I am more than ever an admirer of van der Waals.
Lord Rayleigh (1891)
From Chapter 6, it is obvious that we can calculate changes in U, S, H, A, and G by knowing changes in any two variables from the set {P–V–T} plus CP or CV. This chapter introduces the various ways available for quantitative prediction of the P-V-T properties we desire in a general case. The method of calculation of thermodynamic properties like U, H, and so on. is facilitated by the use of departure functions, which will be the topic of the next chapter. The development of the departure functions is a relatively straightforward application of derivative manipulations. What is less straightforward is the logical development of a connection between P, V, and T. We introduced the concept in Chapter 1 that the pressure, temperature, and density (i.e., V–1) are connected through intermolecular interactions. We must now apply that concept to derive quantitative relationships that are applicable to any fluid at any conditions, not simply to ideal gases. You will see that making the connection between P, V, and T hinges on the transition from the molecular-scale forces and potential energy to the macroscopic pressure and internal energy. Understanding the approximations inherent in a particular equation of state is important because effectively all of the approximations in a thermodynamic model can be traced to the assumed equation of state. Whenever deficiencies are found in a process model, the first place to look for improvement is in revisiting the assumptions of the equation of state.
Understanding the transition from the molecular scale to the macroscopic is a major contribution in our conceptual puzzle of calculating energy, entropy, and equilibrium. We made qualitative connections between the microscopic and macroscopic scales for entropy during our introduction to entropy. For energy, however, we have left a gap that you may not have noticed. We discussed the molecular energy in Chapter 1, but we did not quantify the macroscopic implications. We discussed the macroscopic implications of energy in Chapter 2, but we did not discuss the molecular basis. It is time to fill that gap, and in doing so, link the conceptual framework of the entire text.
From one perspective, the purpose of the examples in Chapter 6 was to explain the need for making the transition from the molecular scale to the macroscopic scale. The purpose of the material following this chapter is to demonstrate the reduction to practice of this conceptual framework in several different contexts. So in many ways, this chapter represents the conceptual kernel for all molecular thermodynamics.
Chapter Objectives: You Should Be Able to…
1. Explain and apply two- and three- parameter corresponding states.
2. Apply an equation of state to solve for the density given T and P, including liquid and vapor roots.
3. Evaluate partial derivatives like those in Chapter 6 using an equation of state for PVT properties.
4. Identify the repulsive and attractive contributions to an equation of state and critically evaluate their accuracy relative to molecular simulations and experimental data.
7.1. Experimental Measurements
The preferred method of obtaining P–V–T properties is from experimental measurements of the desired fluid or fluid mixture. We spend most of the text discussing theories, but you should never forget the precious value of experimental data. Experimental measurements beat theories every time. The problem with experimental measurements is that they are expensive, especially relative to pushing a few buttons on a computer.
To illustrate the difficulty of measuring all properties experimentally, consider the following case. One method to determine the P–V–T properties is to control the temperature of a container of fluid, change the volume of the container in carefully controlled increments, and carefully measure the pressure. The required derivatives are then calculated by numerical differentiation of the data obtained in this manner. It is also possible to make separate measurements of the heat capacity by carefully adding measured quantities of heat and determining changes in P, V, and T. These measurements can be cross-referenced for consistency with the estimated changes as determined by applying Maxwell’s relations to the P-V-T measurements. Imagine what a daunting task this approach would represent when considering all fluids and mixtures of interest. It should be understandable that detailed measurements of this type have been made for relatively few compounds. Water is the most completely studied fluid, and the steam tables are a result of this study. Ammonia, carbon dioxide, refrigerants, and light hydrocarbons have also been quite thoroughly studied. The charts which have been used in earlier chapters are results of these careful measurements. Equations of state permit correlation and extrapolation of experimental data that can be much more convenient and more broadly applicable than the available charts.
The basic procedure for calculating properties involves using derivatives of P-V-T data.
An experimental approach is naturally impractical for all substances due to the large number of fluids needing to be characterized. The development of equations of state is the engineering approach to describing fluid behavior for prediction, interpolation, and extrapolation of data using the fewest number of adjustable parameters possible for the desired accuracy. Typically, when data are analyzed today, they are fitted with elaborate equations (embellishments of the equations of state discussed in this chapter) before determination of interpolated values or derivatives. The charts are generated from the fitted results of the equation of state.
As a summary of the experimental approach to equations of state, a brief review of the historical development of P-V-T measurements may be beneficial. First, it should be recalled that early measurements of P-V-T relations laid the foundation for modern physical chemistry. Knowing the densities of gases in bell jars led to the early characterizations of molecular weights, molecular formulas, and even the primary evidence for the existence of molecules themselves. At first, it seemed that gases like nitrogen, hydrogen, and oxygen were non-condensable and something quite different from liquids like water or wood alcohol (methanol). As technology advanced, however, experiments were performed at higher temperatures and pressures. Carbon dioxide was a very common compound in the early days (known as “carbonic acid” to van der Waals), and it soon became apparent that it showed a high degree of compressibility. Experimental data were carefully measured in 1871 for carbon dioxide ranging to 110 bars, and these data were referenced extensively by van der Waals. Carbon dioxide is especially interesting because it has some very “peculiar” properties that are exhibited near room temperature and at high pressure. At 31°C and about 70 bars, a very small change in pressure can convert the fluid from a gas-like density to a liquid density. Van der Waals showed that the cause of this behavior is the balance between the attractive forces from the intermolecular potential being accentuated at this density range and the repulsive forces being accentuated by the high-velocity collisions at this temperature. This “peculiar” range of conditions is known as the critical region. The precise temperature, pressure, and density where the vapor and the liquid become indistinguishable is called the critical point. Above the critical point, there is no longer an abrupt change in the density with respect to pressure while holding temperature constant. Instead, the balance between forces leads to a single-phase region spanning vapor-like densities and liquid-like densities. With the work of van der Waals, researchers began to recognize that the behavior was not “peculiar,” and that all substances have critical points.1
Fortunately, P,V,T behavior of fluids follows the same trends for all fluids. All fluids have a critical point.
7.2. Three-Parameter Corresponding States
If we plot P versus ρ for several different fluids, we find some remarkably similar trends. As shown in Fig. 7.1 below, both methane and pentane show the saturated vapor density approaching the saturated liquid density as the temperature increases. Compare these figures to Fig. 1.4 on page 23, and note that the P versus ρ figure is qualitatively a mirror image of the P versus V figure. The isotherms are shown in terms of the reduced temperature, Tr ≡ T/Tc. Saturation densities are the values obtained by intersection of the phase envelope with horizontal lines drawn at the saturation pressures. The isothermal compressibility 
is infinite, and its reciprocal is zero, at the critical point (e.g., 191 K and 4.6 MPa for methane). It is also worth noting that the critical temperature isotherm exhibits an inflection point at the critical point. This means that (∂2P/∂ρ2)T = 0 at the critical point as well as (∂P/∂ρ)T = 0. The principle of corresponding states asserts that all fluid properties are similar if expressed properly in reduced variables.
Figure 7.1. Comparison of the PρT behavior of methane (left) and pentane (right) demonstrating the qualitative similarity which led to corresponding states’ treatment of fluids. The lines are calculated with the Peng-Robinson equation to be discussed later. The phase envelope is an approximation sketched through the points available in the plots. The smoothed experimental data are from Brown, G.G., Sounders Jr., M., and Smith, R.L., 1932. Ind. Eng. Chem., 24:513. Although not shown, the Peng-Robinson equation is not particularly accurate for modeling liquid densities.
The isothermal compressibility is infinite at the critical point.
Although the behaviors in Fig. 7.1 are globally similar, when researchers superposed the P-V-T behaviors based on only critical temperature, Tc and critical pressure, Pc, they found the superposition was not sufficiently accurate. For example, one way of comparing the behavior of fluids is to plot the compressibility factor Z. The compressibility factor is defined as
Note: The compressibility factor is not the same as the isothermal compressibility. The similarity in names can frequently result in confusion as you first learn the concepts.
The compressibility factor has a value of one when a fluid behaves as an ideal gas, but will be non-unity when the pressure increases. By plotting the data and calculations from Fig. 7.1 as a function of reduced temperature Tr = T/Tc, and reduced pressure, Pr = P/Pc, the plot of Fig. 7.2 results. Clearly, another parameter is needed to accurately correlate the data. Note that the vapor pressure for methane and pentane differs on the compressibility factor chart as indicated by the vertical lines on the subcritical isotherms. The same behavior is followed by other fluids. For example, the vapor pressures for six compounds are shown in Fig. 7.3, and although they are all nearly linear, the slopes are different. In fact, we may characterize this slope with a third empirical parameter, known as the acentric factor, ω. The acentric factor is a parameter which helps to specify the vapor pressure curve which, in turn, correlates the rest of the thermodynamic variables.2
Figure 7.2. The Peng-Robinson lines from Fig. 7.1 plotted in terms of the reduced pressure at Tr = 0.8, 0.9, 1.0, 1.1, and 1.3, demonstrating that critical temperature and pressure alone are insufficient to accurately represent the P-V-T behavior. Dashed lines are for methane, solid lines for pentane. The figure is intended to make an illustrative point. Accurate calculations should use the compressibility factor charts developed in the next section.
Figure 7.3. Reduced vapor pressures plotted as a function of reduced temperature for six fluids demonstrating that the shape of the curve is not highly dependent on structure, but that the primary difference is the slope as given by the acentric factor.
Critical temperature and pressure are insufficient characteristic parameters by themselves. The acentric factor serves as a third important parameter.
Note: The specification of Tc, Pc, and ω provides two points on the vapor pressure curve. Tc and Pc specify the terminal point of the vapor pressure curve. ω specifies a vapor pressure at a reduced temperature of 0.7. The acentric factor was first introduced by Pitzer et al.3 Its definition is arbitrary in that, for example, another reduced temperature could have been chosen for the definition. The definition above gives values of ω ~ 0 for spherical molecules like argon, xenon, neon, krypton, and methane. Deviations from zero usually derive from deviations in spherical symmetry. Nonspherical molecules are “not centrally symmetric,” so they are “acentric.” In general, there is no direct theoretical connection between the acentric factor and the shape of the intermolecular potential. Rather, the acentric factor provides a convenient experimental vapor pressure which can be correlated with the shape of the intermolecular potential in an ad hoc manner. It is convenient in the sense that its value has been experimentally determined for a large number of compounds and that knowing its value permits a significant improvement in the accuracy of our engineering equations of state.
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