Theorem, equipartition

The discussion in the preceding section is premised on the assumption that subatomic degrees of freedom are not important for accurate calculation of thermodynamic properties. Naturally, these subatomic degrees of freedom are associated with the state of electrons and nuclei in atoms. In principle, Eq. 7.10 should be modihed to include the nuclear and electronic Hamiltonian 

Of course, as we have discussed, the nuclear and electronic degrees of freedom do not contribute to the molecular partition function at ordinary temperatures. For the nuclear term, this is because the two lowest energy levels of the nucleus are separated by an energy of 0(1 MeV). This means that only at temperatures of 0(10 °) K would 1 MeV, 

At this point it is useful to prove the equipartition theorem. This theorem states that for each quadratic degree of freedom in the Hamiltonian, a 

Consider a system with n distinct, variable degrees of freedom, denoted by a vector 4- The Hamiltonian is defined as 

These degrees of freedom could be the positions and momenta of particles. Consider the th degree of freedom and assume that the Hamiltonian can be split additively as follows  

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With this identification of T, the above result reduces to the generalized equipartition theorem ... 

In this Fourier representation the Hamiltonian is quadratic and the equipartition theorem yields for the thennal... 

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is... 

Internal energy is stored as molecular kinetic and potential energy. The equipartition theorem can be used to estimate the translational and rotational contributions to the internal energy of an ideal gas. 

The result is independent of the coefficient ai and is the same for all coordinates and momenta. Hence H = nO. This expression resembles the equipartition theorem according to which each degree of freedom has the average energy kT, half of it kinetic and half potential, and suggests that the distribution modulus 9 be identified with temperature. 

In emphasizing the need for satisfying the equipartition theorem, the linear response theory provides a connection for stationary processes through the fluctuation-dissipation theorem. 

Numerical simulations of these stochastic equations under fast temperature ramping conditions indicate that the correlations in the random forces obtained by way of the adiabatic method do not satisfy the equipartition theorem whereas the proposed iGLE version does. Thus though this new version is phenomenological, it is consistent with the physical interpretation that 0(t) specifies the effective temperature of the nonstationary solvent. 

According to the energy equipartition theorem of classical physics, the three translational kinetic energy modes each acquire average thermal energy kT (where k = R/NA is Boltzmann s constant),... 

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... 

During the course of the calculations the translational and rotational temperatures, TT and TR, respectively, were monitored at each step. These temperatures were defined by the equipartition theorem ... 

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. 

A molecule can move through space along any of three dimensions, so it has three translational degrees of freedom. It follows from the equipartition theorem that the average translational energy of a molecule in a sample at a temperature T is 3 X kT = kT. The molar contribution to the internal energy is therefore NA times this value, or... 

Rotation requires energy and leads to higher heat capacities for complex molecules the equipartition theorem can be used to estimate the molar heat capacities of gas-phase molecules, Eq. 22. 

Hendrik Antoon Lorentz, from Leyden (Holland), presided the conference, whose general theme was the Theory of Radiation and the Quanta. The conference5 was opened with speeches by Lorentz and Jeans, one on Applications of the Energy Equipartition Theorem to Radiation, the other on the Kinetic Theory of Specific Heat according to Maxwell and Boltzmann. In their talks, the authors explored the possibility of reconciling radiation theory with the principles of statistical mechanics within the classical frame. Lord Rayleigh, in a letter read to the... 

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... 

Translation. From the well-known Equipartition Theorem, which assumes that (l/2)ksT of translational energy resides in each "normal mode," we get... 

The equipartition theorem, which describes the correlation structure of the variables of a Hamiltonian system in the NVT ensemble, is a central component of the held of statistical mechanics. Although the intent of this chapter is to introduce aspects of statistical thermodynamics essential for the remainder of this book -and not to be a complete text on statistical mechanics - the equipartition theorem provides an interpretation of the intrinsic variable T that is useful in guiding our intuition about temperature in chemical reaction systems. 

To derive the equipartition theorem we denote the 6N independent momentum and position coordinates by x and seek to evaluate the ensemble average ... 

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