The Equipartition Theorem

On page 208 we noted that the average energies for gases are integral multiples of (l/2)feT for rotational and translational degrees of freedom. This is a manifestation of the principle of equipartition of energy. Where does this principle come from  

When a form of energy depends on a degree of freedom x, the average energy (f of that component is 

When quantum effects are unimportant so that a large number of states are populated, the sums in Equation (11.51) can be approximated by integrals and the average energy ( ) is 

The constant c gives the scale for the energy level spacings, but note that f) does not depend on c. 

Square-law relations hold for many types of degrees of freedom. For translations and rotations, the energy depends on the square of the appropriate quantum number  

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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. 

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. 

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 ... 

The equipartition theorem is based on classical mechanics. Its application to translational motion is in accord with quantum mechanics as well. At ordinary temperatures the rotational results are also in accord with quantum mechanics. (The greatest deviation from the classical result is in the case of hydrogen, H2. At temperatures below 100 K the rotational energy of H2 is significantly below the equipartition value, as predicted by quantum mechanics.)... 

For polyatomic molecules, theory based on the equipartition theorem allows one to calculate only limiting values for Cy by either completely ignoring all vibrational contributions or assuming that the vibrational contributions achieve their full classical value. For monatomic gases and all ordinary diatomic molecules (where the vibrational contribution is not important at room temperature and can be ignored), definite Cy values can be calculated. For a brief discussion of a more accurate calculation of C (vib), see Exp. 37. 

For each of the three runs on He and N2 (and perhaps CO2), calculate Cp j C using Eq. (13). Also calculate the theoretical value of Cpj Cy predicted by the equipartition theorem. In the case of N2 and CO2, calculate the ratio both with and without a vibrational contribution to C of R per vibrational degree of freedom. 

MCAT won t test tJis equipartition theory This is a variation of an equation from Lecture 2. The equipartition theorem is... 

Q.28.2 Use the equipartition theorem to predict the energy of a standing wave in a cavity filled with standing waves of frequency, v, in thermal equilibrium. 

Expanding h(y,z) in Fourier space and using the equipartition theorem allows to calculate the power spectrum of membrane fluctuations [40,48] ... 

We now assume that (Ax) vanishes, because the force A varies in a completely irregular manner. In other words, the random force A and the displacement x are completely uncorrelated. We also assume that the equipartition theorem holds (i.e., the velocity process has reached its equilibrium value, given by the Maxwellian distribution) so that. 

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