The torsion at the C-C bond moves the hydrogen atoms on one methyl group with respect to the hydrogen atoms on the other methyl group. This torsion is described by the three-fold cosine function with a barrier height of 12 kJ/mol, MS Fig. III-1. This barrier is high enough so that a Boltzmann distribution over phi, at ordinary temperatures, exhibits a population that is restricted to the regions with phi near 60, 180, and 240 deg, MS Fig. III-2. These three preferred conformations are the rotational isomers of ethane. They are indistinguishable, each being a staggered conformation. The lifetime of a particular rotational isomer is 2 to 4 orders of magnitude longer than the time for a vibration.
The torsion at the internal C-C bond moves one methyl group with respect to the other methyl group. The torsion is still three-fold and symmetric, as it was in ethane, but now the rotational isomers are of different energy, MS Fig. III-3. The energy minimum at phi = 180 deg is lower than the two equivalent minima near 60 and -60 deg, due to incipient repulsion between the methyl groups in the latter two states. This incipient repulsion also displaces these two minima slightly away from 60 and -60 deg. The population of the three rotational isomers is roughly in the ratio 1:3:1 for g+ : t : g-, with the exact ratio depending on the temperature, MS Fig. III-4.
A simple quantitative analysis of the populations of the three rotational isomers can be constructed using the statistical weight for one of the g states relative to the t state.
sigma = exp [-(Eg - Et)/RT]
This energy difference is often denoted by Esigma. Its value is about 2 kJ/mol. The conformation partition function, i. e., the sum of the statistical weights of all conformations, in the rotational isomeric state approximation is
Z = 1 + 2 sigma
and the normalized temperature-dependent population of each rotational isomer (given by the ratio of its statistical weight to Z) is
pt = 1 / Z
pg+ = pg- = sigma / Z
If the internal C-C bonds were independent, the conformation of n-pentane could be accurately predicted from the conformation of n-butane. There would be 3 X 3 states, with tt being the one of lowest energy. Four states would have energy Esigma (these states have a t state at one internal C-C bond and a g state at the other). Four states (with g state at both bonds) would have energy 2Esigma. This simple picture is not obtained in reality, because the four conformations with g states are not of the same energy, MS Fig. III-5. A higher energy is seen when these two g states are of opposite sign than when they are of the same sign. This result shows that the bonds are interdependent. Most polymers have interdependent bonds, a fact that complicates the quantitative analysis of their conformations.
The nine conformations of n-pentane can be described by the introduction of another energy, Eomega, for the second-order (dependent on two torsions) interaction of the methyl groups. Its value is about 8 kJ/mol, which is 4 times as large as Esigma,
Z = 1 + 4 sigma + 2 sigma2 (1 + omega)
The populations of the nine conformations are obtained as the ratio of the statistical weight for each conformation to Z. At 300 K, the populations are
The rotational isomeric state model provides a tractable method for extending these concepts to the analysis of the conformations of large polymers. The interdependence of the bonds is important, because the second-order interaction strongly penalizes local conformation with two successive g states of opposite sign, and these local conformations produce a strong kink in the chain. The conformational partition function, including sigma and omega for all of the first- and second-order interactions, for an unperturbed polyethylene chain with n C-C bonds is
Z = U1 Un-2 Un
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