Thermodynamic Euler relations

While the evaluation of pi can be simphfied using Euler relations for certain classes of monomer structures, these Euler relations [50] do not apply, for example, to the three polyolefin chains depicted in Fig. lb because they have short side chains. (Of the monomer structures in Fig. la, the Euler relations for pi are valid only for PEE and PHI.) Therefore, the geometrical index pi = /Mi = nPVsi is evaluated by directly enumerating all sets of three sequential bonds (nP ) that traverse a monomer of species i. [46] siunmarizes the details of these calculations and tabulates values of p,- for several monomer structures, so we pass now to a consideration of how thermodynamic properties depend on n and p, in the high molecular weight, incompressible system limit of the LCT. 

The extensive thermodynamic variables are homogeneous functions of degree one in the number of moles, and Euler s theorem can be used to relate the composition derivatives of these variables. 

This relation corresponds to the Euler equation, and the following treatment is similar to that familiar from thermodynamics. The variation of the volume 5T with an infinitesimal change of A,-, SA,-, can be obtained, on one hand, from 5F = E A,8A,. ... 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

This defines a set of equations for the mean field Hamiltonians HPF. These equations have to be solved self-consistently since the thermodynamic values within the angle brackets in (109) involve the mean field Hamiltonians // F. In principle, all // F can be different in practice, we impose symmetry relations. Therefore, we choose a unit cell, compatible with the symmetry of the lattice introduced in Section II,D, and we put Hpf equal to // F whenever P and P belong to the same sublattice. Moreover, we apply unit cell symmetry that relates the mean field Hamiltonians on different sublattices. By using the symmetry-adapted functions introduced in Section II,B, the latter symmetry can be imposed as follows. We select a set of molecules constituting the asymmetric part of the unit cell. Then we assign to all other molecules P Euler angles tip-through which the mean field. Hamiltonian of some molecule P in the asymmetric part has to be rotated in order to obtain HrF. As a result, we... 

The lattice thermodynamics described by Swope and Andersen are formulated as the thermodynamics of a constrained system. A system with a fixed value of M exhibits a free energy Ac, pressure pc, and chemical potential Pc appropriate to a constrained system. Moreover, a Euler equation exists to relate the principal thermodynamic quantities... 

An important set of identities obtained from the Euler reciprocity relation and thermodynamic equations is the set of Maxwell relations. These relations allow you to replace a partial derivative that is difficult or impossible to measure with one that can be measured. One of the Maxwell relations is ... 

Equation (A-29) holds for all values of A. For A = 1, Eq. (A-29) reduces to Eq. (A-23) and we have completed the proof of Euler s theorem. Equation (A-23) is of use in relating extensive thermodynamic properties to the corresponding partial molal properties. 

In Chapter 9 w e will use the Euler relationship to establish the Maxwell relations between the thermodynamic quantities. Here we derive the Euler relationship. Figure 5.12 shows four points at the vertices of a rectangle in the xy plane. Using a Taylor series expansion, Equation (4.22), compute the change in a function Af through tw o different routes. First integrate from point A to point B to point C. Then integrate from point A to point D to point C. Compare the results to find the Euler reciprocal relationship. For Af = f(x + Ax,y + Ay) -f(x,y), the hrst terms of the Taylor series are... 

Some functions depend on more than a single variable. To find extrema of such functions, it is necessary to find where all the partial derivatives are zero. To find extrema of multivariate functions that are subject to constraints, the Lagrange multiplier method is useful. Integrating multivariate functions is different from integrating single-variable functions multivariate functions require the concept of a pathway. State functions do not depend on the pathway of integration. The Euler reciprocal relation relation can be used to distinguish state functions from path-dependent functions. In the next three chapters, we will combine the First and Second Laws with multivariate calculus to derive the principles of thermodynamics. 

We have explored two methods of thermodynamics. The Maxwell relations derive from the Euler expression for state functions. They provide another way to predict unmeasurable quantities from measurable ones. For multicomponent systems, you get another relationship from the fact that many thermodynamic functions are homogeneous. In the following chapters we will develop microscopic statistical mechanical models of atoms and molecules. 

The turbine power can also be related to the flow properties using the first law of thermodynamics, Wy = ihAht — Q, defined as a function of the change of the fluid s total (or stagnation) enthalpy hi, and the rate of heat transfer to the flow Q. For a constant mass flow through the turbine, the total power per unit mass flow can be expressed as the Euler turbine equation ... 

The extensive thermodynamic variables are a first-order homogeneous function of Nj and N, a relation known as Euler s theorem ... 

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