By Fantasy Flight Games, Various
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52) %EjdNj = 0. 53) Utilizing the Lagrange's method of undetermined coefficients, multiplying Eq. 52) and Eq. 53) by a and ß respectively and combining with Eq. 51) yields N, dJV, = 0 . 2 ENERGY DISTRIBUTION OF MOLECULES 39 Let the coefficient of dNj to be 0, the relation which 7V; must satisfy is obtained JjL±\)-a-ßEj = 0. 56) Under such circumstances the factor 1 in the denominator of Eq. 55) can be omitted, and identical result is obtained for both Bose- Einstein and Fermi-Dirac statistics N] 1 —= 7^7 (157) This limiting result is called the Boltzmann limit and can be obtained utilizing the Boltzmann statistics, or the classical statistics.
Thus, the similarity of two flows in rarefaction requires that the ratio of Ma to Re remains the same. For not too high a speed the similarity of two flows requires the equality of Mach number and Reynolds number, and thus, according to Eq. 27), the equality of Knudsen number. Owing to Eq. 27) the similarity of flows requires that any two of the three parameters Ma , Re and Kn are the same. 6 SIMILARITY CRITERIA 15 Re number unchanged but allowing a larger Ma number in a scope where the compressibility can be neglected.
20). Now let us estimate the relative order of magnitude of the three modes of energies. Use d to denote the linear scale of the molecule. The electronic energy ee is associated with the motion of the electron with mass m in the molecular scope d, the order of magnitude of the electron velocity v, according to the Heisenberg uncertainty principle (see Eq. 22)), can be estimated as h v . md The order of magnitude of ee is mv , that is Fig. 1 DIATOMIC MOLECULES £e~-^7ma 29 (1-26) The order of magnitude of the vibrational energy £ v , according to Eq.