Principle of equipartition

If the separation of successive levels is small compared to kT, these summations may be expressed as integrals, as discussed in 12 6. We obtain 

This has already been shown to be the case with regard to the components of translational energy, equations (12 37)-(12 39), and is also approximately true of the energy of rotation about an axis. Substituting (12 108) in (12 107) and evaluating the integrals, which are now of a stcmdard form,t we obtain 

Therefore a molecule has an average energy, of the type in question, equal to ikT and the corresponding contribution to the heat capacity of the gas is ik per molecule or 4.167 J mol . This result is in accordance with equations (12 66) and (12 68) which were concerned with the three degrees of translational freedom. 

Equation (12 109), which expresses the principle of equipartition, is only applicable under the conditions stated, in particular that the separation of the levels is small and that the energy can be expressed as a square term, as in (12 108). These conditions certainly apply to the translational motion and also, at room temperatures and above, to the rotational motion. A linear molecule requires two co-ordinates for the specification of its rotation, and for each of these the contribution to Cy is per mole. A non-linear molecule, on the other hand, requires three co-ordinates for the specification of its rotation and the rotational contribution to Cy is f Jl per mole. The overall value of Cy would thus be 24.943 J K mol S and the corresponding value of Cp 33.267 J mol  

Turning to the case of vibration, the above conditions are not obeyed in either respect the separation of levels is not small compared to kT, at normal temperatures, and the eigenvalues are not proportional to the square of the quantum number. In fact, for a harmonic oscillator, the permissible energies are a linear function of the quantum number v, 

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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. 

Another consequence of Eq. (4.91) is that if we arrange the n subsystems in time instead of in space, then the collection of subsystems constitutes the reaction path of a batch reactor where Vk is the volume of subsystem k. For a specified conversion and time, we should minimize the sum of Jk(AGk/T)Vk. This minimization leads to results similar to Eq. (4.91), and supports the principle of equipartition of forces. Hence, for a given total conversion and reaction time, minimum entropy production results when the driving force A GIT is equal in all n time intervals. Similarly, the conversion is maximum for a given entropy production and reaction time when the driving forces are uniform. 

Using the principle of equipartition of energy, the translational, vibrational and rotational contribution to the heat capacity (Cv) of the hydrogen molecule, indicated ... 

In the limit of long wave-lengths and hi temperatures the principle of equipartition of energy is valid. This leads to the classical formula of... 

Here is the combination of Frank elastic moduli and K22-The last equation has a remarkable feature the different Fourier components of fluctuations are decoupled because they are normal modes for the system. This allows us to apply the principle of equipartition, according to which the energy of each mode is equal to k TI2. Therefore, for each mode with a = 1, 2, the final equation for the mean-square magnitude of the director fluctuations reads ... 

Ug - Uj is the difference in kinetic energy between a sorbate molecule in the gaseous and adsorbed states and as such depends on the nature of the adsorbed phase. Using the principle of equipartition of energy... 

The same result as (13 45) could also be obtained by applying the principle of equipartition ( 12 12) which will be valid under conditions where hvjkT< i. Each oscillator will have a mean energy of kT, as shown in equation (12 111), and the total thermal energy of the crystal is therefore SNk T in agreement with (13 45). 

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 ... 

The thermal motion assumption provides a justification here to use the principle of equipartition of energy, which states that the kinetic energy of each particle on the... 

Planck realized that the difficulty lay in the principle of equipartition which could only be circumvented by a complete departure from classical mechanics. He postulated that an oscillator of frequency a)/2ir, instead of being able to assume all possible energy values, could exist only in one of a set of equally spaced energy levels having the... 

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