Thermodynamics natural variable

The simplicity of the form of these expressions suggests that S and V are natural variables of the internal energy E S,V). These expressions have demonstrated the first and second laws of thermodynamics for our colloidal system. However, it is not easy to perform an experiment at... 

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... 

The phase diagrams of aqueous surfactant systems provide information on the physical science of these systems which is both useful industrially and interesting academically (1). Phase information is thermodynamic in nature. It describes the range of system variables (composition, temperature, and pressure) wherein smooth variations occur in the thermodynamic density variables (enthalpy, free energy, etc.), for macroscopic systems at equilibrium. The boundaries in phase diagrams signify the loci of system variables where discontinuities in these thermodynamic variables exist (2). 

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 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. 

Now we are in a position to generalize on the number of different thermodynamic potentials there are for a system. The number of ways to subtract products of conjugate variables, zero-at-a-time, one-at-a-time, and two-at-a-time, is 2k, where k is the number of conjugate pairs involved. In probability theory the number 2k of ways is referred to as the number of sets of k elements. For a one-phase system involving PV work but no chemical reactions, the number of natural variables is D and the number of different thermodynamic potentials is 2D. If D = 2 (as in dU = TdS — PdV), the number of different thermodynamic potentials is 22 = 4, as we have seen with U, H, A, and G. If D = 3 (as in d(7=TdS — PdL+ n da), the number of different thermodynamic potentials is 23 = 8. 

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. 

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... 

If a system contains two types of species, but the membrane is permeable only to species number 1, the natural variables for the system are T, K //, and N2, where N2 is the number of molecules of type 2 in the system. The thermodynamic potential for this system containing two species is represented by U[T, //,]. The corresponding ensemble is referred to as a semigrand ensemble, and the semigrand partition function can be represented by P(71 K /q, N2). The thermodynamic potential of the system is related to the partition function by... 

In order to discuss phase transformations in this chapter and chemical reactions in the next chapter, we will need to develop the thermodynamics of open systems. In open systems, the number of moles of the various components of the system can change and the thermodynamic fimctions depend on the numbers of moles of these components, as well as on thermodynamic variables. For example, the natural variables for U become (/(.S, V, n, ), where the index i ranges over the components of the system. 

In thermodynamics, we focus on the most important variables needed to describe a system. Although we are interested in the size of a system (or of a phase), we usually do not concern ourselves with the shape of the system. One way in which the shape of a system does influence its thermodynamic properties is through its surface area. The surface of a phase is a different environment than its bulk region. Molecules on the surface of a material do not experience attractive interactions to other molecules in all directions and, therefore, have higher energy than molecules in the bulk of the material. Energy is increased when the surface area of a condensed system (usually a liquid) is increased at constant volume and temperature. Because the Helmholtz free energy has T and V as its natural variables, we can immediately write... 

The condition for equilibrium may be described by any of several thermodynamic functions, such as the minimization of the Gibbs or Helmholtz free energy or the maximization of entropy. If one wishes to use temperature and pressure to characterize a thermodynamic state, one finds that the Gibbs free energy is most easily minimized, inasmuch as temperature and pressure are its natural variables. Similarly, the Helmholtz free energy is most easily minimized if the thermodynamic state is characterized by temperature and volume (density) [4]. 

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. 

Equations of state relate the pressure, temperature, volume, and composition of a system to each other. In this Chapter, we show how to determine other thermodynamic properties of the system from an equation of state. In a typical equation of state, the pressure is given as an explicit function of temperature, volume, and composition. Therefore, the natural variables are the temperature, volume, and composition of the system. That is, once given the volume, temperature, and composition of the system, the pressure is readily calculated from the equation of state. 

Once one of the free energies of a system is known as a function of its natural variables, then all the other thermodynamic properties of the system can be derived. For these equations of states, the Helmholtz free energy is the relevant quantity. In the following, we demonstrate how to determine the Helmholtz free energy from an equation and then proceed to show how to derive other properties from it. 

The free energy- that has temperature, volume, and mole numbers as its natural variables is the Helmholtz free energy. Before we stated that once the Gibb s free energy of a system is known as a function of temperature, pressure, and mole numbers G(T,p, N, N2,..all the thermodynamics of the system are known. This is equivalent to the statement that once the Helmholtz free energy is known as a function of temperature, volume, and mole numbers of the system A(T, V, Ni,N2, -all the thermodynamics of the system are known. The fundamental equation of thermodynamics can be written in terms of the Helmholtz free energy as... 

Changes in one thermodynamic state variable can be related to changes in other state variables. The set of state functions that determines the simplest relationship between these functions is termed a natural function. 

It needs to be emphasized at this point that one could, of course, express each thermodynamic potential in terms of different. sets of (nonnatural) variables. The immediate consequence is that thermodynamic potentials would not necessarily attain a minimum value if the system is in a state of thermodynamic equilibrium. This point is important to realize because it implies that the set of natural variables of a given thermodjmamic potential is distinguished and unique eunong other conceivable. sets of variables. 

The full differentials (or the total changes) of the principal thermodynamic potentials cast in terms of their natural variables are ... 

Standard thermodynamic formalism for the total differential of specific enthalpy in terms of its natural variables (i.e., via Legendre transformation, see equations 29-20 and 29-24b) allows one to calculate the pressure coefQdent of specific enthalpy via a Maxwell relation and the definition of the coefficient of thermal expansion, a. 

Step 2. Use the total differential of specific enthalpy in terms of its natural variables, via Legendre transformation of the internal energy from classical thermodynamics, to re-express the pressure gradient in the momentum balance in terms of enthalpy, entropy, and mass fractions. Then, write the equation of change for kinetic energy in terms of specific enthalpy and entropy. 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

Extensive thermodynamic state functions such as the internal energy U depend linearly on mass or mole numbers of each component. This claim is consistent with Euler s theorem for homogeneous functions of the first degree with respect to molar mass. If a mixture contains r components and exists as a single phase, then U exhibits r - - 2 degrees of freedom and depends on the following natural variables, all of which are extensive ... 

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