# Anomalous Humidity Dependence in Photoacoustic Spectroscopy of CO Explained by Kinetic Cooling
## 3 Results
### Simulation Results  
Based on the quantitative kinetic model described in the previous section, the photothermal heat dissipation \(H\) was simulated in accordance with the experimental conditions corresponding to Figure 2. Simulations were carried out in MATLAB (MathWorks, Inc., Natick, MA, USA) using the finite element method to stepwise calculate the temporal evolution of \(H\) in response to wavelength modulated absorption. The MATLAB script for the simulation can be found in the electronic supporting information. The initial excited state populations of all molecular species at \(t=0\) was set to zero and the simulation was carried out over several modulation periods until all populations \(N\) reached a steady (yet periodically modulated) state. Wavelength modulated excitation was included by inserting a time dependent wavenumber in \(\Gamma\) in Equation (15)  
\[\Gamma=\Gamma(\widetilde{v}(t)) \tag{17}\]  
\[\widetilde{v}(t)=\widetilde{v}_{0}+\Delta\widetilde{v}\cos(2\pi f_{mod}t) \tag{18}\]  
The relevant constants of the R9 transition of CO in Equation (15) are \(A_{21}=17.91\) s\({}^{-1}\), \(g_{2}=21\), and \(g_{1}=19\)[27]. From the rates \(N(t)\) of all molecular species, the heat \(H(t)\) was calculated using Equation (14). The Fourier transform of the last modulation period of \(H(t)\) at the second harmonic of the modulation frequency yielded the complex valued heat \(\hat{H}(2f_{mod})\), which is directly proportional to the photoacoustic signal. The resulting amplitude and phase of \(\hat{H}\) for H\({}_{2}\)O concentrations from 0% to 2.2%v are given in Figure 4. The simulations closely resemble the experimental data for both sample pressures. We point out that no fitting was involved in simulating \(\hat{H}\) and all collision numbers and other parameters were taken directly from literature (compare Table 1). From the amplitudes shown in Figure 4 (left panels), it can be seen that the proportionality constant between the simulation of \(H\) and the measured QEPAS signal was different for 200 mbar and 800 mbar (\(\sim\)factor two difference). Since QEPAS signals were normalized by the quality factor of the tuning fork, this cannot be attributed to the pressure dependence of the latter. Rather, a pressure dependent  
Figure 4: Simulation of the photothermal response \(\hat{H}\) and measured QEPAS signal of CO in N\({}_{2}\) at varying humidities and conditions identical to those in Figure 2. The QEPAS phase was inverted and shifted with respect to Figure 2 to resemble the simulations (see text). resonant enhancement of the acoustic resonator may explain this discrepancy [28]. The phase of the QEPAS signal in Figure 4 was inverted and shifted to overlap with the phase of the simulated heat amplitude. The necessity of inverting and shifting the QEPAS phase can be attributed to the \"arbitrary\" sign of the recorded piezoelectric voltage and the phase shifts associated with the tuning fork and the acoustic resonator. Note, however, that the phase was not scaled. Further simulations were carried out for excitation frequencies of 32.8 kHz and 1.5 kHz, corresponding to the resonance frequency of a standard quartz tuning fork and a typical excitation frequency of PAS when conventional microphones are used, respectively. The simulated humidity dependent amplitude and phase of \(\hat{H}(32.8\,kHz)\) and \(\hat{H}(1.5\,kHz)\) are shown in Figure 5 for different sample pressures. Intriguingly, the minimum in amplitude was more pronounced at high pressure for 32.8 kHz, while the opposite trend was observed for an excitation frequency of 1.5 kHz. Additionally, the minimum for 1.5 kHz shifted toward a lower humidity for 1.5 kHz while it stayed constant at a humidity of 0.19%v for 32.8 kHz as well as 12.46 kHz (compare Figure 4). An interpretation of the physical origin of these observations was beyond the scope of this manuscript.