The rotation of a ethane molecule is modeled as the motion of a

quantum mechanical rigid rotator. The rotation takes place about

an axis through the center of mass of the molecule and

Perpendicular to the line joining the two atoms. We take I as the

moment of inertia of the molecule about this axis; where is the reduced mass of the two atoms and r0 is the equilibrium

value of the distance between the nuclei.

Quantum mechanics states the quantized energy levels characteristic temperature for rotation quantum mechanics also give about the energy level states for

level “l” corresponding to different possible directions of the

angular momentum vector. Given these results, we can write

down the partition function for rotation We write x = l(l+ 1) and treat x as a continuous variable. Note that

dx = (2l+ 1)dl this approximation As potential Energy U can be written as For Thus, at significantly high temperatures, the rigid rotator exhibits

the equipartition of energy between two degrees of freedom.

At very low temperatures such that For , the partition function,

we can write above Equation can be expanded as Retaining only the first two terms, we can write Since the exponential term is small compared with unity, we can

use the

approximate relation , and obtain the result So Potential Energy can be written in form of So as heat capacity can be written in form of As partition function Q can be written as Nv is greater than pV because of intermolecular repulsion

between two molecules so when one molecule sit at its centre it

will cover volume more than its original volume so Nv is greater

than pV.

Like two sphere have volume v But these will cover volume as shown below As we can see that some amount of extra volume they cover so

Nv is greater than pV As total no. of molecules is Nv so no. of pair of molecule for attration =

Each pair have attraction energy of E

So total attraction energy of this model will be

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