Simple models for molecules and crystals

Let us first look at a monoatomic perfect gas consisting of N atoms. The internal energy of the gas is 

The molar internal energy of a monoatomic ideal gas is therefore 

The internal energy is, as indicated above, connected to the number of degrees of freedom of the molecule that is the number of squared terms in the Hamiltonian function or the number of independent coordinates needed to describe the motion of the system. Each degree of freedom contributes jRT to the molar internal energy in the classical limit, e.g. at sufficiently high temperatures. A monoatomic gas has three translational degrees of freedom and hence, as shown above, Um =3/2RT andCy m =3/2R. 

The statistical treatment of the vibrational degrees of freedom of crystals is far more difficult compared to gases. Let us initially consider a monoatomic crystal. An atom in a crystal vibrates about its equilibrium lattice position. In the simplest approach, three non-interacting superimposed linear harmonic oscillators represent the vibrations of each atom. The total energy, given by the sum of the kinetic and potential energies for the harmonic oscillators, is 

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