A system of s classical oscillators

We now derive an expression for the density of states of a set of s classical harmonic oscillators. 

First proceed by considering the case s = 2. Recall that for a single oscillator, the density of states is (/ip) 1 and the total states for H E is e Ihv. For two oscillators E 

This result is easily proven by induction, adding one more oscillator and integrating as previously. The density of states is found by differentiation, viz. 

If a molecule comprised of s classical oscillators is in contact with a thermostat at temperature T, the probability P( C)d that it will carry vibrational energy in the range e, C + d is obtained from the Boltzmann distribution law by substituting S2( )d for the degeneracy, viz. 

It should be noted further that the Laplace transform of the density of states if equivalent to the partition function. 

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Figure 9.1 The calculated microcanonical translational product energy distributions for a system of s classical oscillators at the indicated excess energies. The equivalent canonical RED at a system energy k jT = 400 cm is a simple exponential function, exp(—EJkaf ).

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