Extensive variable conjugate intensity

The fundamental equation for U is in agreement with the statement of the preceding section that for a homogeneous mixture of Ns substances, the state of the system can be specified by Ns + 2 properties, at least one of which is extensive. The total number of variables involved in equation 2.2-8 is 2NS + 5. Ns + 3 of these variables are extensive (U, S, V, and (nj), and Ns + 2 of the variables are intensive (T, P, /.q ). Note that except for the internal energy, these variables appear in pairs, in which one property is extensive and the other is intensive these are referred to as conjugate pairs. These pairs are given later in Table 2.1 in Section 2.7. When other kinds of work are involved, there are more than 2Ns + 5 variables in the fundamental equation for U (see Section 2.7). 

Note that the Legendre transform has interchanged the roles of the conjugate intensive /r(H + ) and extensive nc(H) variables in the last term of equation 4.1-9. The number D of natural variables of G is Ns + 2, just as it was for G, but the chemical potential of the hydrogen ion is now a natural variable instead of the amount of the hydrogen component (equation 4.1-7). 

Energy = Intensive variable x Conjugate extensive variable. (2.1)... 

In conclusion, entropy is the physical quantity that represents the capacity of distribution of energy over the energy levels of the individual constituent particles in the system. The extensive variable entropy S and the intensive variable the absolute temperature Tare conjugated variables, whose product TdS represents the heat reversibly transferred into or out of the system. In other words, the reversible transfer of heat into or out of the system is always accompanied by the transfer of entropy. 

In these equations we see the regularity that the partial differential of these four thermodynamic potentials with respect to their respective extensive variables gives us their conjugated intensive variables and vice versa. We thus obtain minus the affinity of an irreversible process in terms of the partial differentials of U, H, F, and G with respect to the extent of reaction affinity is an extensive variable. 

When we consider changes of H with P we must invoke Eq. (1.22.6) which addresses the change from the extensive variable V to its intensive conjugate, P. Thus, we write... 

The sign of the derivatives can be determined simply by writing down the variables in two rows, the intensive variables above, the extensive variables below so that conjugate pairs are in the same column. The signs attached to the intensive variables are those of the corresponding terms in the fundamental equation (4.23) for the internal energy. 

We summarize in Table 2.1 the physical dimensions of various energy forms. We emphasize that the list is not complete. In square brackets the physical dimension of the extensive variable has been given. The intensive variable has the energy-conjugated physical dimension. In the following sections, we discuss briefly some issues of the enagy forms detailed in Table 2.1. 

From the foregoing it is appealing to relate generalized susceptibilities xj as the differential of an arbitrary extensive variable X with respect to its energy-conjugated intensive variable... 

The replacement of an arbitrary number of extensive variables i i = 1,..., m) by their conjugate intensive variables m yields new energy functions [/i ) of the system... 

Eq. (3.4) that d = 0, namely, that E is constant Consequently, E must be a function of the extensive variables (i.e., a state function). The expression E=E (ci, 62, 62,. . . , 6 ), known as the th6rmodynamic pot6titial Junction, precisely characterizes the system. It follows from the mathematical analysis that the intensive variables conjugated to the independent, extensive variables can be derived from the latter if the thermodynamic potential function is known. Because the energy is a state function, d is a total differential from Eq. (3.1), it therefore follows that... 

Where entropy S and volume V are the extensive variables and temperature T and pressure p are the conjugated intensive variables (the signs indicated follow from the definition of the positive direction of the energy fiow, namely, one directed into the system). Thus, the thermodynamic potential of this system is E=E(S, V) if this function is known, the system can be regarded as completely described. But the variables S and V cannot be controlled in a simple, direct manner. Thus, it would be preferable to select the pressure p and the temperature T as variables because they can easily be set experimentally. Which quantity is then the proper thermodynamic potential ... 

It follows from the third law that for every extensive variable e and every conjugated intensive variable i, the following equations apply at T= 0 K ... 

If we choose an intensive variable Y, which is the conjugate of the extensive variable X, we know that the variation of the generalized chemical potential of a component with that variable Y is of the form ... 

Let us consider a partial molar variable. In a uniform solution, this variable is a function of the variables of the problem, i.e. some intensive and extensive variables belong to other conjugate couples of J. If we have one mole of a solution overall, by having the molar fractions jc, as the composition... 

We introduce the subject by considering a thermodynamic function of state Y which involves + 1 independent extensive variables XQ,x, ...,Xr which are conjugate to a corresponding setpo, Pi, , Pr of intensive variables. We are interested in constructing a new thermodynamic function of state Z, in which the first n+ extensive quantities xi (i = 0,1,..., n) are replaced by their intensive counterparts p, (i = 0,1,..., ). To achieve this objective we introduce the (partial) Legendre transform as follows ... 

In a mouthful of words, a sign change occurs when one switches from taking second derivatives of a function of state (F) with respect to extensive variables, to taking second derivatives of the Legendre transformed function of state (Z) with respect to the conjugate intensive variables. This mathematical relationship must be clearly kept in mind what now follows. 

It is recalled to mind that in case of a transition from thermodynamics to statistical mechanics the macroscopic equilibrium state of the system in consideration must be preset completely and precisely. Of the microsc< ic data those are presupposed as known which are necessary to calculate all possible mechanical states. If the macroscopic state of the system is fixed-by the values of m intensive variables. .. and n—m extensive variables Uej+i. .. 17 , the extensive variables. .. U, canonically conjugated to the variables. .. t , are the very quantities which, from the mechanical standpoint, only statements on the average behavior can be obtained from. If, for instance, the thermodynamical state of a system is fixed by certain values of the variables T (temperature), V (volume), and. .. N,... (numbers of moles), the statistical mechanics can give only mean or most probable values for the internal energy U. If the thermodynamical state is preset by fixed values of the variables T, p (pressure), and. .. Nf. from the viewpoint of statistical mechanics only mean or most probable values are possiUe regarding the internal energy V and the volume V. The addi-... 

The second class of control variables comprises derivative strength-type ( intensive ) properties Rb such as temperature and pressure. Each Rt is related through the fundamental equation (8.72) to a conjugate extensity Xt by a derivative relationship ( equation of state ) of the form [cf. (3.32), (4.33)]... 

This shows that the natural variables of G for a one-phase nonreaction system are T, P, and n . The number of natural variables is not changed by a Legendre transform because conjugate variables are interchanged as natural variables. In contrast with the natural variables for U, the natural variables for G are two intensive properties and Ns extensive properties. These are generally much more convenient natural variables than S, V, and k j. Thus thermodynamic potentials can be defined to have the desired set of natural variables. 

This equation 5.20, called the Gibbs-Duhem equation, is unique among a variety of the thermodynamic equations of state in that the characteristic variables are all intensive quantities, each multiplied by its conjugate extensive quantity. 

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