don't have the one tru back of the envelope koan) molecule is 0.044 kg / 6.02 x 10^23 molecules/mole and we get that the velocity is ~ 400 m/s.
The speed of light is 3 x 10 ^8 m/s so v/c is 1 x 10^-6. For a 600 cm-1 transition (CO2 bend) this is about 1 x 10^-3 cm-1 or 30 MHz. The Doppler width varies directly with the frequency of the transition, so a transition at 6000 cm-1 would have a Doppler width that is ~300 MHz at room temperature.
Finally the line shape associated with collisional line broadening is also Lorentzian. The natural and collision broadened line shapes can be simply combined by setting the line width equal to the sum of the radiative and collisional terms. The collisional term is proportional to the total pressure in the binary collision limit (atmospheric, unless you deep down in Jupiter). The higher the pressure, the more collisions. Remember this.
Combining the Gaussian Doppler broadening with the Lorentzian radiative and collisional terms is trickier. The solution was first found by Armstrong, and is called the Voigt line profile. However, it should be clear that if the collisional broadening is >> than the Doppler broadening (0.001 cm-1 @ 300 K for the CO2 bend) and the Doppler broadening is >> the natural line width, we can neglect the foofaw and treat the line profile as a Lorentzian whose width is aP where a is a constant for broadening of CO2 lines by air (there is some dependence on rotational state, some non-linear component, but remember this is back of the envelope)
The integral of the Lorentzian profile across all frequencies is unity (1). The total absorption of the line whether broadened or not will be Abs = A PCO2 L where A is the line absorption, PCO2 the partial pressure of CO2 and L the path length. PCO2 = VMR P where P is the total pressure and VMR is the volume mixing ratio.
At line center (substitute x = xo) into the Lorentzian formula the magnitude of the maximum is
The maximum absorption is Abs x L(xo). Substituting for Abs and Gamma we get
A is the integrated line absorbance for unit pressure, α the linear line broadening coefficient and VMR the volume mixing ratio.
If you go to very low pressures, the Doppler broadening approaches the pressure broadening and this approximation no longer works, but for tropospheric and stratospheric pressures it is fine.