# Anomalous Humidity Dependence in Photoacoustic Spectroscopy of CO Explained by Kinetic Cooling
## 3 Results
### Collisional Relaxation in the CO, N\({}_{2}\), and H\({}_{2}\)O System  
To explain the peculiar trend of the QEPAS signal of CO with humidity, the processes involved in the excitation and relaxation of CO in N\({}_{2}\) and H\({}_{2}\)O are summarized in Figure 3. The number density \(N_{CO}\) of CO molecules in the excited vibrational state changes along a coherent pathway (absorption and stimulated emission) and a non-coherent pathway (inelastic collisions). Spontaneous emission as well stimulated emission are neglected in the following (linear regime). The non-coherent processes involved in the relaxation of CO are rotational-translational (R-T), vibrational-vibrational (V-V), and vibrational-translational (V-T) transfers. The collision numbers \(\hat{Z}_{A-B}\) and the collision rates \(Z_{A-B}\) for the involved collisional energy transfers are summarized in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline
**Process** & **Z\({}_{\text{A-B}}\) [10\({}^{9}\) s\({}^{-1}\)atm\({}^{-1}\)]** & \(\hat{Z}\) & **Z\({}_{\text{A-B}}\)/\(\hat{Z}\) [s\({}^{-1}\)atm\({}^{-1}\)]** & \(\Delta E\) [cm\({}^{-1}\)] \\ \hline R-T, CO (N\({}_{2}\)) & 7.0 & 3 & \(2.3\times 10^{9}\) & 37 \\ V-V, CO \(\rightarrow\) N\({}_{2}\) & 7.0 & \(4.9\times 10^{4}\) & \(1.4\times 10^{5}\) & \(-188\) \\ V-V, N\({}_{2}\)\(\rightarrow\) CO & 7.0 & \(2.0\times 10^{4}\) & \(3.5\times 10^{5}\) & 188 \\ V-V, CO \(\rightarrow\) H\({}_{2}\)O & 6.0 & \(1.1\times 10^{3}\) & \(5.5\times 10^{6}\) & 548 \\ V-V, N\({}_{2}\)\(\rightarrow\) H\({}_{2}\)O & 5.7 & \(2.6\times 10^{4}\) & \(2.2\times 10^{5}\) & 736 \\ V-T, N\({}_{2}\) (N\({}_{2}\)) & 6.8 & \(4\times 10^{9}\) & \(1.7\) & 2331 \\ V-T, H\({}_{2}\)O (N\({}_{2}\)) & 5.7 & \(4.7\times 10^{2}\) & \(1.2\times 10^{7}\) & 1595 \\ V-T, H\({}_{2}\)O (H\({}_{2}\)O) & 4.5 & 4 & \(1.1\times 10^{9}\) & 1595 \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of processes involved in the relaxation of CO. Collision frequencies \(Z_{A-B}\) were normalized to the partial pressure of the collision partner B. “R-T, A (B)” and “V-T, A (B)” denote R–T and V–T relaxation of A in collisions with B. “V-V, A \(\rightarrow\) B” denotes the V–V transfer from A to B. \(\Delta E\) denotes the energy released in the process (negative for energy uptake). Collision numbers \(\hat{Z}\) were obtained from [2, 19, 20, 21, 22]. All values are given for T = 298 K.  
Figure 3: Energy levels of CO, N\({}_{2}\), and H\({}_{2}\)O involved in the collisional relaxation of CO. Straight arrows indicate energy transfer in collisions and the kind of transfer is given in parenthesis: {R-T, A} denotes rotational relaxation, {V-T, A (B)} denotes the vibrational-translational relaxation of A in collisions with B, and {V-V, A \(\rightarrow\) B} denotes V–V transfer from A to B. The rates \(\gamma_{A-B}\) for each energy transfer, normalized to the partial pressure of the collision partner, is given for T = 298 K, compare Table 1.  
The fastest step is R-T transfer from the rotational level J = 10, which leads to the rapid establishment of rotational thermal equilibrium within the excited vibrational state. The same is true for the vibrational ground state for which the rotational energy level J = 9 is rapidly repopulated. The net kinetic energy released in R-T transfer equals the difference between the excitation wavenumber (2179.77 cm\({}^{-1}\)) and the wavenumber of the vibrational mode (2143 cm\({}^{-1}\)). V-V transfer occurs between CO and N\({}_{2}\) (bidirectional), from CO to H\({}_{2}\)O and from N\({}_{2}\) to H\({}_{2}\)O. In the absence of H\({}_{2}\)O, V-V transfer to N\({}_{2}\) is much faster than V-T relaxation of CO and is hence the prevailing process. At low concentrations and partial pressure \(p\) of CO (ppm range), the collision rate \(Z_{N_{2}-CO}\) is so low that, on the timescale of photoacoustic modulation, V-V transfer from N\({}_{2}\) back to CO can be neglected. Therefore, V-V transfer only occurs in the endothermic direction from CO to N\({}_{2}\), yielding a kinetic cooling effect.