Weighted Averaging A Very Important Concept

At this point, we would like to proceed to apply the KMTG to experimentally measurable quantities, but we need a firmer foundation for the velocities and speeds of atoms/molecules in the gas phase. The velocity based on the phenomenological ideal gas law is suspect because we know it may not apply to high pressure and/or low temperature, so we need a more rigorous method. The concept/principle of weighted averaging occurs in kinetics, statistical thermodynamics, and in quantum mechanics, so we think this is more than just a math interlude it is a unifying principle. 

FIGURE 3.2 Hypothetical grade distribution in a class of 15 students. 

Wc now come to a key idea, which the student should make sure he or she understands since we will use it over and over in later applications. We introduce the symbol ( ) to denote an averaging process, in this case a weighted average as shown in Table 3.1. 

Weighted Average Grade of Class No. of Students Grade 

We will use these formulas many times in this text so that you will become accustomed to using them. In the case at hand, we first note that the integrands have squared exponents to the base e, but the power of x in the integrand can be odd or even. Note also that in the exponent of the Boltzmann factor, the value of a is squared relative to the value in the final answer. The formula for the odd case (2p + 1) can be checked if desired using integration-by-parts for a low value of the exponent, but the even case (2p) is more difficult and one can wonder, where the factor of y/ii 

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