Total Partition Function of System

For a system of distinguishable particles the total partition function of the system is the product of all the individual partition functions, i.e. 

However, if the particles are indistinguishable, as the atoms in a gas, the number of possible configurations is significantly reduced and the partition function for an ensemble of atoms or molecules is therefore 

Consider a particle (a molecule or atom) for which the different degrees of freedom are independent and the energy is simply the sum of the energies contained in the different degrees of freedom. We can then write the partition function as the product of the partition functions for the various degrees of freedom. For an atom this is rather trivial  

Several relevant physical parameters can be extracted for this partition function, such as  

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