Equipartition principle

Some options for achieving a thermodynamic optimum are to improve an existing design so the operation will be less irreversible and to distribute the irreversibilities uniformly over space and time. This approach relates the distribution of irreversibilities to the minimization of entropy production based on linear nonequilibrium thermodynamics. For a transport of single substance, the local rate of entropy production is 

The total flow is the integral over time and space of the local flow 

The difference between the general case and the average value is (Hohmann, 1971) 

The square bracket on the right of the above equation is the difference between the mean square and the square mean of the force distribution, and is the variance of X. We therefore have 

The entropy production Pw of a process with a uniform driving force is smaller than that of a nonuniform situation with the same size, and duration of the same average driving force, and the same overall load J. Equations (4.94) and (4.95) show that the local flow and the local rate of entropy production l will be constant whenX is uniform. 

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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. 

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. 

This expression reflects the quantum-mechanical version of equipartition. Substituting the partition functions for different modes into (6) the classical equipartition principle can readily be recovered. 

Quantum theory therefore correctly predicts the equipartition principle in the classical limit. 

In /z-space the most probable kinetic energy ep — mCp = kT, which differs from the average energy according to the equipartition principle,... 

According to the calorimetrists view, if we consider an electron as a particle that takes part in chemical reactions as any other chemical element, then it is fair to agree on Af7/°(c-,g) = 0, at any temperature. Moreover, because we regard the electron as a regular monoatomic perfect gas, Boltzmann statistics or the equipartition principle [1] imply that C° = Cy + R = 5R/2 (three translational degrees of freedom) or X = — //(, )c = 298.15 x C" =... 

Table 5.1 Upper limits of DH°(A-B) - DHq(A-B) for some molecules, estimated by the equipartition principle (EP), compared with the correct values [17]. Data in kj mol-1.

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]. 

This chapter establishes a direct relation between lost work and the fluxes and driving forces of a process. The Carnot cycle is revisited to investigate how the Carnot efficiency is affected by the irreversibilities in the process. We show to what extent the constraints of finite size and finite time reduce the efficiency of the process, but we also show that these constraints still allow a most favorable operation mode, the thermodynamic optimum, where the entropy generation and thus the lost work are at a minimum. Attention is given to the equipartitioning principle, which seems to be a universal characteristic of optimal operation in both animate and inanimate dynamic systems. 

The unknown coupling constant B can be related to the mean-square fluctuation of the bilayer position o 2 = (u(x,y)2) by using the equipartition principle... 

Example 5.5 Equipartition principle in separation processes Extraction Since the minimization of entropy production is not always an economic criterion, it is necessary to relate the overall entropy production and its distribution to the economy of the process. To do this, we may consider various processes with different operating configurations. For example, by modifying an existing design, we may attain an even distribution of forces and hence an even distribution of entropy production. 

Example 5.7 Equipartition principle Heat exchanger For a heat exchanger operating at steady state, the total entropy generation P is obtained by integrating over the surface area... 

Example 5.12 Equipartition principle in an electrochemical cell with a specified duty We desire the electrode to transfer a specified amount of electricity Q over a finite time t0... 

The equipartition principle is mainly used to investigate binary distillation columns, and should be extended to multicomponent and nonideal mixtures. One should also account for the coupling between driving forces since heat and mass transfer coupling may be considerable and should not be neglected especially in diabatic columns. 

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