# Anomalous Humidity Dependence in Photoacoustic Spectroscopy of CO Explained by Kinetic Cooling
## 3 Results
### Rate Equations and Quantitative Model  
Based on the scheme of collisional energy transfers shown in Figure 3, the rate equations for the number density \(N\) of all involved molecular species can be defined: CO, CO\({}^{\star}\), N\({}_{2}\), N\({}_{2}^{\star}\), H\({}_{2}\)O, and H\({}_{2}\)O\({}^{\star}\). Here, \"\(\star\)\" denotes the vibrationally excited state (\(\
u_{2}\) bending mode of H\({}_{2}\)O). Thermal rotational equilibrium is assumed at any time and the small contribution from rotational relaxation to the dissipated heat will be included later. The rate equation for \(N_{CO}\) can be written as a sum of coherent excitation of CO, \(\dot{N}_{CO}^{\dagger}\), and collisional relaxation of vibrationally excited CO\({}^{\star}\), \(\dot{N}_{CO}^{\dagger}\). \[\dot{N}_{CO^{\star}}=-\dot{N}_{CO}=-\dot{N}_{CO}^{\dagger}+\dot{N}_{CO^{\star}} ^{\downarrow} \tag{5}\]  
As before, stimulated emission was neglected (\(N_{CO}\gg N_{CO^{\star}}\)). According to Figure 3, \(\dot{N}_{CO}^{\downarrow}\) holds contributions from V-V transfer between CO and N\({}_{2}\) as well as H\({}_{2}\)O.  
\[\dot{N}_{CO^{\prime}}^{\downarrow}=\dot{N}_{CO-N_{2}}^{V-V}+\dot{N}_{CO-H_{2}O }^{V-V} \tag{6}\]  
\[\dot{N}_{CO-N_{2}}^{V-V}=-N_{CO^{\prime}}\frac{Z_{CO-N_{2}}}{\dot{Z}_{CO-N_{2} }}+N_{N_{2}}^{N_{2}}\frac{Z_{N_{2}-CO}}{\dot{Z}_{N_{2}-CO}} \tag{7}\]  
\[\dot{N}_{CO-H_{2}O}^{V-V}=-N_{CO^{\prime}}\frac{Z_{CO-H_{2}O}}{\dot{Z}_{CO-H_{ 2}O}} \tag{8}\]  
Herein, Equation (2) was used to describe the rates of V-V transfer in terms of collision rates and collision numbers. As outlined above, the second term in Equation (7) describing the V-V transfer from N\({}_{2}^{\star}\) back to CO can be neglected if the concentration of CO is small. Analogous to Equation (6), the rate equations for the number densities of N\({}_{2}^{\star}\) and H\({}_{2}\)O\({}^{\star}\) can be written as  
\[\dot{N}_{N_{2}}=\dot{N}_{N_{2}-H_{2}O}^{V-V}-\dot{N}_{CO-N_{2}}^{V-V}+\dot{N}_{N _{2}}^{V-T} \tag{9}\]  
\[\dot{N}_{N_{2}-H_{2}O}^{V-V}=-N_{N_{2}}^{N_{2}}\frac{Z_{N_{2}-H_{2}O}}{\dot{Z}_{ N_{2}-H_{2}O}} \tag{10}\]\[\dot{N}_{N_{2}^{*}}^{V-T}=-N_{N_{2}^{*}}\frac{Z_{N_{2}-N_{2}}}{\dot{Z}_{N_{2}-N_{ 2}}} \tag{11}\]  
and  
\[\dot{N}_{H_{2}O^{*}}=-\dot{N}_{CO-H_{2}O}^{V-V}-\dot{N}_{N_{2}-H_{2}O}^{V-V}+ \dot{N}_{H_{2}O^{*}}^{V-T} \tag{12}\]  
\[\dot{N}_{H_{2}O^{*}}^{V-T}=-N_{H_{2}O}\left(\frac{Z_{H_{2}O-N_{2}}}{\dot{Z}_{H_{ 2}O-N_{2}}}+\frac{Z_{H_{2}O-H_{2}O}}{\dot{Z}_{H_{2}O-H_{2}O}}\right) \tag{13}\]  
The collision rates \(Z_{A-B}\) are given by Equation (1) with the kinetic radii \(r_{H_{2}O}=132\) pm, \(r_{N_{2}}=182\) pm, and \(r_{CO}=188\) pm and molar masses \(M_{CO}=M_{N_{2}}=0.028\) kg mol\({}^{-1}\) and \(M_{H_{2}O}=0.018\) kg mol\({}^{-1}\)[23]. The collision rates and collision numbers for the involved energy transfers are summarized in Table 1.  
As outlined in Section 2.2, the energy difference \(\Delta E\) between the levels involved in V-V and V-T transfer is released or taken up as heat (compare Table 1). Hence, the thermal energy \(H\) dissipated per unit time and volume is directly proportional to the rates \(\dot{N}\) given in Equations (6)-(8). \[ \begin{split} H=c\cdot h\cdot(\dot{N}_{CO-N_{2}}^{V-V}\cdot 188 \text{ cm}^{-1}-\dot{N}_{CO-H_{2}O}^{V-V}\cdot 548\text{ cm}^{-1}-\dot{N}_{N_{2}-H_{2}O}^{V-V}\cdot 736\text{ cm}^{-1}-\dot{N}_{N_{2} ^{*}}^{V-T}\\ \cdot 2331\text{ cm}^{-1}-\dot{N}_{H_{2}O^{*}}^{V-T}\cdot 1595\text{ cm}^{-1}-\dot{N}_{CO}^{\uparrow}\cdot 37\text{ cm}^{-1})\end{split}  \tag{14}\]  
As demonstrated in Section 3.4, numerically solving the rate Equations (5)-(8) and inserting them in Equation (14) allows for \(H\) to be simulated in response to modulated excitation and, due to its direct proportionality to \(H\), the photoacoustic signal. The last term in Equation (14) corresponds to the establishment of thermal rotational equilibrium in the excited vibrational state of CO (R-T relaxation).