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Thermal desorption is the process by which molecules adsorbed to a substrate are released due to increasing kinetic energy. The substance which adsorbs and desorbs is referred to as the adsorbate. The substance to which the adsorbate adsorbs is called the substrate. The two types of adsorption to a surface are chemisorption and physisorption. Chemisorption involves the actual formation of bonds between the adsorbate and the substrate. The second type of adsorption, and the type thoroughly discussed in this mathematical discussion, is physisorption. This type of adsorption involves van der Waals interactions such as dispersion forces or a dipolar interaction. Since these interactions are very weak, a physisorbed molecule can be easily desorbed simply by raising the temperature, thus increasing the kinetic energy and breaking the interacting forces between the adsorbate and substrate.
Accurate and extensive data acquisition is fundamental in order to observe the desorption phenomenon. While observing temperature and pressure with time, a sudden increase in pressure should be observed as the temperature rises to a point where there is enough thermal energy to increase the kinetic energy of the adsorbate to the point where it will break free of the substrate. In order to achieve accurate data, calibration curves must be set up for temperature and pressure. This can be done by taking two measurements for each parameter: one with the digital voltage output, and one with an independent measuring apparatus. For example, for temperature, a calibrated digital thermometer could be used in conjunction with the computer's digital voltage data. Calibrating in this way also allows the conversion of the digital data from the computer (in voltage) to the corresponding appropriate units.
The simplifying assumption must be made that temperature increases linearly with time. In this way, a constant heating rate can be measured. By graphing temperature data versus time data, the slope of a least squares fit line will yield the heating rate.
1) T=TI+at where a=dT/dt
a would equal the heating rate in Kelvins per second. In this way, several different heating rates can be calculated for several experimental runs which vary this parameter.
In order to observe the desorption phenomenon, a plot of pressure versus temperature and dP/dt versus temperature should be made for each run. Because the system is closed, and heat is applied, the background pressure is linearly increasing with temperature. This means that in a graph of P vs. T, the graph will have a positive and near linear slope. However, a slight lump in the graph should appear where the desorption phenomenon occurs. This is due to a greater increase in pressure over a shorter range of time from the desorbing adsorbate in addition to the standard linear pressure increase with temperature. A more decisive graph however, is the dP/dt vs. T graph. This graph should reach a maximum at the temperature at which corresponds to the greatest increase in pressure. Since the Temperature and time are linearly related, dP/dt or dP/dT will both yield a maximum at the temperature at which the desorption rate is the maximum (T*).
From the data in these graphs, much important data can be compiled. The heating rate (a) and temperature of max desorption (T*) are essential for further analysis. By plotting a graph of ln(T*2/a) vs 1/T*, the activation energy of desorption and Arrhenius prefactor can be determined from the following relation:
2) ln(T*2/a)=Ed/RT*+ln(Ed/RA)
Here, Ed is the activation energy of desorption, A is the Arrhenius prefactor, and R is the gas constant. The slope of the aforementioned graph will be equivalent to Ed/R, and the y-intercept will be equivalent to ln(Ed/RA). Thus Ed and A can be easily determined (see Appendix I for the ln(T*2/a) vs 1/T* graph for this experiment).
The rate constant for the desorption process can also be determined by using the following relation:
3) kd=A e(-Ed/RT)
Here A is the Arrhenius prefactor, Ed is the activation energy of desorption, R is the gas constant, and T is any specified temperature. Since the rate constant is dependent upon temperature, a specific temperature value must be used to get a kd value valid only at that temperature.
The half-life of the adsorbate is the time it takes for the adsorbate coverage on the substrate to decrease by half. This can also be calculated at different temperatures according to the following relation:
4) t1/2=ln2/A e(Ed/RT) = t0 e(Ed/RT)
Here, A is the Arrhenius prefactor, Ed is the activation energy of desorption, R is the universal gas constant, and T is any specified temperature. The vibrational frequency of the physisorbed bond can be approximated by the following relation:
5) vibrational frequency » 1/t0
In this equation, t0 is taken from equation 4 above.
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