Activation Energies for Ideal Gases from the Maxwell Boltzmann Distribution

Standard thermodynamic treatments of molecular systems, defined in terms of average properties, require two definitions of equilibrium but cannot explain fully the behaviour of molecular systems where the equilibrium properties are determined by the distribution of molecular energies. They have no explanation for the initiation, propagation and termination of reactions. The treatment proposed here addresses these problems for ideal gas reactions. It shows that the standard explanation, that the kinetic energies of molecular collisions are the sole causes of reactions, is incorrect. It is based on the average properties from the MBD （Maxwell Boltmnann distribution） and its standard deviation, o. It states the principle that explains why two definitions of equilibrium are needed. It provides criteria for the propagation of a reaction, and shows how the mole fraction of reactants needed for propagation, Yi~, can be calculated from the heat capacities of the components. The best heat capacity correlations, due to Harmens, were used to calculate value of Yig.. The energies of the molecules needed to propagate reactions are calculated from the CDF （cumulative distribution function） of the MBD. An alternative method, producing very similar values, is discussed in the following paper. Once the values ofyign are known, the molecular kinetic energies at the ignition temperature can be calculated from the MBD and its CDF. Calculations for four ideal gases show the molecular kinetic energies are much too small, compared to the bond energies, to explain propagation of reactions for four ideal gas mixtures. A proposed explanation, involving excited electronic states also explains why mixtures have ignition temperatures. Correlations for the heat capacities of the reactants which include contributions from electrons in excited states, are needed to establish this.

Methods for Calculating Activation Energies of Ideal Gases

The previous paper Ref. [1] showed how to calculate activation energies for ideal gas reactions from the CDF （cumulative distribution function） of the MBD （Maxwell Boltzmann Distribution） and the heat capacity data of the components. The results presented here show comparisons of activation energies of four ideal gases calculated in that way with those calculated from the ND （Normal Distribution） and its CDF. The evaluation of the CDF for the MBD in Ref. [1] required extensive numerical integrations for each substance. In this paper this method of calculating activation energies is generalised, by showing the CDF is a unique function, independent of temperature and composition, enabling the CDF to be presented graphically or in tabular form. These activation energies are compared to those calculated from the ND and its CDF. The MBD is related to the ND because it has a generating function which is shown here to have the simple form （1-kT）-1.5. The activation energies obtained from the CDF of the ND are shown to agree ca. 5-7% with those obtained directly from the MBD. Because existing thermodynamic treatments are based on average properties, they cannot give either a complete account of thermodynamic controlled and kinetic controlled equilibrium states or explain transitions between them. Complete treatments must include effects from the MBD which are the causes of kinetic controlled equilibrium. The basis for a complete treatment is outlined, which includes the standard deviations and activation energies.

Effect of Spatial Dimension and External Potential on Joule-Thomson Coefficients of Ideal Bose Gases

Based on the form of the n-dimensional generic power-law potential, the state equation and the heat capacity, the analytical expressions of the Joule-Thomson coefficient （3TC） for an ideal Bose gas are derived in n-dimensional potential. The effect of the spatial dimension and the external potential on the JTC are discussed, respectively. These results show that： （i） For the free ideal Bose gas, when n/s ≤ 2 （n is the spatial dimension, s is the momentum index in the relation between the energy and the momentum）, and T → Tc （Tc is the critical temperature）, the JTC can obviously improve by means of changing the throttle valve＇s shape and decreasing the spatial dimension of gases. （ii） For the inhomogeneous external potential, the discriminant △= [1 - y∏^ni=1（kT/εi）^1/tiГ（1/ti＋1）] （k is the Boltzmann Constant, T is the thermodynamic temperature, ε is the external field＇s energy）, is obtained. The potential makes the JTC increase when △ 〉 0, on the contrary, it makes the JTC decrease when A 〈△. （iii） In the homogenous strong external potential, the JTC gets the maximum on the condition of kTεi〈〈1.