A Appendix Density of states and partition functions

We begin with the density of internal states. First, we need the number, Ni(Ei), of internal states, whose energy is less than or equal to i. If the reactants are an atom in its groimd electronic state and a diatomic molecule, then the only internal energy is the rovibrational energy of the diatom. We need to coimt how many states have a rovibrational energy less than or equal to Ei. We can formally write this cormting as 

The density of internal states is defined such that pi Ei)AE is the number of internal states in the narrow energy range E to Ei + 6Ei. For the systems of interest to us, the number of states per energy interval is sufficiently high so that the number of states is essentially a smooth function. Therefore the density of states is also a continuous function and is to be understood as 

Following Dirac we call the derivative the delta function  

To practice handling delta functions see Problem C. Taking the derivative of (A.6.1) with respect to eneigy, we get the formal definition of the density of states 

The molecular partition function Q(V) is defined as the normalization factor for the Boltzmann distribution at the temperature T. Specifically, the probability Pi of a single internal state with energy Ei is 

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