Vibrations equipartition

Quantum mechanical vibrations do not obey a square-law potential. Quantum 1brational energies depend only linearly on the vibrational quantum number, Ev = (v + l/2)hv. Therefore the equipartition theorem for vibrations is different from Equation (11.53). Using Appendix D, Equation (D.9), the vibrational equipartition theorem is... 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

The equipartition principle is a classic result which implies continuous energy states. Internal vibrations and to a lesser extent molecular rotations can only be understood in terms of quantized energy states. For the present discussion, this complication can be overlooked, since the sort of vibration a molecule experiences in a cage of other molecules is a sufficiently loose one (compared to internal vibrations) to be adequately approximated by the classic result. 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

The mechanical modes whereby molecules may absorb and store energy are described by quadratic terms. For translational kinetic energy it involves the square of the linear momentum (E = p2/2m), for rotational motion it is the square of angular momentum (E = L2121) and for vibrating bodies there are both kinetic and potential energy (kx2/2) terms. The equipartition principle states that the total energy of a molecule is evenly distributed over all available quadratic modes. 

By the equipartition principle it now follows that each rotational degree of freedom can absorb energy of kT while each vibrational mode can absorb kT. By the same principle the heat capacity of an ideal gas... 

This interval is considerably more than the spacing between rotational levels, and since AEy kT at room temperature it is safe to conclude that most molecules in such a sample exist in the lowest vibrational state with only their zero-point energies. Here is the real reason for the breakdown of the classical equipartition principle. 

Although the upper limits of DH° (A-B) - /)//,) (A-B), set by the equipartition principle, must be regarded with caution (see table 5.1), they are indeed applicable to many molecules because, as stated, the vibrational degrees of freedom are not totally frozen at 298.15 K. For instance, when A and B are heavy atoms, like cesium, the vibration frequency is small enough to ensure that the vibration mode is considerably excited, for example, DH° Cs-Cs) -DH Cs-Cs) is only 1.4 kJ mol-1 [17]. 

J.C. Maxwell saw these atoms as capable of internal vibrations 9). Atoms had some form of internal stracture and were not static If there were internal modes of motion in the atoms, Boltzmarm predicted that equipartition of energy would lead to an increased heat capacity. Maxwell worried about the missing heat capacity due to these motions as well. Maxwell died before an explanation of the... 

From a molecular viewpoint, we know that heat capacity is closely connected to internal modes of molecular vibration. According to the classical equipartition theorem (Sidebar 3.8), a nonlinear polyatomic molecule of Aat atoms has ftmodes = 3Aat — 6 independent internal modes of vibration, each of which would contribute equally to heat capacity... 

A fundamental theorem of classical mechanics called the equipartition theorem (which we shall not derive here) states that the average energy of each degree of freedom of a molecule in a sample at a temperature T is equal to kT. In this simple expression, k is the Boltzmann constant, a fundamental constant with the value 1.380 66 X 10-21 J-K l. The Boltzmann constant is related to the gas constant by R = NAk, where NA is the Avogadro constant. The equipartition theorem is a result from classical mechanics, so we can use it for translational and rotational motion of molecules at room temperature and above, where quantization is unimportant, but we cannot use it safely for vibrational motion, except at high temperatures. The following remarks therefore apply only to translational and rotational motion. 

If equipartition is obtained, and the 138 kcal./mole exothermicity of reaction (31) is divided between two 02(3S ) molecules, the maximum vibrational energy possible would be 69 kcal./mole which corresponds to the experimental result. Reaction (32), 69 kcal./mole endothermic, would then be barely possible, but... 

This is the classical equipartition theorem. It states that each rotation (which only contributes one term to the sum) adds RT/2 to the energy, whereas each vibration (which contributes two terms) adds RT to the energy. From Eq. (73), each of the... 

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