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Then (Vincenti and Kruger, 1965, Chapter IV) the total energy of the molecule is given by: ˛ ¼ ˛0 þ ˛00 þ ˛000 : For gases with weakly interacting particles we can identify and separate energy components as: ˛ ¼ ˛tr þ ˛rot þ ˛vib þ ˛el : 36 Introduction to Plasmas and Plasma Dynamics Now since we have deﬁned: X Q¼ eÀ˛i =ðkT Þ where i0tr; rot; vib; el; i and: eÀ˛=ðkT Þ ¼ eÀð˛tr þ˛rot þ˛vib þ˛el Þ=ðkT Þ ¼ eÀ˛tr =ðkT Þ $eÀ˛rot =ðkT Þ $eÀ˛vib ==ðkT Þ $eÀ˛el ==ðkT Þ ; then: Q¼ P eÀð˛tr Þi =ðkT Þ $ i;tr X eÀð˛rot Þi =ðkT Þ $ i;rot X eÀð˛vib Þi =ðkT Þ $ X i;vib ¼ Qtr $Qrot $Qvib $Qel : eÀð˛el Þi =ðkT Þ i;el Furthermore: Q hQtr $Qint where Qint ¼ Qrot $Qvib $Qel : We have been considering energy per molecule, ˛; we can extend this to energy per unit mass, as: energy energy 1 molecule E 1 e ¼˛ $ ¼ $ : mass molecule m mass N m Directly: !
Probability of a single event is wW1, and the probability of a second, independent event is wW2, and the , probability of both events occurring is W1,2 ¼ W1$W2. In general, W ¼ PN! i Ni ! where W is the number arrangements of N with Ni and εi energy; therefore, the maximum disorder state has maximum arrangements. As: the number with property the total number the number of arrangement ¼ : the total number of arrangements Probability ¼ ðgiven stateÞ Therefore: Probability of a given state ¼ PW . Accordingly, for W ðpro: of all W Þ i two given (component) states of a gas molecular system: S1 ¼ f(W1), S2 ¼ f(W2); then for the total system S ¼ S1 þ S2 and W ¼ W1 W2.
Pergamon, New York. , 1958. Kinetic Theory of Gases. McGraw-Hill, New York. , 1963. Introduction to Chemical Physics. McGraw-Hill, New York. , 1975. Introduction to Physical Gas Dynamics. Krieger, Huntington, NY. CHAPTER 3 Molecular Energy Distribution and Ionization in Gases INTRODUCTION The previous chapter developed the functional form for the equilibrium distribution of molecular speeds that occur relative to an average temperature for a gas. This behavior is for kinetic theory that assumed only random translational motion and the related kinetic energy.