Natural variables equations

Equation (6.111) is sometimes called the combined first and second laws of thermodynamics, and this equation suggests that S and V are natural independent variables for U. Conversely, we can say that U and V are natural variables for S. One can also... 

In this chapter we introduce a more useful equation for the surface tension. This we do in two steps. First, we seek an equation for the change in the Gibbs free energy. The Gibbs free energy G is usually more important than F because its natural variables, T and P, are constant in most applications. Second, we have just learned that, for curved surfaces, the surface tension is not uniquely defined and depends on where precisely we choose to position the interface. Therefore we concentrate on planar surfaces from now on. 

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

This fundamental equation for the entropy shows that S has the natural variables U, V, and n . The corresponding criterion of equilibrium is (dS) 0 at constant U, V, and n . Thus the entropy increases when a spontaneous change occurs at constant U, V, and ,. At equilibrium the entropy is at a maximum. When U, V, and , are constant, we can refer to the system as isolated. Equation 2.2-13 shows that partial derivatives of S yield 1/T, P/T, and pJT, which is the same information that is provided by partial derivatives of U, and so nothing is gained by using equation 2.2-13 rather than 2.2-8. Since equation 2.2-13 does not provide any new information, we will not discuss it further. 

Since F is the symbol for the number of independent intensive variables for a system, it is also useful to have a symbol for the number of natural variables for a system. To describe the extensive state of a system, we have to specify F intensive variables and in addition an extensive variable for each phase. This description of the extensive state therefore requires D variables, where D = F + p. Note that D is the number of natural variables in the fundamental equation for a system. For a one-phase system involving only PV work, D = Ns + 2, as discussed after equation 2.2-12. The number F of independent intensive variables and the number D of natural variables for a system are unique, but there are usually multiple choices of these variables. The choice of independent intensive variables F and natural variables D is arbitrary, but the natural variables must include as many extensive variables as there are phases. For example, for the one-phase system described by equation 2.2-8, the F = Ns + 1 intensive variables can be chosen to be T, P, x, x2,xN. and the D = Ns + 2 natural variables can be chosen to be T, P, ni, n2,..., or T, P, xx, x2,..., xN and n (total amount in the system). 

Since there are D = 3 natural variables, there are 23 — 1=7 possible Legendre transforms. The Legendre transforms defining H, A, and G are given in equations 2.5-1 to 2.5-3, and the four remaining Legendre transforms are... 

These four Legendre transforms introduce the chemical potential as a natural variable. The last thermodynamic potential U T, P, /<] defined in equation 2.6-6 is equal to zero because it is the complete Legendre transform for the system, and this Legendre transform leads to the Gibbs-Duhem equation for the system. 

Thus we have demonstrated the remarkable fact that equation 2.8-1 makes it possible to calculate all the thermodynamic properties for a monotomic ideal gas without electronic excitation. Here we have considered an ideal monatomic gas. but this illustrates the general conclusion that if any thermodynamic potential of a one-component system can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be calculated. 

The preceding section was based on the fundamental equation for G in terms of the extent of reaction, but in order to identify the D natural variables for a one-reaction system at equilibrium, we need to apply the condition for equilibrium ZvjjUi = 0 (equation 3.1-6) that is due to the reaction. That is done by using each independent equilibrium condition to eliminate one chemical potential from equation 2.5-5. This is more easily seen for a simple reaction ... 

This form of the fundamental equation for G applies to a system at chemical equilibrium. Note that the number D of natural variables of G is now C + 2, rather than Ns + 2 as it was for a nonreaction system (see Section 2.5). There are... 

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). 

In discussing one-phase systems in terms of species, the number D of natural variables was found to be Ns + 2 (where the intensive variables are T and P) and the number F of independent intensive variables was found to be Ns + 1 (Section 3.4). When the pH is specified and the acid dissociations are at equilibrium, a system is described in terms of AT reactants (sums of species), and the number D of natural variables is N + 3 (where the intensive variables are T, P, and pH), as indicated by equation 4.1-18. The number N of reactants may be significantly less than the number Ns of species, so that fewer variables are required to describe the state of the system. When the pH is used as an independent variable, the Gibbs-Duhem equation for the system is... 

The preceding paragraph applies to a system in which there are no biochemical reactions. Now we consider systems with reactions that are at equilibrium (Alberty, 1992d). For a chemical reaction system, we saw (Section 3.4) that D = C + 2 and F = C + 1 for a one-phase system. For a biochemical reaction system at equilibrium, we need the fundamental equation written in terms of apparent components to show how many natural variables there are. When the reaction conditions 2 vj//f = 0 for the biochemical reactions in the system are used to eliminate one for each independent reaction from equation 4.1-18, the following fundamental equation for G in terms of apparent components is obtained ... 

This shows that the natural variables for G for this system before phase equilibrium is established are T, P, nAx, and nAp. When A is transferred from one phase to the other, dnAa = — dnAp. Substituting this conservation relation into equation 8.1-1 yields... 

This form of the fundamental equation, which applies at equilibrium, indicates that the natural variables for this system are T, nAx, and nAfi. Alternatively, P, nAx, and nAp could be chosen. Specification of the natural variables gives a complete description of the extensive state of the system at equilibrium, and so the criterion of spontaneous change and equilibrium is dG < 0 at constant 7( nAz, and nA/l or... 

The number C of components is equal to the number of terms in the summations in equation 8.1-12 minus the number N% of independent equilibria between phases, that is, C = 2NS — Ns = Ns. Equation 8.1-13 shows that there are D = C + 2 = jVs + 2 natural variables. The Gibbs-Duhem equations for the two phases are... 

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. 

Gibbs considered the statistical mechanics of a system containing one type of molecule in contact with a large reservoir of the same type of molecules through a permeable membrane. If the system has a specified volume and temperature and is in equilibrium with the resevoir, the chemical potential of the species in the system is determined by the chemical potential of the species in the reservoir. The natural variables of this system are T, V, and //. We saw in equation 2.6-12 that the thermodynamic potential with these natural variables is U[T, //] using Callen s nomenclature. The integration of the fundamental equation for yields... 

The constraints shown in Equation 8.10 and Equation 8.11 are a consequence of the nature of mixture problems. In the example illustrated by these equations, each variable represents the relative proportion of a particular ingredient in a mixture blended from q components. For example, a mixture of three components, where the first component makes up 25% of the total, the second component makes up 15% of the total, and the third component makes up 60% of the total, is said to be a ternary mixture. The respective values of the mixture variables are x, = 0.15, x2 = 0.25, x3 = 0.60, giving xx + x2 + x3 = 1. Depending on the number of mixture variables, the mixture could be binary, ternary, quaternary, etc. 

Natural Variables Legendre Transforms Isomer Group Thermodynamics Gibbs-Duhem Equation References... 

These relations are often called equations of state because they relate different state properties. Since the variables T, P, and [nj] play this special role of yielding the other thermodynamic properties, they are referred to as the natural variables of G. Further information on natural variables is given in the Appendix of this chapter. In writing partial derivatives, subscripts are omitted to simplify the notation. The second type of interrelations are Maxwell equations (mixed partial derivatives). Ignoring the VdP term, equation 3.1-1 has two types of Maxwell relations ... 

These equations are often referred to as equations of state because they provide relations between state properties. If G could be determined experimentally as a function of T, P, n, and pH, then S, V, /i,, and c(H) could be calculated by taking partial derivatives. This illustrates a very importnat concept when a thermodynamic potential can be determined as a function of its natural variables, all the other thermodynamic properties can be obtained by taking partial derivatives of this function. However, since there is no direct method to determine G, we turn to the Maxwell relations of equation 3.3-10. 

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