The Partition Function Monatomic Gases

The previous section made it clear that all thermodynamic state functions are in some way related to the partition function q. This means that in order to know these state functions, we need to know what q is. How  

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recall that q is simply a sum of negative exponentials of the discrete energy levels  

Technically this is an infinite sum, because there are an infinite number of possible energy levels for any particle (which is a general conclusion of both classical and quantum mechanics). However, because q is defined in terms of negative exponentials, each successive term gets smaller, so the potentially infinite number of terms in the summation does not automatically imply that q = °o. 

Second, if the energy levels are close enough together, then each term in the summation is infinitesimally close to the previous term, and also infinitesimally close to the next term. It can be well approximated that rather than a sum of discrete terms, q can be written as an integral of a continuous function  

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