The Equation of State

Finally, we calculate the internal energy U and the specific heat c (at constant length) and Cj (at constant external force). For this purpose, we go beyond the quasi harmonic approximation by including the specific anhar-monic term F p given by (5.42). From (5.18,39-42), we obtain 

Here we have approximated (5.42) by replacing f, g and h by their unrenormalized values Fq, gQ, From (5.19a), we find for the classical 

C normally increases linearly with T at high temperatures. From (5.19b,50,51), one finds 

In the following, we consider an isotropic continuous medium. In this case, the free energy F depends only on the volume V and not on the individual coordinates of the atoms in the unit cell. For the equation of state, we can then write 

Here the anharmonic nature appears only in the equilibrium distances a changed by thermal expansion, see (5.52), and the altered vibrational frequencies w corresponding to the renormalized force constant f, see (5.55,57). We therefore approximate the atomic motion by a system of uncoupled normal vibrations but with different equilibrium positions and vibrational frequencies. This is, of course, an approximation since the anharmonic terms of the potential energy really lead to interactions of the normal vibrations. From (5.68,69), we obtain 

There is a growing technological need for accurate equation-of-state data for fluid metals over wide ranges of temperature and pressure. The practical motivations for measuring the equations of state are reinforced by basic scientific considerations related to the MNM transition. We might expect to find the most dramatic manifestations of the transition in the electronic properties. This is indeed the case. But the fact that the cohesive forces are of a different character in metals and nonmetals should also lead to signs of the MNM transition, perhaps in subtler form, in the equation of state. 

Isochores in the p — T plane for cesium are presented in Fig. 3.17. The isochores are linear within the accuracy and experimental range of 

Values of / , along the saturated liquid-vapor coexistence curve of liquid cesium and rubidium are shown in a reduced plot as a function of the reduced temperatures in Fig. 3.18. It is clear once again that the two alkali metals behave very similarly close to the critical temperature, suggesting the possibility that the alkalis obey a law of corresponding states. The properties of cesium and rubidium and, in particular, the mechanism of the MNM transition are similar in many respects and we might suspect that the interparticle forces are of similar form for these 

For readers in need of actual pVT data, we have tabulated numerical values of jy and p, above 1000 °C in Tables A.1 and A.2 for cesium and rubidium, respectively. 

The isochores of the alkali metals, or any other liquid for that matter, are nonlinear in principle. If one could measure with sufficient accuracy or obtain data over a sufficiently wide range of temperature and pressure, the isochores would always exhibit curvature. The curvature (temperature dependence of the slope ( dpldT)y) is a consequence of the state-dependence of the interatomic interaction. It reflects, in particular, softening of the repulsive part of the potential as the temperature is increased. If we choose to represent the repulsive potential by a hard sphere diameter, the softening appears as a reduction of the diameter at higher temperatures. 

Here h — 6.626 x Js) is Planck s constant and kB(= 1-381 x 10 JK ) is Boltzmann s constant. We shall not present the derivatives of Planck s formula (which requires statistical mechanics) our focus will be on thermodynamic aspects of radiation. Finally, we note that total energy of thermal radiation is 

When the function u(v, T) obtained using classical electromagnetic theory was used in this integral, the total energy density, u(T), turned out to be infinite. The Planck formula (11.1.3), however, gives a finite value for u(T). 

It was clear, even from the classical electromagnetic theory, that a field which interacts with matter and imparts energy and momentum must itself carry energy and momentum. Classical expressions for the energy and momentum associated with the electromagnetic field can be found in texts on electromagnetic theory. To understand the thermodynamic aspects of radiation, we need an equation of state, i.e. an equation that gives the pressure exerted by thermal radiation and its relation to the temperature. 

Using classical electrodynamics it can be shown [1] that the pressure exerted by radiation is related to the energy density u by 

Although thermal radiation is a gas of photons, it differs from an ideal gas. At a fixed temperature T, as the volume of thermal radiation expands, the total energy increases (unlike in an ideal gas, in which it remains constant). As the volume increases, the heat that must be supplied to such a system to keep its temperature constant is thermal radiation entering the system. This heat keeps the energy density constant. The change in entropy due to this heat flow is given 

For systems in thermodynamic equilbrium there exists an equation which connects the intensive thermodynamic variables (T,p,v,Xi,. . . , x, i). This is called the equation of state. A macroscopic theory like thermodynamics cannot predict this equation, and its form must be determined by experiment or molecular theory. The equation of state may be written in many forms. For a system of r components, it may be implicit 

Equations (7-4) describe a surface in T-p-v space. This surface in the three-dimensional temperature, pressure, and molar-volume space is a representation of all the experimental or statistical mechanical information concerning the equation of state of the system. It is usually 

Although many of the detailed features of the isotherms appearing in Fig. 7-1 depend on the system under investigation, a number of remarks can be made concerning their general character which apply to all systems so far studied. As was stated in Chap. 1, it is observed experimentally that 

No analytical expression for the equation of state of real systems, in terms of known functions, has been found. Hence, many approximations have been employed. Many of these apply to the vapor phase but none of them is completely satisfactory. We shall now present some of the approximate equations of state for gases. 

Ideal-gas Equation. The simplest form that we can assume for the equation of state of a real gas is 

Before starting the calculating of the formulas, we shall introduce several new variables, combinations of other quantities which prove to be useful for one reason or another. As a matter of fact, we shall work with quite a number of variables, some of which can be taken to be independent, others dependent, and it is necessary to recognize at the outset the nature of the relations between them. In the next section we consider the equation of state, the empirical relation connecting certain thermodynamic variables. 

The Equation of State.—In considering the properties of matter, our system is ordinarily a piece of material enclosed in a container and 

In any case, even with much more complicated systems, the work done1 will have an analogous form for Eq. (1.1) is simply a force (P) times a displacement (rfF), and we know that work can always be put in such a form. If there is occasion to set up the thermodynamic formulas for a more general type of force than a pressure, we simply set up dW in a form corresponding to Eq. (1.1), and proceed by analogy with the derivations which we shall give here. 

The equation of state-does not include all the experimental information which we must have about a system or substance. Ve need to tnow also its heat, capacity or specific heat, as a function of temperature. Suppose, for instance, that we know the specific heat at constant pressure Cp as a function of temperature at a particular pressure. Then we can find the difference of internal energy, or of entropy, between any two states. From the first state, we can go adiabatically to the pressure at which we know Cp, In this process, since no heat is absorbed, the change of internal energy equals the work done, which we can compute from the equation of state. Then we absorb heat at constant pressure, until we reach the point from which another adiabatic process will carry us to the desired end point. The change of internal energy can be found for the process at constant pressure, since there we know CP) from which we can 

Given the equation of state and specific heat, we see that we can obtain all but two of the quantities P, V, T, Uy S, provided those two are known. We have shown this if two of the three quantities P, F, T art known but if U and S are determined by these quantities, that means simply that two out of the five quantities are independent, the rest dependent. It is then possibie to use any two as independent variables. For instance, in thermodynamics it is not unusual to use T and S, or V and S, as independent variables, expressing everything else as functions of them. 

---

The fugacity coefficient can be found from the equation of state using the thermodynamic relation (Beattie, 1949) ... 

Hydrocarbon mixtures are most often modeled by the equations of state of Soave, Peng Robinson, or Lee and Kesler. 

In 1972, Soave published a method of calculating fugacities based on a modification of the Redlich and Kwong equation of state which completely changed the customary habits and became the industry standard. In spite of numerous attempts to improve it, the original method is the most widespread. For hydrocarbon mixtures, its accuracy is remarkable. For a mixture, the equation of state is ... 

The equation of state for an ideal gas, that is a gas in which the volume of the gas molecules is insignificant, attractive and repulsive forces between molecules are ignored, and molecules maintain their energy when they collide with each other. 

The first equation (1) is the equation of state and the second equation (2) is derived from the measurement process. Finally, G5 (r,r ) is a row-vector that takes the three components of the anomalous ciurent density vector Je (r) = normal component of the induced magnetic field. This system is non hnear (bilinear) because the product of the two unknowns /(r) and E(r) is present. 

A 1.5% by weight aqueous surfactant solution has a surface tension of 53.8 dyn/cm (or mN/m) at 20°C. (a) Calculate a, the area of surface containing one molecule. State any assumptions that must be made to make the calculation from the preceding data, (b) The additional information is now supplied that a 1.7% solution has a surface tension of 53.6 dyn/cm. If the surface-adsorbed film obeys the equation of state ir(o - 00) = kT, calculate from the combined data a value of 00, the actual area of a molecule. 

Derive the equation of state, that is, the relationship between t and a, of the adsorbed film for the case of a surface active electrolyte. Assume that the activity coefficient for the electrolyte is unity, that the solution is dilute enough so that surface tension is a linear function of the concentration of the electrolyte, and that the electrolyte itself (and not some hydrolyzed form) is the surface-adsorbed species. Do this for the case of a strong 1 1 electrolyte and a strong 1 3 electrolyte. 

A film at low densities and pressures obeys the equations of state described in Section III-7. The available area per molecule is laige compared to the cross-sectional area. The film pressure can be described as the difference in osmotic pressure acting over a depth, r, between the interface containing the film and the pure solvent interface [188-190]. 

Contact angle will vary with liquid composition, often in a regular way as illustrated in Fig. X-13 (see also Ref. 136). Li, Ng, and Neumann have studied the contact angles of binary liquid mixtures on teflon and found that the equation of state that describes... 

Isotherms Based on the Equation of State of the Adsorbed Film... 

The equation of state for a solid film is often ic= b - aa (note Section IV-4D). Derive the corresponding adsorption isotherm equation. Plot the data of Problem 11 according to your isotherm equation. 

It is a universal experimental observation, i.e. a law of nature , that the equations of state of systems 1 and 2 are then coupled as if the wall separating them were diathemiic rather than adiabatic. In other words, there is a relation... 

If tlie arbitrary constant C is set equal to nRy where n is the number of moles in the system and R is the gas constant per mole, then the themiodynamic temperature T = 9j where 9j is the temperature measured by the ideal-gas themiometer depending on the equation of state... 

While volume is a convenient variable for the calculations of theoreticians, the pressure is nomially the variable of choice for experimentalists, so there is a corresponding equation in which the equation of state is expanded in powers of p ... 

The equation of state of a fluid relates the pressure (P), density (p) and temperature (7),... 

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

Figure A2.3.4 The equation of state P/pkT- 1, calculated from the virial series and the CS equation of state for hard spheres, as a fimction of q = where pa is the reduced density.

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

Ebeling W and Grigoro M 1980 Analytical calculation of the equation of state and the critical point in a dense classical fluid of charged hard spheres Phys. (Leipzig) 37 21... 

In the microcanonical ensemble, the signature of a first-order phase transition is the appearance of a van der Waals loop m the equation of state, now written as T(E) or P( ). The P( ) curve switches over from one... 

Toxvaerd S 1990. Molecular Dynamics Calculation of the Equation of State of Alkanes. Journal of Chemical Physics 93 4290-4295. 

It is, however, possible to calculate the tensile strength of a liquid by extrapolation of an equation of state for the fluid into the metastable region of negative pressure. Burgess and Everett in their comprehensive test of the tensile strength hypothesis, plot the theoretical curves of T /T against zjp, calculated from the equations of state of van der Waals, Guggenheim, and Berthelot (Fig. 3.24) (7], and are the critical temperature and critical... 

This expression is sometimes called the equation of state for an elastomer in analogy to... 

The work term IF is restricted to the mechanical work deflvered to the outside via normal and shear forces acting on the boundary. Electrochemical work, ie, by electrolysis of the fluid, is excluded. Evaluation of the integral requires knowledge of the equation of state and the thermodynamic history of the fluid... 

In each of these expressions, ie, the Soave-Redhch-Kwong, 9gj j (eq. 34), Peng-Robinson, 9pj (eq. 35), and Harmens, 9 (eq. 36), parameter 9, different for each equation, depends on temperature. Numerical values for b and 9(7) are deterrnined for a given substance by subjecting the equation of state to the critical derivative constraints of equation 20 and by requiring the equation to reproduce values of the vapor—Hquid saturation pressure,... 

The second common procedure for VLE calculations is the equation-of-state approach. Here, fugacity coefficients replace the fugacities for both Hquid and vapor phases, and equation 220 becomes equation 226 ... 

The density of Hquid carbon monoxide at various temperatures is Hsted in Table 4 (5,7). The density of gaseous carbon monoxide (7) can be calculated direcdy from the equation of state using the compressibihty factor at the temperature and pressure of interest. 

The pressure and the density of a gas are related by an equation of state. If the maximum pressure permitted within the centrifuge bowl is not too high, the equation of state for an ideal gas will suffice. The relationship between the pressure and density of an ideal gas is given by the weU-known equation ... 

---