ABSTRACT
A reaction needs minimum energy to break chemical bindings to occur. These energies come from different forms and configurations with electron transfer and geometrical rearrangements.
Consider a closed system with N total number of molecules which is the sum of a set of ni molecules. Thus, for each set ni, an energy εci is associated. These energies result from the translation movement of the mass center and vibration and rotation molecules movement. In fact, each molecule has its own trajectory and the fraction of molecules with energy εci follows the Boltzmann distribution function which affirms that “the energy of a system in equilibrium is distributed exponentially according to the different degrees of freedom, satisfying the Boltzmann law.’’ Therefore, the mean energy is:
ε= ∑
) ∑
) (12.1)
where kB is the Boltzmann constant. The denominator of this equation contains the partition function, which is the
sum of all states and energy levels. In the case of an ideal monoatomic gas, there is only translation movement of the mass center, and the mean energy can be calculated as follows:
ε= kBT2 ∂(fp) ∂T
(12.2)
where
fp = ∑
) (12.3)
In this case, the partition function can be calculated, resulting in the following equation:
fp =V [
2πmkBT h2
]3/2 (12.4)
Substituting Equations 12.4 and 12.3 into Equation 12.2, we obtain the mean
energy:
ε= 3 2 kBT (12.5)
Note that the mean energy depends only on the temperature. For diatomic molecules, there are several degrees of freedom, being three of trans-
lation of the mass center and three relative to the rotation (2) and vibration (1) movements, as shown in Figure 12.1.
