Category: Introductory Chemical Engineering

All the effects of nature are only the mathematical consequences of a small number of immutable laws.

P.-S. LaPlace

Maxwell’s relations make it clear that changes in any one variable can be represented as changes in some other pair of variables. In chemical processes, we are often concerned with the changes of enthalpy and entropy as functions of temperature and pressure. As an example, recall the operation of a reversible turbine between some specified inlet conditions of T and P and some specified outlet pressure. Using the techniques of Unit I, we typically determine the outlet T and q which match the upstream entropy, then solve for the change in enthalpy. Applying this approach to steam should seem quite straightforward at this stage. But what if our process fluid is a new refrigerant or a multicomponent natural gas, for which no thermodynamic charts or tables exist? How would we analyze this process? In such cases, we need to have a general approach that is applicable to any fluid. A central component of developing this approach is the ability to express changes in variables of interest in terms of variables which are convenient using derivative manipulations. The other important consideration is the choice of “convenient” variables. Experimentally, P and T are preferred; however, V and T are easier to use with cubic equations of state.

These observations combine with the observation that the approximations in equations of state themselves exhibit a certain degree of “fluidity.” In other words, the “best” approximations for one application may not be the best for another application. Responding to this fluidity requires engineers to revisit the approximations and quickly reformulate the model equations for U, H, and S. Fortunately, the specific derivative manipulations required are similar regardless of the equation of state since equations of state are either in the {T,P} or {T,V} form. The formalism of departure functions streamlines the each formulation.

An equation of state describes the effects of pressure on our system properties, including the low pressure limit of the ideal gas law. However, integration of properties over pressure ranges is relatively complicated because most equations of state express changes in thermodynamic variables as functions of density instead of pressure, whereas we manipulate pressure as engineers. Recall that our engineering equations of state are typically of the pressure-explicit form:

 Experimentally, P and T are usually specified. However, equations of state are typically density (volume) dependent.

and general equations of state (e.g., cubic) typically cannot be rearranged to a volume explicit form:

Therefore, development of thermodynamic properties based on {V,T} is consistent with the most widely used equations of state, and deviations from ideal gas behavior will be expressed with the density-dependent formulas for departure functions in Sections 8.1–8.5. In Section 8.6, we present the pressure-dependent form useful for the virial equation. In Section 8.8, we show how reference states are used in tabulating thermodynamic properties.

Chapter Objectives: You Should Be Able to…

1. Choose between using the integrals in Section 8.5 or 8.6 for a given equation of state.

2. Evaluate the integrals of Section 8.5 or 8.6 for simple equations of state.

3. Combine departure functions with ideal gas calculations to determine numerical values of changes in state properties, and use a reference state.

4. Solve process thermodynamics problems using a tool like Preos.xlsx or PreosProps.m rather than a chart or table. This skill requires integration of several concepts covered by other topical objectives including selection of the correct root, and reading/interpreting the output file.

8.1. The Departure Function Pathway

Suppose we desire to calculate the change in U in a process which changes state from (V_L, T_L) to (V_H, T_H). Now, it may seem unusual to pose the problem in terms of T and V, since we stated above that our objective was to use T and P. The choice of T and V as variables is because we must work often with equations of state that are functions of volume. The volume corresponding to any pressure is rapidly found by the methods of Chapter 7. We have two obvious pathways for calculating a change in U using {V, T} as state variables as shown in Fig. 8.1. Path A consists of an isochoric step followed by an isothermal step. Path B consists of an isothermal step followed by an isochoric step. Naturally, since U is a state function, ΔU for the process is the same by either path. Recalling the relation for dU(T,V), ΔU may be calculated by either.

Figure 8.1. Comparison of two alternate paths for calculation of a change of state.

Path A:

or Path B:

We have previously shown, in Example 7.6 on page 269, that C_V depends on volume for a real fluid. Therefore, even though we could insert the equation of state for the integrand of the second integral, we must also estimate C_V by the equation of state for at least one of the volumes, using the results of Example 7.6. Not only is this tedious, but estimates of C_V by equations of state tend to be less reliable than estimates of other properties.

To avoid this calculation, we devise an equivalent pathway of three stages. First, imagine if we had a magic wand to turn our fluid into an ideal gas. Second, the ideal gas state change calculations would be pretty easy. Third, at the final state we could turn our fluid back into a real fluid. Departure functions represent the effect of the magic wand to exchange the real fluid with an ideal gas. Being careful with signs of the terms, we may combine the calculations for the desired result:

The calculation can be generalized to any fundamental property from the set {U,H,A,G,S}, using the variable M to denote the property

 Departure functions permit us to use the ideal gas calculations that are easy, and incorporate a departure property value for the initial and final states.

The steps can be seen graphically in Fig. 8.2. Note the dashed lines in the figure represent the calculations from our “magic wand” effect of turning on/off the nonidealities.

Figure 8.2. Illustration of calculation of state changes for a generic property M using departure functions where M is U, H, S, G, or A.

Note how all the ideal gas terms in Eqns. 8.5 and 8.6 cancel to yield the desired property difference. A common mistake is to get the sign wrong on one of the terms in these equations. Make sure that you have the terms in the right order by checking for cancellation of the ideal gas terms. The advantage of this pathway is that all temperature calculations are done in the ideal gas state where:

and the ideal gas heat capacities are pressure- (and volume-) independent (see Example 6.9 on page 242).

To derive the formulas to be used in calculating the values of enthalpy, internal energy, and entropy for real fluids, we must apply our fundamental property relations once and our Maxwell’s relations once.

8.2. Internal Energy Departure Function

Fig. 8.3 schematically compares a real gas isotherm and an ideal gas isotherm at identical temperatures. At a given {T,P} the volume of the real fluid is V, and the ideal gas volume is V^ig = RT/P. Similarly, the ideal gas pressure is not equal to the true pressure when we specify {T,V}. Note that we may characterize the departure from ideal gas behavior in two ways: 1) at the same {T,V}; or 2) at the same {T,P}. We will find it convenient to use both concepts, but we need nomenclature to distinguish between the two departure characterizations. When we refer to the departure of the real fluid property and the same ideal gas property at the same {T,P}, we call it simply the departure function, and use the notation U – U^ig. When we compare the departure at the same {T,V} we call it the departure function at fixed T,V, and designate it as (U – U^ig)_TV.¹

Figure 8.3. Comparison of real fluid and ideal gas isotherms at the same temperature, demonstrating the departure function, and the departure function at fixed T,V.

To calculate the change in internal energy along an isotherm for the real fluid, we write:

 The departure for property M is at fixed T and, P and is given by (M–M^ig). The departure at fixed T,V is also useful (particularly in Chapter 15) and is denoted by (M–M^ig)_TV.

For an ideal gas:

Since the real fluid approaches the ideal gas at infinite volume, we may take the difference in these two equations to find the departure function at fixed T,V,

where (U – U^ig)_{T, V = ∞} drops out because the real fluid energy approaches the ideal gas at infinite volume (low pressure). We have obtained a calculation with the real fluid in our desired state (T,P,V); however, we are referencing an ideal gas at the same volume rather than the same pressure. To see the difference, consider methane at 250 K, 10 MPa, and 139 cm³/mole. The volume of the ideal gas should be V^ig = 8.314·250/10 = 208 cm³/mole. To obtain the departure function denoted by (U – U^ig) (which is referenced to an ideal gas at the same pressure), we must add a correction to change the ideal gas state to match the pressure rather than the volume. Note in Fig. 8.3 that the real state is the same for both departure functions—the difference between the two departure functions has to do with the volume used for the ideal gas part of the calculation. The result is

We have already solved for (∂U^ig/∂V)_T (see Example 6.6 on page 238), and found that it is equal to zero. We are fortunate in this case because the internal energy of an ideal gas does not depend on the volume. When it comes to properties involving entropy, however, the dependency on volume requires careful analysis. Then the systematic treatment developed above is quite valuable.

Making these substitutions, we have

If we transform the integral to density, the resultant expression is easier to integrate for a cubic equation of state. Recognizing dV = –dρ/ρ², and as V → ∞, ρ → 0, thus,

The above equation applies the chain rule in a way that may not be obvious at first:

We now have a compact equation to apply to any equation of state. Knowing Z = Z(T, ρ), (e.g., Eqn. 7.15, the Peng-Robinson model), we simply differentiate once, cancel some terms, and integrate. This a perfect sample application of the multivariable calculus that should be familiar at this stage in the curriculum. More importantly, we have developed a systematic approach to solving for any departure function. The steps for a system where Z = Z(T, ρ) are as follows.

1. Write the derivative of the property with respect to volume at constant T. Convert to derivatives of measurable properties using methods from Chapter 6.

2. Write the difference between the derivative real fluid and the derivative ideal gas.

3. Insert integral over dV and limits from infinite volume (where the real fluid and the ideal gas are the same) to the system volume V.

4. Add the necessary correction integral for the ideal gas from V to V^ig. (This will be more obvious for entropy.)

5. Transform derivatives to derivatives of Z. Evaluate the derivatives symbolically using the equation of state and integrate analytically.

6. Rearrange in terms of density and compressibility factor to make it more compact.

Some of these steps could have been omitted for the internal energy, because (∂U^ig/∂V)_T = 0. Steps 1 through 4 are slightly different when Z = Z(T, P) such as with the truncated virial EOS. To see the importance of all the steps, consider the entropy departure function.

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Example 8.1. Internal energy departure from the van der Waals equation

Derive the internal energy departure function for the van der Waals equation. Suppose methane is compressed from 200 K and 0.1 MPa to 220 K and 60 MPa. Which is the larger contribution in magnitude to ΔU, the ideal gas contribution or the departure function? Use C_P from the back flap and ignore temperature dependence.

Solution

For methane, a = 230030 J-cm³/mol² and b = 43.07 cm³/mol were calculated by the critical point criteria in Example 7.7 on page 271. Deriving the departure function, –T(dZ/dT)ρ = –aρ/RT, because the repulsive part is constant with respect to T. Substituting,

Because T_r > 1 there is only one real root. A quick but crude computation of ρ is to rearrange as Zbρ = bP/RT = bρ/(1 – bρ) – (a/bRT)(bρ)².

At state 2, 220 K and 60 MPa,
60·43.07/(8.314·220) = bρ/(1 – bρ) – 230030/(43.07·8.314·220)(bρ)².
Taking an initial guess of bρ = 0.99 and solving iteratively gives bρ = 0.7546, so
(U₂ – U₂^ig)/RT = –230030·0.7546/(43.07·8.314·220) = –2.203.

At state 1, 200 K and 0.1 MPa,
0.1·43.07/(8.314·200) = bρ/(1 – bρ) – 230030/(43.07·8.314·200)(bρ)².
Taking an initial guess of bρ = 0.99 and solving iteratively gives bρ = 0.00290, so
(U₁ – U₁^ig)/RT = –230030·0.00290/(43.07·8.314·200) = –0.00931.

ΔU = –2.203(8.314)220 + (4.3 – 1)·8.314(220 – 200) + 0.00931(8.314)200 = –4030 + 549 + 15 = –3466 J/mol. The ideal gas part (549) is 14% as large in magnitude as the State 2 departure function (–4030) for this calculation. Clearly, State 2 is not an ideal gas.

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Note that we do not need to repeat the integral for every new problem. For the van der Waals equation, the formula (U–U^ig)/(RT) = –(aρ)/(RT) may readily be used whenever the van der Waals fluid density is known for a given temperature.

8.3. Entropy Departure Function

To calculate the entropy departure, adapt Eqn. 8.11,

Inserting the integral for the departure at fixed {T, V}, we have (using a Maxwell relation),

Since , we may readily integrate the ideal gas integral (note that this is not zero whereas the analogous equation for energy was zero):

Recognizing V^ig = RT/P, V/V^ig = PV/RT = Z,

where Eqn. 8.15 has been applied to the relation for the partial derivative of P.

Note the ln(Z) term on the end of this equation. It arises from the change in ideal gas  represented by the integral in Eqn. 8.16. Changes in states like this may seem pedantic and arcane, but they turn out to be subtle details that often make a big difference numerically. In Example 7.4 on page 266, we determined vapor and liquid roots for Z. The vapor root was close to unity, so ln(Z) would make little difference in that case. For the liquid root, however, Z = 0.016, and ln(Z) makes a substantial difference. These arcane details surrounding the subject of state specification are the thermodynamicist’s curse.

8.4. Other Departure Functions

The remainder of the departure functions may be derived from the first two and the definitions,

 The departures for U and S are the building blocks from which the other departures can be written by combining the relations derived in the previous sections.

where we have used PV^ig = RT for the ideal gas in the enthalpy departure. Using H – H^ig just derived,

8.5. Summary of Density-Dependent Formulas

Formulas for departures at fixed T,P are listed below. These formulas are useful for an equation of state written most simply as Z = f(T,ρ) such as cubic EOSs. For treating cases where an equation of state is written most simply as Z = f (T,P) such as the truncated virial EOS, see Section 8.6.

Useful formulas at fixed T,V include:

8.6. Pressure-Dependent Formulas

Occasionally, our equation of state is difficult to integrate to obtain departure functions using the formulas from Section 8.5. This is because the equation of state is more easily arranged and integrated in the form Z = f (T,P), such as the truncated virial EOS. For treating cases where an equation of state is written most simply as Z = f(T,ρ) such as a cubic EOS, see Section 8.5. We adapt the procedures given earlier in Section 8.2.

1. Write the derivative of the property with respect to pressure at constant T. Convert to derivatives of measurable properties using methods from Chapter 6.

2. Write the difference between the derivative real fluid and the derivative ideal gas.

3. Insert integral over dP and limits from P = 0 (where the real fluid and the ideal gas are the same) to the system pressure P.

4. Transform derivatives to derivatives of Z. Evaluate the derivatives symbolically using the equation of state and integrate analytically.

5. Rearrange in terms of density and compressibility factor to make it more compact.

We omit derivations and leave them as a homework problem. The two most important departure functions at fixed T,P are

The other departure functions can be derived from these using Eqns. 8.20 and 8.21. Note the mathematical similarity between P in the pressure-dependent formulas and ρ in the density-dependent formulas.

8.7. Implementation of Departure Formulas

The tasks that remain are to select a particular equation of state, take the appropriate derivatives, make the substitutions, develop compact expressions, and add up the change in properties. The good news is that many years of engineering research have yielded several preferred equations of state (see Appendix D) which can be applied generally to any application with a reasonable degree of accuracy. For the purposes of the text, we use the Peng-Robinson equation or virial equation to illustrate the principles of calculating properties. However, many applications require higher accuracy; new equations of state are being developed all the time. This means that it is necessary for each student to know how to adapt the departure function method to new situations as they come along.

The following example illustrates the procedure with an equation of state that is sufficiently simple that it can be applied with either the density-dependent formulas or the pressure-dependent formulas. Although the intermediate steps are a little different, the final answer is the same, of course.

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Example 8.2. Real entropy in a combustion engine

A properly operating internal combustion engine requires a spark plug. The cycle involves adiabatically compressing the fuel-air mixture and then introducing the spark. Assume that the fuel-air mixture in an engine enters the cylinder at 0.08 MPa and 20°C and is adiabatically and reversibly compressed in the closed cylinder until its volume is 1/7 the initial volume. Assuming that no ignition has occurred at this point, determine the final T and P, as well as the work needed to compress each mole of air-fuel mixture. You may assume that  for the mixture is 32 J/mole-K (independent of T), and that the gas obeys the equation of state,

PV = RT + aP

where a is a constant with value a = 187 cm³/mole. Do not assume that C_V is independent of ρ. Solve using density integrals.

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 C_P or C_V. 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, T_r ≡ T/T_c. 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, T_c and critical pressure, P_c, 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

 The compressibility factor.

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 T_r = T/T_c, and reduced pressure, P_r = P/P_c, 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 T_r = 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 T_c, P_c, and ω provides two points on the vapor pressure curve. T_c and P_c 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.

Forming an intermediate state between liquids, in which we assume no external pressure, and gases, in which we omit molecular forces, we have the state in which both terms occur. As a matter of fact, we shall see further on, that this is the only state which occurs in nature.

van der Waals (1873, ch2)

In Unit I we focused predominantly on a relatively small number of pure fluids. But the number of chemical compounds encountered when considering all possible applications is vast, and new compounds are being invented and applied every day. Imagine how many charts and tables would be necessary to have properties available for all those compounds. Then imagine how many charts would be necessary to represent the properties of all the conceivable mixtures of those compounds. Clearly, we cannot address all problems by exactly the same techniques as applied in Unit I. We must still use the energy and entropy balance, but we need to be able to represent the physical properties of pure compounds and mixtures in some condensed form, and we desire to predict physical properties based on very limited data.

As one might expect, an excellent shorthand is offered by the language of mathematics. When we sought values in the steam tables, we noticed that specification of any two variables was sufficient to determine the variable of interest (e.g, S or H). This leads to an excellent application of the calculus of two variables. Changes in each value of interest may be expressed in terms of changes in whatever other two variables are most convenient. It turns out that the most convenient variables mathematically are temperature and density, and that the most convenient variables experimentally are temperature and pressure.

There is a limit to how condensed our mathematical analysis can be. That limit is dictated by how much physical insight is required to represent the properties of interest to the desired accuracy. With no physical insight, we can simply measure the desired values, but that is impractical. With maximum physical insight, we can represent all the properties purely in terms of their fundamental electronic structure as given by the periodic table and their known molecular structure. The current state-of-the-art lies between these limits, but somewhat closer to the fundamental side. By developing a sophisticated analysis of the interactions on the molecular scale, we can show that three carefully selected parameters characterizing physical properties are generally sufficient to characterize properties to the accuracy necessary in most engineering applications. This analysis leads to an equation of state, which is then combined with the necessary mathematics to provide methods for computing and predicting physical properties of interest. The development of van der Waals’ equation of state provides an excellent case study in the development of engineering models based on insightful physics and moderately clever extrapolation. Note that before van der Waals the standard conception was that the vapor phase was represented by what we now refer to as an ideal gas, and the liquid was considered to be an entirely different species. Van der Waals’ analysis led to a unification of these two conceptions of fluids that also set the stage for the analysis of interfaces and other inhomogeneous fluids. Van der Waals’ approach lives on in modern research on inhomogeneous fluids.

Forming an intermediate state between liquids, in which we assume no external pressure, and gases, in which we omit molecular forces, we have the state in which both terms occur. As a matter of fact, we shall see further on, that this is the only state which occurs in nature.

van der Waals (1873, ch2)

In Unit I we focused predominantly on a relatively small number of pure fluids. But the number of chemical compounds encountered when considering all possible applications is vast, and new compounds are being invented and applied every day. Imagine how many charts and tables would be necessary to have properties available for all those compounds. Then imagine how many charts would be necessary to represent the properties of all the conceivable mixtures of those compounds. Clearly, we cannot address all problems by exactly the same techniques as applied in Unit I. We must still use the energy and entropy balance, but we need to be able to represent the physical properties of pure compounds and mixtures in some condensed form, and we desire to predict physical properties based on very limited data.

As one might expect, an excellent shorthand is offered by the language of mathematics. When we sought values in the steam tables, we noticed that specification of any two variables was sufficient to determine the variable of interest (e.g, S or H). This leads to an excellent application of the calculus of two variables. Changes in each value of interest may be expressed in terms of changes in whatever other two variables are most convenient. It turns out that the most convenient variables mathematically are temperature and density, and that the most convenient variables experimentally are temperature and pressure.

There is a limit to how condensed our mathematical analysis can be. That limit is dictated by how much physical insight is required to represent the properties of interest to the desired accuracy. With no physical insight, we can simply measure the desired values, but that is impractical. With maximum physical insight, we can represent all the properties purely in terms of their fundamental electronic structure as given by the periodic table and their known molecular structure. The current state-of-the-art lies between these limits, but somewhat closer to the fundamental side. By developing a sophisticated analysis of the interactions on the molecular scale, we can show that three carefully selected parameters characterizing physical properties are generally sufficient to characterize properties to the accuracy necessary in most engineering applications. This analysis leads to an equation of state, which is then combined with the necessary mathematics to provide methods for computing and predicting physical properties of interest. The development of van der Waals’ equation of state provides an excellent case study in the development of engineering models based on insightful physics and moderately clever extrapolation. Note that before van der Waals the standard conception was that the vapor phase was represented by what we now refer to as an ideal gas, and the liquid was considered to be an entirely different species. Van der Waals’ analysis led to a unification of these two conceptions of fluids that also set the stage for the analysis of interfaces and other inhomogeneous fluids. Van der Waals’ approach lives on in modern research on inhomogeneous fluids.

There cannot be a greater mistake than that of looking superciliously upon practical applications of science. The life and soul of science is its practical application.

In the first four chapters, we have concentrated on applications of the first and second laws to simple systems (e.g., turbine, throttle). The constraints imposed by the second law should be clear. In this chapter, we show how the analyses we have developed for one or two operations at a time can be assembled into complex processes. In this way, we provide several specific examples of ways that operations can be connected to create power cycles, refrigeration cycles, and liquefaction cycles. We can consider these processes as paradigms for general observations about energy and entropy constraints.

Chapter Objectives: You Should Be Able to…

1. Write energy and entropy balances around multiple pieces of equipment using correct notation including mass flow rates.

2. Simplify energy balances by recognizing when streams have the same properties (e.g., splitter) or flow rates (heat exchanger inlet/outlet).

3. Apply the correct strategy for working through a power cycle with multiple reheaters and feedwater preheaters.

4. For ordinary vapor compression cycles, locate condenser/evaporator P or T given one or the other and plot the process outlet and P-H diagram.

5. Successfully approach complex processes by simplifying the E-balance and S-balance, solving for unknowns.

5.1. The Carnot Steam Cycle

We saw in Example 4.4 on page 145 how a Carnot cycle could be set up using steam as a working fluid. The addition of heat at constant temperature and the macroscopic definition of entropy establish a correspondence between temperature and heat addition/removal. Steam is especially well suited for isothermal heat exchange because boiling and condensation are naturally isothermal and exchange large amounts of heat. To review, we could plot this cycle in T-S coordinates and envision a flow process with a turbine to produce work during adiabatic expansion and some type of compressor for the adiabatic compression as shown in Fig. 5.1. The area inside the P-V cycle represents the work done by the gas in one cycle, and the area enclosed by the T-S path is equal to the net intake of energy as heat by the gas in one cycle.

Figure 5.1. Illustration of a Carnot cycle based on steam in T-S coordinates.

The Carnot cycle has a major advantage over other cycles. It operates at the highest temperature available for as long as possible, reducing irreversibilities at the boundary because the system approaches the reservoir temperature during the entire heat transfer. In contrast, other cycles may only approach the hot reservoir temperature for a short segment of the heat transfer. A similar argument holds regarding the low temperature reservoir. Unfortunately, it turns out that it is impossible to make full use of the advantages of the Carnot cycle in practical applications. When steam is used as the working fluid, the Carnot cycle is impractical for three reasons: 1) It is impractical to stay inside the phase envelope because higher temperatures correlate with higher pressure. Higher pressures lead to smaller heat of vaporization to absorb heat. Since the critical point of water is only ~374°C, substantially below the temperatures from combustion, the temperature gradient between a fired heater and the steam would be large; 2) expanding saturated vapor to low-quality (very wet) steam damages turbine blades by rapid erosion due to water droplets; 3) compressing a partially condensed fluid is much more complex than compressing an entirely condensed liquid. Therefore, most power plants are based on modifications of the Rankine cycle, discussed below. Nevertheless, the Carnot cycle is so simple that it provides a useful estimate for checking results from calculations regarding other cycles.

5.2. The Rankine Cycle

In a Rankine cycle, the vapor is superheated before entering the turbine. The superheat is adjusted to avoid the turbine blade erosion from low-quality steam. Similarly, the condenser completely reduces the steam to a liquid that is convenient for pumping.

In Fig. 5.2, state 4′ is the outlet state for a reversible adiabatic turbine. We use the prime (′) to denote a reversible outlet state as in the previous chapter. State 4 is the actual outlet state which is calculated by applying the efficiency to the enthalpy change.

Figure 5.2. Rankine cycle.

 The prime denotes a reversible outlet state.

Because a real turbine always generates entropy, state 4 will always be to the right of 4′ on a T-S diagram. States 4 and 4′ can be inside or outside the phase envelope. Efficiencies are greater if state 4 is slightly inside the phase envelope because the enthalpy change will be larger for the same pressure drop due to the large enthalpy of vaporization; however, to avoid turbine blade damage, quality is kept above 90% in most cases.

Note in Fig. 5.2 that the superheater between the boiler and the turbine is not drawn, and only a single unit is shown. In actual power plants, separate superheaters are used; however, for the sake of simplicity in our discussions the boiler/superheater steam generator combination will be represented by a single unit in the schematic.

 Most plants will have separate boilers and superheaters. We show just a boiler for simplicity.

Turbine calculation principles were covered in the last chapter. Now we recognize that the net work is the sum of the work for the turbine and pump and that some of the energy produced by the turbine is needed for the pump. In general, the thermal efficiency is given by:

The boiler input can be calculated directly from the enthalpy out of the pump and the desired turbine inlet. The key steps are illustrated in Example 5.1.

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Example 5.1. Rankine cycle

A power plant uses the Rankine cycle. The turbine inlet is 500°C and 1.4 MPa. The outlet is 0.01 MPa. The turbine has an efficiency of 85% and the pump has an efficiency of 80%. Determine:

a. The work done by the turbine (kJ/kg)

b. The work done by the pump, the heat required, and the thermal efficiency;

c. The circulation rate to provide 1 MW net power output.

Solution

We will refer to Fig. 5.2 for stream numbers. The recommended method for solving process problems is to establish a table to record values as they are determined. In this text we will show values in the tables with bold borders if they have been determined by balance calculations. The turbine outlet can be read from the temperature table without interpolation. Cells with standard borders refer to properties determined directly from the problem statement

 Boldfaced table cells show calculations that were determined by balances. We follow this convention in the following examples.

Because the turbine inlet has two state variables specified, the remainder of the state properties are found from the steam tables and tabulated in the property table. We indicate a superheated vapor with “supV” compressed liquid with “compL.”

a. Stepping forward across the turbine involves the same specifications as part (c) of Example 4.13 on page 168. The properties from 4 and 4′ are transferred from that example to the property table. The work done is –959 kJ/kg.

b. The outlet of the condenser is taken as saturated liquid at the specified pressure, and those values are entered into the table. We must calculate  and . So we need H₆ and W_S,pump which are determined by calculating the adiabatic work input by the pump to increase the pressure from state 5. Although the reversible calculation for the pump is isentropic, we may apply Eqn. 2.61 without direct use of entropy, and then correct for efficiency. For the pump,

Thus, the work of the pump is small, resulting in H₆ = 191.8 + 1.8 = 193.6 kJ/kg. The net work is W’_S,net = –959.0 + 1.8 = 957.2 kJ/kg. The only source of heat for the cycle is the boiler/superheater. All of the heat input is at the boiler/superheater. The energy balance gives Q_H = (H₃ – H₆) = 3281.2 kJ/kg. The thermal efficiency is

If we neglected the pump work, the efficiency would 29.23%. Note that the pump work has only a small effect on the thermal efficiency but is included for theoretical rigor.

c. For 1 MW capacity, , the circulation rate is

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The cycle in Fig. 5.2 is idealized from a real process because the inlet to the pump is considered saturated. In a real process, it will be subcooled to avoid difficulties (e.g., cavitation¹) in pumping. In fact, real processes will have temperature and pressure changes along the piping between individual components in the schematic, but these changes will be considered negligible in the Rankine cycle and all other processes discussed in the chapter, unless otherwise stated. These simplifications allow focus on the most important concepts, but the simplifications would be reconsidered in a detailed process design.

5.3. Rankine Modifications

Two modifications of the Rankine cycle are in common use to improve the efficiency. A Rankine cycle with reheat increases the boiler pressure but keeps the maximum temperature approximately the same. The maximum temperatures of the boilers are limited by corrosion concerns. This modification uses a two-stage turbine with reheat in-between. An illustration of the modified cycle is shown in Fig. 5.3. Crudely, adding multiple stages with reheat leads to the maximum temperature being applied as much as possible, while avoiding extremely wet steam during expansion. This moves the process efficiency in the direction of a Carnot cycle. The implication of this modification is shown in Example 5.2.

Figure 5.3. Rankine cycle with reheat.

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Example 5.2. A Rankine cycle with reheat

Consider a modification of Example 5.1. If we limit the process to a 500°C boiler/superheater with reheat, we can develop a new cycle to investigate an improvement in efficiency and circulation rate. Let us operate a cycle utilizing two reversible turbines with η_E = 0.85 and a pump with η_C = 0.8. Let the feed to the first turbine be steam at 500°C and 6 MPa. Let the feed to the second stage be 1.4 MPa and 500°C (the same as Example 5.1). Determine the improvement in efficiency and circulation rate relative to Example 5.1.

Solution

Refer to Fig. 5.3 for stream numbers. First, let us find state 3. The inlet state values are entered in the table. P₄ = P₅ because we neglect the heat exchanger pressure drop. Upon expansion through the first reversible turbine, we look at the S^satV at 1.4 MPa and find it lower that S_4’. Therefore, the reversible state is superheated. Using {S,P} to find H,

Correcting for efficiency,

ΔH_I = W_{S, I} = 0.85(3001.2 – 3423.1) = 0.85(–421.9) = –358.6 kJ/kg

H₄ = 3423.1 – 358.6 = 3064.5 kJ/kg

State 5 was used in Example 5.1 (as state 3). Solving the energy balance for the reheater,

Q_reheat = (H₅ – H₄) = 3474.8 – 3064.5 = 410.3 kJ/kg

Turbine II was analyzed in Example 5.1. We found W_S,II = –959.0 kJ/kg and the total work output is W_S,turbines = (–358.6 –959.0) = –1317.6 kJ/kg. The pump must raise the pressure to 6 MPa. Using Eqn. 2.61, and correcting for efficiency,

State 7 is the same as state 5 in Example 5.1 and has been tabulated in the property table. H₈ = H₇ + W_S,pump = 191.8 + 7.6 = 199.4 kJ/kg. The net work is thus

W_{S, net} = –1317.6 + 7.6 = –1310 kJ/kg

The heat for the boiler/superheater is given by Q_b/s = H₃ – H₈ = 3423.1 – 199.4 = 3223.7 kJ/kg.

The thermal efficiency is

The efficiency has improved by , and the circulation rate has been decreased by 27%.

S = k ln W

L. Boltzmann

We have discussed energy balances and the fact that friction and velocity gradients cause the loss of useful work. It would be desirable to determine maximum work output (or minimum work input) for a given process. Our concern for accomplishing useful work inevitably leads to a search for what might cause degradation of our capacity to convert any form of energy into useful work. As an example, isothermally expanding an ideal gas from Vⁱ to 2Vⁱ can produce a significant amount of useful work if carried out reversibly, or possibly zero work if carried out irreversibly. If we could understand the difference between these two operations, we would be well on our way to understanding how to minimize wasted energy in many processes. Inefficiencies are addressed by the concept of entropy.

Entropy provides a measure of the disorder of a system. As we will see, increased “disorder of the universe” leads to reduced capability for performing useful work. This is the second law of thermodynamics. Conceptually, it seems reasonable, but how can we define “disorder” mathematically? That is where Boltzmann showed the way:

S = klnW

where S is the entropy, W is the number of ways of arranging the molecules given a specific set of independent variables, like T and V; k is known as Boltzmann’s constant.

For example, there are more ways of arranging your socks around the entire room than in a drawer, basically because the volume of the room is larger than that of the drawer. We will see that ΔS = Nkln(V₂/V₁) in this situation, where N is the number of socks and Nk = nR, where n is the number of moles, V is the volume, and R is the gas constant. In Chapter 1, we wrote U^ig = 1.5NkT without thinking much about who Boltzmann was or how his constant became so fundamental to the molecular perspective. This connection between the molecular and macroscopic scales was Boltzmann’s major contribution.

Chapter Objectives: You Should Be Able to…

1. Explain entropy changes in words and with numbers at the microscopic and macroscopic levels. Typical explanations involve turbines, pumps, heat exchangers, mixers, and power cycles.

2. Simplify the complete entropy balance to its most appropriate form for a given situation and solve for the productivity of a reversible process.

3. Sketch and interpret T-S, T-V, H-S, and P-H diagrams for typical processes.

4. Use inlet and outlet conditions and efficiency to determine work associated with turbines/compressors.

5. Determine optimum work interactions for reversible processes as benchmarks for real systems.

6. Sketch and interpret T-S, T-V, H-S, and P-H diagrams for typical processes.

4.1. The Concept of Entropy

Chapters 2 and 3 showed the importance of irreversibility when it comes to efficient energy transformations. We noted that prospective work energy was generally dissipated into thermal energy (stirring) when processes were conducted irreversibly. If we only had an “irreversibility meter,” we could measure the irreversibility of a particular process and design it accordingly. Alternatively, we could be given the efficiency of a process relative to a reversible process and infer the magnitude of the irreversibility from that. For example, experience might show that the efficiency of a typical 1000 kW turbine is 85%. Then, characterizing the actual turbine would be simple after solving for the reversible turbine (100% efficient).

In our initial encounters, entropy generation provides this measure of irreversibility. Upon studying entropy further, however, we begin to appreciate its broader implications. These broader implications are especially important in the study of multicomponent equilibrium processes, as discussed in Chapters 8–16. In Chapters 5–7, we learn to appreciate the benefits of entropy being a state property. Since its value is path independent, we can envision various ways of computing it, selecting the path that is most convenient in a particular situation.

Entropy may be contemplated microscopically and macroscopically. The microscopic perspective favors the intuitive connection between entropy and “disorder.” The macroscopic perspective favors the empirical approach of performing systematic experiments, searching for a unifying concept like entropy. Entropy was initially conceived macroscopically, in the context of steam engine design. Specifically, the term “entropy” was coined by Rudolf Clausius from the Greek for transformation.¹ To offer students connections with the effect of volume (for gases) and temperature, this text begins with the microscopic perspective, contemplating the detailed meaning of “disorder” and then demonstrating that the macroscopic definition is consistent.

 Entropy is a useful property for determining maximum/minimum work.

Rudolf Julius Emanuel Clausius (1822–1888), was a German physicist and mathematician credited with formulating the macroscopic form of entropy to interpret the Carnot cycle and developed the second law of thermodynamics.

To appreciate the distinction between the two perspectives on entropy, it is helpful to define the both perspectives first. The macroscopic definition is especially convenient for solving problems process problems, but the connection between this definition and disorder is not immediately apparent.

Macroscopic definition—Intensive entropy is a state property of the system. For a differential change in state of a closed simple system (no internal temperature gradients or composition gradients and no internal rigid, adiabatic, or impermeable walls),² the differential entropy change of the system is equal to the heat absorbed by the system along a reversible path divided by the absolute temperature of the system at the surface where heat is transferred.

where dS is the entropy change of the system. We will later show that this definition is consistent with the microscopic definition.

Microscopic definition—Entropy is a measure of the molecular disorder of the system. Its value is related to the number of microscopic states available at a particular macroscopic state. Specifically, for a system of fixed energy and number of particles, N,

where p_i is the number of microstates in the i^th macrostate, k = R/N_A. We define microstates and macrostates in the next section.

The microscopic perspective is directly useful for understanding how entropy changes with volume (for a gas), temperature, and mixing. It simply states that disorder increases when the number of possible arrangements increases, like the socks and drawers mentioned in the introduction. Similarly, molecules redistribute themselves when a valve is opened until the pressures have equilibrated. From the microscopic approach, entropy is a specific mathematical relation related to the number of possible arrangements of the molecule. Boltzmann showed that this microscopic definition is entirely consistent with the macroscopic property inferred by Rudolf Clausius. We will demonstrate how the approaches are equivalent.

Entropy is a difficult concept to understand, mainly because its influence on physical situations is subtle, forcing us to rely heavily on the mathematical definition. We have ways to try to make some physical connection with entropy, and we will discuss these to give you every opportunity to develop a sense of how entropy changes. Ultimately, you must reassure yourself that entropy is defined mathematically, and like enthalpy, can be used to solve problems even though our physical connection with the property is occasionally less than satisfying.

In Section 4.2, the microscopic definition of entropy is discussed. On the microscopic scale, S is influenced primarily by spatial arrangements (affected by volume and mixing), and energetic arrangements (occupation) of energy levels (affected by temperature). We clarify the meaning of the microscopic definition by analyzing spatial distributions of molecules. To make the connection between entropy and temperature, we outline how the principles of volumetric distributions extend to energetic distributions. In Section 4.3, we introduce the macroscopic definition of entropy and conclude with the second law of thermodynamics.

 The microscopic approach to entropy is discussed first, then the macroscopic approach is discussed.

The second law is formulated mathematically as the entropy balance in Section 4.4. In this section we demonstrate how heat can be converted into work (as in an electrical power plant). However, the maximum thermal efficiency of the conversion of heat into work is less than 100%, as indicated by the Carnot efficiency. The thermal efficiency can be easily derived using entropy balances. This simple but fundamental limitation on the conversion of heat into work has a profound impact on energy engineering. Section 4.5 is a brief section, but makes the key point that pieces of an overall process can be reversible, even while the overall process is irreversible.

In Section 4.6 we simplify the entropy balance for common process equipment, and then use the remaining sections to demonstrate applications of system efficiency with the entropy balance. Overall, this chapter provides an understanding of entropy which is essential for Chapter 5 where entropy must be used routinely for process calculations.

4.2. The Microscopic View of Entropy

Probability theory is nothing but common sense reduced to calculation.

LaPlace

To begin, we must recognize that the disorder of a system can change in two ways. First, disorder occurs due to the physical arrangement (distribution) of atoms, and we represent this with the configurational entropy.³ There is also a distribution of kinetic energies of the particles, and we represent this with the thermal entropy. For an example of kinetic energy distributions, consider that a system of two particles, one with a kinetic energy of 3 units and the other of 1 unit, is microscopically distinct from the same system when they both have 2 units of kinetic energy, even when the configurational arrangement of atoms is the same. This second type of entropy is more difficult to implement on the microscopic scale, so we focus on the configurational entropy in this section.⁴

 Configurational entropy is associated with spatial distribution. Thermal entropy is associated with kinetic energy distribution.

Entropy and Spatial Distributions: Configurational Entropy

Given N molecules and M boxes, how can these molecules be distributed among the boxes? Is one distribution more likely than another? Consideration of these issues will clarify what is meant by microstates and macrostates and how entropy is related to disorder. Our consideration will focus on the case of distributing particles between two boxes.

 Distinguishability of particles is associated with microstates. Indistinguishability is associated with macrostates.

First, let us suppose that we distribute N = 2 ideal gas⁵ molecules in M = 2 boxes, and let us suppose that the molecules are labeled so that we can identify which molecule is in a particular box. We can distribute the labeled molecules in four ways, as shown in Fig. 4.1. These arrangements are called microstates because the molecules are labeled. For two molecules and two boxes, there are four possible microstates. However, a macroscopic perspective makes no distinction between which molecule is in which box. The only macroscopic characteristic that is important is how many particles are in a box, rather than which particle is in a certain box. For macrostates, we just need to keep track of how many particles are in a given box, not which particles are in a given box. It might help to think about connecting pressure gauges to the boxes. The pressure gauge could distinguish between zero, one, and two particles in a box, but could not distinguish which particles are present. Therefore, microstates α and δ are different macrostates because the distribution of particles is different; however, microstates β and γ give the same macrostate. Thus, from our four microstates, we have only three macrostates.

Figure 4.1. Illustration of configurational arrangements of two molecules in two boxes, showing the microstates. Not that β and γ would have the same macroscopic value of pressure.

To find out which arrangement of particles is most likely, we apply the “principle of equal a priori probabilities.” This “principle” states that all microstates of a given energy are equally likely. Since all of the states we are considering for our non-interacting particles are at the same energy, they are all equally likely.⁶ From a practical standpoint, we are interested in which macrostate is most likely. The probability of a macrostate is found by dividing the number of microstates in the given macrostate by the total number of microstates in all macrostates as shown in Table 4.1. For our example, the probability of the first macrostate is 1/4 = 0.25. The probability of the evenly distributed state is 2/4 = 0.5. That is, one-third of the macrostates possess 50% of the probability. The “most probable distribution” is the evenly distributed case.

Table 4.1. Illustration of Macrostates for Two Particles and Two Boxes

What happens when we consider more particles? It turns out that the total number of microstates for N particles in M boxes is M^N, so the counting gets tedious. For five particles in two boxes, the calculations are still manageable. There will be two microstates where all the particles are in one box or the other. Let us consider the case of one particle in box A and four particles in box B. Recall that the macrostates are identified by the number of particles in a given box, not by which particles are in which box. Therefore, the five microstates for this macrostate appear as given in Table 4.2(a).

Table 4.2. Microstates for the Second and Third Macrostates for Five Particles Distributed in Two Boxes

The counting of microstates for putting two particles in box A and three in box B is slightly more tedious, and is shown in Table 4.2(b). It turns out that there are 10 microstates in this macrostate. The distributions for (three particles in A) + (two in B) and for (four in A) + (one in B) are like the distributions (two in A) + (three in B), and (one in A) + (four in B), respectively. These three cases are sufficient to determine the overall probabilities. There are M^N = 2⁵ = 32 microstates total summarized in the table below.

Note now that one-third of the macrostates (two out of six) possess 62.5% of the microstates. Thus, the distribution is now more peaked toward the most evenly distributed states than it was for two particles where one-third of the macrostates possessed 50% of the microstates. This is one of the most important aspects of the microscopic approach. As the number of particles increases, it won’t be long before 99% of the microstates are in one-third of the macrostates. The trend will continue, and increasing the number of particles further will quickly yield 99% of the microstates in that one-tenth of the macrostates. In the limit as N→∞ (the “thermodynamic limit”), virtually all of the microstates are in just a few of the most evenly distributed macrostates, even though the system has a very slight finite possibility that it can be found in a less evenly distributed state. Based on the discussion, and considering the microscopic definition of entropy (Eqn. 4.2), entropy is maximized at equilibrium for a system of fixed energy and total volume.⁷

A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore the deep impression which classical thermodynamics made upon me.

Albert Einstein

Having established the principle of the energy balance for individual systems, it is straightforward to extend the principle to a collection of several individual systems working together to form a composite system. One of the simplest and most enlightening examples is the Carnot cycle which is a benchmark system used to evaluate conversion of heat into work. The principle of the first law is perhaps most powerful when applied from an overall perspective. In other words, it is not necessary to deal with individual operations in order to draw conclusions about the overall system. Clever selections of composite systems or subsystems can then permit valuable insights about key behaviors and where to focus greater analysis. Implementing this overall perspective often requires dealing with multicomponent and reacting systems. For example, calorie counting for dietary needs must consider at least glucose, oxygen, CO₂, and water. It is not necessary at this stage to have a precise estimate of the mixture properties, but the reality of mixed systems must be acknowledged approximately. To illustrate practical implications, we consider distillation systems, reacting systems, and biological systems.

Chapter Objectives: You Should Be Able to…

1. Understand the steps of a Carnot engine and Carnot heat pump.

2. Analyze cycles to compute the work and heat input per cycle.

3. Apply the concepts of constant molar overflow in distillation systems.

4. Apply the concepts of ideal gas mixtures and ideal mixtures to energy balances.

5. Apply mole balances for reacting systems using reaction coordinates for a given feed properly using the stoichiometric numbers for single and multiple reactions.

6. Properly determine the standard heat of reaction at a specified temperature.

7. Use the energy balance properly for a reactive system.

3.1. Heat Engines and Heat Pumps – The Carnot Cycle

In this section we introduce the Carnot cycle as a method to convert heat to work. Many power plants work on the same general principle of using heat to produce work. In a power plant, heat is generated by coal, natural gas, or nuclear energy. However, only a portion of this energy can be used to generate electricity, and the Carnot cycle analysis will be helpful in understanding those limitations. Before we start the analysis, let us define the ratio of net work produced to the heat input as the thermal efficiency using the symbol η_θ:

 Thermal efficiency.

We wish to make the thermal efficiency as large as possible. We prove in the next chapter that the Carnot engine matches the highest thermal efficiency for an engine operating between two isothermal reservoirs. Maximizing thermal efficiency is a design goal that pervades Units I and II. We reconsider the thermal efficiency each time we add a layer of sophistication in our analysis. The concept of entropy in Chapter 4 will help us to generalize from ideal gases to steam or other fluids with available tables and charts. The calculus of classical thermodynamics in Chapter 5 will help to generalize to any substance, making our own tables and charts in Chapters 6–9. Evaluating the thermal efficiency in many situations is a skill that any engineer should have. Chemical engineering embraces a broad scope of … “chemicals.”

The Carnot Engine

The Carnot cycle was conceived by Sadi Carnot as a route to convert heat into work. In the previous chapter, we developed the energy balances and work calculations for reversible isothermal and adiabatic processes. The Carnot engine combines them in a cycle. Consider a piston in the vicinity of both a hot reservoir and a cold reservoir as illustrated in Fig. 3.1. The insulation on the piston may be removed to transfer heat from the hot reservoir during one step of the process, and also removed from the cold side to transfer heat to the cold reservoir during another step of the process. Carnot conceived of the cycle consisting of the steps shown schematically on the P-V diagram beginning from point a. Between points a and b, the gas undergoes an isothermal expansion, absorbing heat from the hot reservoir. From point b to c, the gas undergoes an adiabatic expansion. From point c to d, the gas undergoes an isothermal compression, rejecting heat to the cold reservoir. From point d to a, the gas undergoes an adiabatic compression to return to the initial state.

Figure 3.1. Schematic of the Carnot engine, and the Carnot P-V cycle when using a gas as the process fluid.

 The Carnot cycle is one method of constructing a heat engine.

Nicolas Léonard Sadi Carnot (1796–1832) was a French scientist who developed the Carnot cycle to demonstrate the maximum conversion of heat into work.

Let us consider the energy balance for the gas in the piston. Because the process is cyclic and returns to the initial state, the overall change in U is zero. The system is closed, so no flow work is involved. This work performed, W_EC = ∫PdV for the gas, is equal to the work done on the shaft plus the expansion/contraction work done on the atmosphere for each step. By summing the work terms for the entire cycle, the net work done on the atmosphere in a complete cycle is zero since the net atmosphere volume change is zero. Therefore, the work represented by the shaded portion of the P-V diagram is the useful work transferred to the shaft.

You can see from the shaded area in Fig. 3.1 that –W_{S, net} > 0; therefore, since Q_H > 0 and Q_C < 0, . The ratio  is negative, and to maximize η we seek to make Q_C as small in magnitude as possible.

The heat transferred and work performed in the various steps of the process are summarized in Table 3.1. For this calculation we assume the gas within the piston follows the ideal gas law with temperature-independent heat capacities. We calculate reversible changes in the system; thus, we neglect temperature and velocity gradients within the gas (or we perform the process very slowly so that these gradients do not develop).

Table 3.1. Illustration of Carnot Cycle Calculations for an Ideal Gas.^a

a. The Carnot cycle calculations are shown here for an ideal gas. There are no requirements that the working fluid is an ideal gas, but it simplifies the calculations.

Note: The temperatures T_H and T_C here refer to the hot and cold temperatures of the gas, which are not required to be equal to the temperatures of the reservoirs for the Carnot engine to be reversible. In Chapter 4 we will show that if these temperatures do equal the reservoir temperatures, the work is maximized.

Comparing adiabats b → c and d → a, the work terms must be equal and opposite since the temperature changes are opposite. The temperature change in an adiabatic process is related to the volume change in Eqn. 2.63. In that equation, when the temperature ratio is inverted, the volume ratio is inverted. Therefore, we reason that for the two adiabatic steps, V_b/V_a = V_c/V_d. Using the ratio in the formulas for the isothermal steps, the ratio of heat flows becomes

Inserting the ratio of heat flows into Eqn. 3.4 results in the thermal efficiency.

 The thermal efficiency of a heat engine is determined by the upper and lower operating temperatures of the engine.

You must use absolute temperature when applying Eqn. 3.6. We can skip the conversion to absolute temperature in the numerator of the last term because the subtraction means that the 273.15 (for units of K) in one term is canceled by the other. There is no such cancellation in the denominator.

Eqn. 3.6 indicates that we cannot achieve η_θ = 1 unless the temperature of the hot reservoir becomes infinite or the temperature of the cold reservoir approaches 0 K. Such reservoir temperatures are not practical for real applications. For real processes, we typically operate between the temperature of a furnace and the temperature of cooling water. For a typical power-plant cycle based on steam as the working fluid, these temperatures might be 900 K for the hot reservoir and 300 K for the cold reservoir, so the maximum thermal efficiency for the process is near 67%, theoretically. Most real power plants operate with thermal efficiencies closer to 30% to 40% owing to inherent inefficiencies in real processes.

Perspective on the Heat Engine

The Carnot cycle provides a quick and convenient guideline for processes that seek to convert heat flow into work. The striking conclusion we will prove in Chapter 4 is that it is impossible to convert all of the heat flow from the hot reservoir into work at reasonable temperatures. From an overall perspective, the detailed steps of the Carnot engine can be ignored. The amount of work is given by Eqn. 3.6. We can simply state that heat comes in, heat goes out, and the difference is the net work. This situation is represented by Fig. 3.2(a). As an alternate perspective, any process with a finite temperature gradient should produce work. If it does not, then it must be irreversible, and “wasteful.” This situation is represented by Fig. 3.2(b). These observations are similar to previous statements about gradients and irreversibility, but Eqn. 3.6 establishes a quantitative connection between the temperature difference and the reversible work possible.

Figure 3.2. The price of irreversibility. (a) Overall energy balance perspective for the reversible heat engine. (b) Zero work production in a temperature gradient without a heat engine, Q_H = Q_C.

When you can measure what you are speaking about, and express it in numbers, you know something about it. When you cannot measure it, your knowledge is meager and unsatisfactory.

The energy balance is based on the postulate of conservation of energy in the universe. This postulate is known as the first law of thermodynamics. It is a “law” in the same sense as Newton’s laws. It is not refuted by experimental observations within a broadly defined range of conditions, but there is no mathematical proof of its validity. Derived from experimental observation, it quantitatively accounts for energy transformations (heat, work, kinetic, potential). We take the first law as a starting point, a postulate at the macroscopic level, although the conservation of energy in elastic collisions does suggest this inference in the absence of radiation. Facility with computation of energy transformations is a necessary step in developing an understanding of elementary thermodynamics. The first law relates work, heat, and flow to the internal energy, kinetic energy, and potential energy of the system. Therefore, we precede the introduction of the first law with discussion of work and heat.

 The energy balance is also known as the first law of thermodynamics.

Chapter Objectives: You Should Be Able to…

1. Explain why enthalpy is a convenient property to define and tabulate.

2. Explain the importance of assuming reversibility in making engineering calculations of work.

3. Calculate work and heat flow for an ideal gas along the following pathways: isothermal, isochoric, adiabatic.

4. Simplify the general energy balance for problems similar to the homework problems, textbook examples, and practice problems.

5. Properly use heat capacity polynomials and latent heats to calculate changes in U, H for ideal gases and condensed phases.

6. Calculate ideal gas or liquid properties relative to an ideal gas or liquid reference state, using the ideal gas law for the vapor phase properties and heats of vaporization.

2.1. Expansion/Contraction Work

There is a simple way that a force on a surface may interact with the system to cause expansion/contraction of the system in volume. This is the type of surface interaction that occurs if we release the latch of a piston, and move the piston in/out while holding the cylinder in a fixed location. Note that a moving boundary is not sufficient to distinguish this type of work—there must be movement of the system boundaries relative to one another. For expansion/contraction interactions, the size of the system must change. This distinction becomes significant when we contrast expansion/contraction work to flow work in Section 2.3.

How can we relate this amount of work to other quantities that are easily measured, like volume and pressure? For a force applied in the x direction, the work done on our system is

dW = F_applied dx = –F_system dx

where we have used Newton’s principle of equal and opposite forces acting on a boundary to relate the applied and system forces. Since it is more convenient to use the system force in calculations, we use the latter form, and drop the subscript with the understanding that we are calculating the work done on the system and basing the calculation on the system force. For a constant force, we may write

W = – FΔx

If F is changing as a function of x then we must use an integral of F,

For a fluid acting on a surface of constant area A, the system force and pressure are related,

where the subscript EC refers to expansion/contraction work.

In evaluating this expression, a nagging question of perspective comes up. It would be a trivial question except that it causes major headaches when we later try to keep track of positive and negative signs. The question is essentially this: In the discussion above, is positive work being done on the system, or is negative work being done by the system? When we add energy to the system, we consider it a positive input into the system; therefore, putting work into the system should also be considered as a positive input. On the other hand, when a system does work, the energy should go down, and it might be convenient to express work done by the system as positive. The problem is that both perspectives are equally valid—therefore, the choice is arbitrary. Since various textbooks choose differently, there is always confusion about sign conventions. The best we can hope for is to be consistent during our own discussions. We hereby consider work to be positive when performed on the system. Thus, energy put into the system is positive. Because volume decreases when performing work of compression, the sign on the integral for work is negative,

where P and V are of the system. Clarification of “reversible” is given in Section 2.4 on page 42. By comparing Eqn. 2.3 with the definitions of work given by Eqns. 2.1 and 2.2, it should be obvious that the dV term results from expansion/contraction of the boundary of the system. The P results from the force of the system acting at the boundary. Therefore, to use Eqn. 2.3, the pressure in the integral is the pressure of the system at the boundary, and the boundary must move. A system which does not have an expanding/contracting boundary does not have expansion/contraction work.¹

 Expansion/Contraction work is associated with a change in system size.

2.2. Shaft Work

In a flowing system, we know that a propeller-type device can be used to push a fluid through pipes—this is the basis of a centrifugal pump. Also, a fluid flowing through a similar device could cause movement of a shaft—this is the basis for hydroelectric power generation and the water wheels that powered mills in the early twentieth century. These are the most commonly encountered forms of shaft work in thermodynamics, but there is another slight variation. Suppose an impeller was inserted into a cylinder containing cookie batter and stirred while holding the piston at a fixed volume. We would be putting work into the cylinder, but the system boundaries would neither expand nor contract. All of these cases exemplify shaft work. The essential feature of shaft work is that work is being added or removed without a change in volume of the system. We show in Section 2.8, page 54, that shaft work for a reversible flow process can be computed from

Note that Eqns. 2.3 and 2.4 are distinct and should not be interchanged. Eqn. 2.4 is restricted to shaft work in an open system and Eqn. 2.3 is for expansion/contraction work in a closed system. We later show how selection of the system boundary in a flow system relates the two types of terms on page 54.

 Shaft work characterizes the work of a turbine or pump.

2.3. Work Associated with Flow

In engineering applications, most problems involve flowing systems. This means that materials typically flow into a piece of equipment and then flow out of it, crossing well-defined system boundaries in the process. Thus, we need to introduce an additional characterization of work: the work interaction of the system and surroundings when mass crosses a boundary. For example, when a gas is released out of a tank through a valve, the exiting gas pushes the surrounding fluid, doing work on the surroundings. Likewise, when a tank valve is opened to allow gas from a higher pressure source to flow inward, the surroundings do work on the gas already in the system. We calculate the work in these situations most easily by first calculating the rate at which work is done.

Let us first consider a fluid entering a system as shown in Fig. 2.1. We have dW = Fdx, and the work interaction of the system is positive since we are pushing fluid into the system. The rate of work is , but  is velocity, and F = P·A. Further rearranging, recognizing , and that the volumetric flow rate may be related to the mass specific volume and the mass flow rate, ,

 Work associated with fluid flowing in/out of boundaries is called flow work.

Figure 2.1. Schematic illustration of flow work.

where PV are the properties of the fluid at the point where it crosses the boundary, and  is the absolute value of the mass flow rate across the boundary. When fluid flows out of the system, work is done on the surroundings and the work interaction of the system is

where  is the absolute value of the mass flow across the boundary, and since work is being done on the surroundings, the work interaction of the system is negative. When flow occurs both in and out, the net flow work is the difference:

where  and  are absolute values of the mass flow rates. For more streams, we simply follow the conventions established, and add inlet streams and subtract outlet streams.

2.4. Lost Work versus Reversibility

 Real processes involve “lost work.”

In order to properly understand the various characteristic forms that work may assume, we must address an issue which primarily belongs to the upcoming chapter on entropy. The problem is that the generation of disorder reflected by entropy change results in conversion of potentially useful work energy into practically useless thermal energy. If “generation of disorder results in lost work,” then operating in a disorderly fashion results in the lost capability to perform useful work, which we abbreviate by the term: “lost work.” It turns out that the most orderly manner of operating is a hypothetical process known as a reversible process. Typically, this hypothetical, reversible process is applied as an initial approximation of the real process, and then a correction factor is applied to estimate the results for the actual process. It was not mentioned in the discussion of expansion/contraction work, but we implicitly assumed that the process was performed reversibly, so that all of the work on the system was stored in a potentially useful form. To see that this might not always be the case, and how this observation relates to the term “reversible,” consider the problem of stirring cookie batter. Does the cookie batter become unmixed if you stir in the reverse direction? Of course not. The shaft work of stirring has been degraded to effect the randomness of the ingredients. It is impossible to completely recover the work lost in the randomness of this irreversible process. Any real process involves some degree of stirring or mixing, so lost work cannot be eliminated, but we can hope to minimize unnecessary losses if we understand the issue properly.

Consider a process involving gas enclosed in a piston and cylinder. Let the piston be oriented upward so that an expansion of the gas causes the piston to move upward. Suppose that the pressure in the piston is great enough to cause the piston to move upward when the latch is released. How can the process be carried out so that the expansion process yields the maximum work? First, we know that we must eliminate friction to obtain the maximum movement of the piston.

Friction decreases the work available from a process. Frequently we neglect friction to perform a calculation of maximum work.

“Aside from the logical and mathematical sciences, there are three great branches of natural science which stand apart by reason of the variety of far reaching deductions drawn from a small number of primary postulates. They are mechanics, electromagnetics, and thermodynamics.

These sciences are monuments to the power of the human mind; and their intensive study is amply repaid by the aesthetic and intellectual satisfaction derived from a recognition of order and simplicity which have been discovered among the most complex of natural phenomena… Yet the greatest development of applied thermodynamics is still to come. It has been predicted that the era into which we are passing will be known as the chemical age; but the fullest employment of chemical science in meeting the various needs of society can be made only through the constant use of the methods of thermodynamics.”

Lewis and Randall eloquently summarized the broad significance of thermodynamics as long ago as 1923. They went on to describe a number of the miraculous scientific developments of the time and the relevant roles of thermodynamics. Historically, thermodynamics has guided the development of steam engines, refrigerators, nuclear power plants, and rocket nozzles, to name just a few. The principles remain important today in the refinement of alternative refrigerants, heat pumps, and improved turbines, and also in technological advances including computer chips, superconductors, advanced materials, fermentations, biological cycles, and bioengineered pharmaceuticals. These latter-day “miracles” might appear to have little to do with power generation and refrigeration cycles at first thought. Nevertheless, as Lewis and Randall point out, the implications of the postulates of thermodynamics are far-reaching and will continue to be important in the development of even newer technologies. Much of modern thermodynamics focuses on characterization of the properties of mixtures, as their constituents partition into stable phases or inhomogeneous domains, and react. The capacity of thermodynamics to bring “quantitative precision in place of the old, vague ideas”1 is as germane today as it was then.

Before overwhelming you with the details that comprise thermodynamics, we outline a few “primary postulates” as clearly as possible and put them into the context of what we will refer to as classical equilibrium thermodynamics. In casual terms, our primary premises can be expressed as follows:

1. You can’t get something for nothing. (Energy is conserved.)
2. Maintaining order requires work. (Entropy generation leads to lost work.)2

Occasionally, it may seem that we are discussing principles that are much more sophisticated. But the fact is that all of our discussions can be reduced to these fundamental principles. The first principle is a casual statement of the first law of thermodynamics (conservation of energy) which will be introduced in Chapters 2 and 3. The second principle is a casual statement of the second law of thermodynamics (entropy balance) which will be introduced in Chapter 4. When you find yourself in the midst of a difficult problem, it may be helpful to remember the underlying principles. We will see that coupling these two principles with some slightly sophisticated reasoning (mathematics included) leads to many clear and reliable insights about a wide range of subjects from energy crises to high-tech materials, from environmental remediation to biosynthesis. The bad news is that the level of sophistication required is not likely to be instantly assimilated by the average student. The good news is that many students have passed this way before, and the proper trail is about as well marked as one might hope.

There is less-than-universal agreement on what comprises “thermodynamics.” If we simply take the word apart, “thermo” sounds like “thermal,” which ought to have something to do with heat, temperature, or energy. “Dynamics” ought to have something to do with movement. And if we could just leave the identification of thermodynamics as the study of “energy movements,” it would be sufficient for the purposes of this text. Unfortunately, such a definition would not clarify what distinguishes thermodynamics from, say, transport phenomena or kinetics, so we should spend some time clarifying the definition of thermodynamics in this way before moving on to the definitions of temperature, heat, energy, and so on.

The definition of thermodynamics as the study of energy movements has evolved considerably to include classical equilibrium thermodynamics, quantum thermodynamics, statistical thermodynamics, and irreversible thermodynamics as well as nonequilibrium thermodynamics. Classical thermodynamics has the general connotation of referring to the implications of constraints related to multivariable calculus as developed by J.W. Gibbs. We spend a significant effort applying these insights in developing generalized equations for the thermodynamic properties of pure substances. Statistical thermodynamics focuses on the idea that knowing the precise states of 1023 atoms is not practical and prescribes ways of computing the average properties of interest based on very limited measurements. We touch on this principle in our introduction to entropy, in our kinetic theory and molecular dynamics, and in the formulation of the internal energy relative to the intermolecular potential energy. We generally refrain from detailed formulation of all the statistical averages, however, maintaining the focus on simple concepts of molecular interactions. Irreversible thermodynamics and nonequilibrium thermodynamics emphasize the ways that local concentrations of atoms and energy evolve over periods of time. At this point, it becomes clear that such a broad characterization of thermodynamics would overlap with transport phenomena and kinetics in a way that would begin to be confusing at the introductory level. Nevertheless, these fields of study represent legitimate subtopics within the general realm of thermodynamics.

1.1. Introduction
These considerations should give you some idea of the potential range of applications possible within the general study of thermodynamics. This text will try to find a happy medium. One general unifying principle about the perspective offered by thermodynamics is that there are certain properties that are invariant with respect to time. For example, the process of diffusion may indicate some changes in the system with time, but the diffusion coefficient is a property which only depends on a temperature, density, and composition profile. A thermodynamicist would consider the diffusion process as something straightforward given the diffusion coefficient, and focus on understanding the diffusion coefficient. A transport specialist would just estimate the diffusion coefficient as best as he could and get on with it. A kineticist would want to know how fast the diffusion was relative to other processes involved. In more down-to-earth terms, if we were touring about the countryside, the thermodynamicists would want to know where we were going, the transport specialists would want to know how long it takes to get there, and the kineticists would want to know how fast the fuel was running out.

In thermodynamics we utilize a few basic concepts: energy, entropy, and equilibrium. The ways in which these are related to one another and to temperature, pressure, and density are best understood in terms of the connections provided by molecular mechanisms. These connections, in turn, can be summarized by the thermodynamic model (e.g., ideal gas), our quantitative description of the substance. Showing how energy and entropy couple with molecular characteristics to impact chemical process applications is the primary goal of this text. These insights should stick with you long after you have forgotten how to estimate any particular thermodynamic property, a heat capacity or activity coefficient, for example. We will see how assuming a thermodynamic model and applying the rules of thermodynamics leads to precise and extremely general insights relevant to many applications. A general theme throughout the text (and arguably throughout engineering) is: observe, predict, test, and evaluate. The prediction phase usually involves a model equation. Testing and evaluation expose limitations of the prospective model, which leads to a new cycle of observation, prediction… We terminate this hierarchy at an introductory level, but it never really ends. Extending this hierarchy is the source of innovation that must serve you for the next 50 years.

Chapter Objectives: You Should Be Able to…

1. Explain the definitions and relations between temperature, molecular kinetic energy, molecular potential energy and macroscopic internal energy, including the role of intermolecular potential energy and how it is modeled. Explain why the ideal gas internal energy depends only on temperature.
2. Explain the molecular origin of pressure.
3. Apply the vocabulary of thermodynamics with words such as the following: work, quality, interpolation, sink/reservoir, absolute temperature, open/closed system, intensive/extensive property, subcooled, saturated, superheated.
4. Explain the advantages and limitations of the ideal gas model.
5. Sketch and interpret paths on a P-V diagram.
6. Perform steam table computations like quality determination, double interpolation.

1.2. The Molecular Nature of Energy, Temperature, and Pressure
Energy is a term that applies to many aspects of a system. Its formal definition is in terms of the capability to perform work. We will not quantify the potential for work until the next chapter, but you should have some concept of work from your course in introductory physics. Energy may take the form of kinetic energy or potential energy, and it may refer to energy of a macroscopic or a molecular scale.

Energy is the sum total of all capacity for doing work that is associated with matter: kinetic, potential, submolecular (i.e., molecular rearrangements by reaction), or subatomic (e.g., ionization, fission).

Kinetic energy is the energy associated with motion of a system. Motion can be classified as translational, rotational, or vibrational.

Temperature is related to the “hotness” of a substance, but is fundamentally related to the kinetic energy of the constituitive atoms.

Potential energy is the energy associated with a system due to its position in a force field.

In the study of “energy movements,” we will continually ask, “How much energy is here now, and how much is there?” In the process, we need to establish a point for beginning our calculations. According to the definition above, we might intuitively represent zero internal energy by a perfect vacuum. But then, knowing the internal energy of a single proton inside the vacuum would require knowing how much energy it takes to make a proton from nothing. Since this is not entirely practical, this intuitive choice is not a good engineering choice usually. This is essentially the line of reasoning that gives rise to the convention of calculating energy changes relative to a reference state. Thus, there is no absolute reference point that is always the most convenient; there are only changes in energy from one state to another. We select reference conditions that are relevant throughout any particular process of interest. Depending on the complexity of the calculation, reference conditions may vary from, say, defining the enthalpy (to be defined later) of liquid water to be zero at 0.01°C (as in the steam tables) to setting it equal to zero for the molecular hydrogen and oxygen at 1 bar and 298.15 K (as in the heat of reaction), depending on the situation. Since this text focuses on changes in kinetic energy, potential energy, and energies of reaction, we need not specify reference states any more fundamental than the elements, and thus we do not consider subatomic particles.