Distribution of molecular kinetic energies

A plot of the Maxwell distribution for the same gas at several different temperatures shows that the average speed increases as the temperature is raised (Fig 4.27). We knew that already (Section 4.9) but the curves also show that the spread of speeds widens as the temperature increases. At low temperatures, most molecules of a gas have speeds close to the average speed. At high temperatures, a high proportion have speeds widely different from their average speed. Because the kinetic energy of a molecule in a gas is proportional to the square of its speed, the distribution of molecular kinetic energies follows the same trends. 

Distribution of relative kinetic energies of collision in the direction along the line of molecular centers at different temperatures. 

Fig. 8-4. Effect of temperature on atomic (or molecular) kinetic energy distribution.

Figure 1.9 Molecular energies follow the Maxwell-Boltzmann distribution energy distribution of nitrogen molecules (as y) as a function of the kinetic energy, expressed as a molecular velocity (as x). Note the effect of raising the temperature, with the curve becoming flatter and the maximum shifting to a higher energy...

The average value of the square velocity has been used in Equation (12) to allow for the fact that a distribution of molecular velocities exists. The nature of the averaging procedure to be used in this case is well established from physical chemistry. We also know from physical chemistry that the average kinetic energy per molecule (KE) per degree of freedom is... 

Kinetic Theory. In the kinetic theory and nonequilibrium statistical mechanics, fluid properties are associated with averages of pruperlies of microscopic entities. Density, for example, is the average number of molecules per unit volume, times the mass per molecule. While much of the molecular theory in fluid dynamics aims to interpret processes already adequately described by the continuum approach, additional properties and processes are presented. The distribution of molecular velocities (i.e., how many molecules have each particular velocity), time-dependent adjustments of internal molecular motions, and momentum and energy transfer processes at boundaries are examples. 

Evaporation and vapor pressure are both explained on a molecular level by the kinetic-molecular theory developed in Section 9.6 to account for the behavior of gases. The molecules in a liquid are in constant motion, but at a variety of speeds depending on the amount of kinetic energy they have. In considering a large sample, molecular kinetic energies follow a distribution curve like that... 

Osorio et al. [134] performed TOF-MS measurements of TNT and RDX on soil surfaces. They used tunable UV radiation from a 130 fs laser to monitor the kinetic energy distribution of N0/N02 photofragments released by the dissociation of TNT and RDX. Analysis of the kinetic energy distribution of the photofragments revealed differences in the processes for NO and NOz ejection in different substrates. Mullen et al. [135] detected triacetone triperoxide (TATP) by laser photoionization. Mass spectra in two time regimes were acquired using nanosecond (5 ns) laser pulses at 266 and 355 nm and femtosecond (130 fs) laser pulses at 795, 500, and 325 nm. The major difference observed between the two time regimes was the detection of the parent molecular ion when femtosecond laser pulses were employed. 

Even though an increase in temperature results in an increase in the average kinetic energy, not all of the gas molecules will have the same amount of kinetic energy. This is why the term average kinetic energy is used. The two curves in Figure 2.5 show the distribution of molecular speeds of a gas at two... 

Finally, the molecule can be translationally, vibrationally, and rota-tionally excited by the distribution of the kinetic recoil energy of the daughter nucleus among the available degrees of freedom. It is apparent from these considerations that the general theoretical treatment of the molecular excitation and fragmentation caused by the /8 decay is quite difficult, even in the case of very simple molecules. Among several theoretical treatments, we will illustrate the time-dependent perturbation theory applied by Cantwell (1956) to the decay of molecular tritium. 

It is useful to have a complete picture of the entire distribution of molecular speeds. This turns out to be important when we study chemical kinetics (see Chapter 18), where we will need to know what fraction of a sample of molecules has kinetic energy above the minimum necessary for a chemical reaction. In particular, we would like to know what fraction of molecules, AN/N, have speeds between u and u + Am. This fraction gives the speed distribution function f u) ... 

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