Hypothetical paths

More than one phase can coexist within the system at equilibrium. When this phenomenon occurs, a phase boundary separates the phases from each other. One of the major topics in chemical thermodynamics, phase equilibrium, is used to determine the chemical compositions of the different phases that coexist in a given mixture at a specified temperature and pressure. 

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The first technique is very intuitive. Out of the few proteins that could be crystallized in a number of different conformations, adenylate kinase is probably the best-studied example. By combining nine observed crystal structures and interpolating between them, a movie was constructed that visualized a hypothetical path of its hinge-bending transition (jVonrhein et al. 1995]). 

FIGURE 7-7 Structure of the P0 glycoprotein protomer. (A) In this ribbon diagram of the extracellular domain of P0, each P strand is labeled with a letter and two antiparallel P sheets are formed. The disulfide bridge is indicated in dark orange and a hypothetical path for disordered amino acids 103-106 is shown in black in the FG loop. (B) Lattice formation by P0. A view of the intraperiod line, or extracellular apposition, of myelin in the PNS. The orange tetramer sets emanate from one bilayer and the blue tetramer interacts with all four of them. This is a view perpendicular to the plane of the myelin membrane. 

The immobilization of enzymes may introduce a new problem which is absent in free soluble enzymes. It is the mass-transfer resistance due to the large particle size of immobilized enzyme or due to the inclusion of enzymes in polymeric matrix. If we follow the hypothetical path of a substrate from the liquid to the reaction site in an immobilized enzyme, it can be divided into several steps (Figure 3.2) (1) transfer from the bulk liquid to a relatively unmixed liquid layer surrounding the immobilized enzyme (2) diffusion through the relatively unmixed liquid layer and (3) diffusion from the surface of the particle to the active site of the enzyme in an inert support. Steps... 

Figs. 1 and 2. Reaction in a catalyst pore. For the inactive catalyst (Fig. 2), reactant molecules can penetrate deeply into the pore structure before they react. For the active catalyst (Fig. 1), only the region near the pore mouth is useful since molecules react after a few collisions with the pore wall. A hypothetical path for a molecule traveling down a pore by Knudsen flow is shown. 

We emphasize that the reactions used in the analysis do not have to be the reactions actually occurring— we only need any convenient hypothetical path that connects products to reactants. In fact, we don t even need reactions at all, so long as we can achieve a balance on every element present. Further, "elements" need not be atoms they can be groups of atoms that may or may not constitute real molecules. Our procedure for identifying and balancing reactions reduces to the stuff equation for material, reformulated to apply to elements. We consider reactions in closed systems here and reactions in open systems in 7.5. 

FIGURE 21. AI2O3-NTMP-H2O SBD showing (a) the surface composition of FPL surfaces after 30 min immersion in aqueous solutions of NTMP at concentrations ranging from 0.1 to 500 ppm, with increases going from left to right, and (b) the hypothetical path representing no displacement of water. (From Reference 3.)... 

For a hypothetical path, assume that, as a first step, the reactants (one benzene and 7.5 oxygen molecules) simply break up into the component elements, that is, 6C, 3H2, and (15/2)02. (Actually, the oxygen does not break up.) For the second step, these elements recombine to form the products. Since the first step is the opposite of the formation reaction for one mole of benzene, its heat of reaction is the negative of benzene s heat of formation. The heat of reaction for the second step is the sum of the C02(g) heat of formation times 6 and the H20(l) heat of formation times 3. Adding the heats of reaction for each step yields the overall heat of reaction. 

The material in Chapter 1 forms the conceptual foundation on which we will construct our understanding of thermodynamics. We will formulate thermodynamics by identifying the state that a system is in and by looking at processes by which a system goes from one state to another. We are interested in both closed systems, which can attain thermodynamic equilibrium, and open systems. The state postulate and the phase rule allow us to identify which independent, intensive thermodynamic properties we can choose to constrain the state of the system. If we also know the amount of matter present, we can determine the extensive properties in the system. Thermodynamic properties are also called state functions. Since they do not depend on path, we may devise a convenient hypothetical path to calculate the change in their values between two states. Conversely, other quantities, such as heat or work, are path functions. 

Create an appropriate hypothetical path to solve these problems with available data. 

Hypothetical paths are constructed to make the calculation easier (or, in some cases, possible at all ). In fact, the ability to construct hypothetical paths between states allows for efficient collection and organization of experimental data. Often a path is constructed so that we can use available physical data. Much of the methodology of this textbook is based on using judicious choices of hypothetical paths to develop theory or obtain... 

Figure 2.3 Plot of a process that takes a system from state 1 to state 2 in Tv space. Three alternative paths are shown the real path as well as two convenient hypothetical paths.

The properties on the left-hand side of Equation (2.12a) depend only on the initial and final states. They can be calculated using the real path or any hypothetical path we create. The terms on the right-hand side are process dependent and the real path of the system must be used. 

Figure 2.3 illustrates a common hypothetical path used to calculate Au. In this case, T and v are chosen as independent properties. In step 1, we must know the temperature dependence ofu to calculate Au, as we go from Ti to T2 at constant volume. This information is often obtained in the form of heat capacity (or specific heat). Similarly, the temperature dependence of h used to find Ah can be found through reported heat capacity values. Therefore, heat capacity data are crucial in this problem-solving methodology. In the next section, we will explore how heat capacities are experimentally determined and how they are reported. 

Values of heat capacity for gases are almost always reported for the ideal gas state. Thus, when doing calculations using these data, you must choose a hypothetical path where the change in temperature occurs when the gas behaves ideally. 

Figure 2.12 Hypothetical path to calculate Ahvap at temperature T from data avadable at Tb and heat capacity data.

We now need a path to calculate AH. A convenient choice is illustrated by the solid lines in Figure E2.15B. That figure also shows the overall energy balance constraint of Equation (E15.2) as a dashed fine. Because enthalpy of reaction data are available at 298 K (Appendix A.3), we choose a hypothetical path where we first completely combust propane at 298 K, then... 

SOLUTION Since enthalpy is a thermodynamic property, we can construct a hypothetical path that utilizes the available data. The enthalpy of reaction at any temperature T can then be found from the path illustrated in Figure E2.17. The reactants are first brought to 298 K. They are then allowed to react under standard conditions to make the desired products. The products are then brought back up to the system temperature, T. Adding these three steps gives the following integral ... 

We have also developed these equations in intensive forms, on a mass and a molar basis, and for differential increments. Given a physical problem, we must determine which form to use and which terms in these equations are important and which terms are negligible or zero. We must also identify whether the ideal gas model or property tables are needed to solve the problem. For some processes, it is convenient to define a hypothetical path so that we can use available data to solve the problem. 

This section illustrates how to calculate the change in entropy of an ideal gas between two states if P and T for each state are known. We will define the initial state as state 1, at Pi and Ti, and the final state as state 2, at P2 and T. Since entropy is a state function, we can construct any path that is convenient between state 1 and state 2 to calculate As. Figure 3.6 illustrates such a hypothetical path. We choose a reversible process for our hypothetical path so that we can apply the definition of entropy. The first step consists of isothermal expansion, while the second step is isobaric heating. To find As, we will calculate the entropy change for each step and add them together. Details of the analysis for each step follow. 

Figure 3.6 Plot of a process in which a system goes from state 1 to state 2 in TP space. The change in entropy is calculated along a reversible hypothetical path.

Develop hypothetical paths to calculate the change in a thermodynamic property between two states, using appropriate property data. Appropriate data may include heat capacity data, pressure or volume explicit equations of state, or thermal expansion coefficients and isothermal compressibilities. 

This theme will commonly recur as we solve problems in thermodynamics. To solve for the hypothetical path depicted in Figure 5.2, we also need to calculate the changes in steps 1 and 3. This task will be made possible by developing a web of thermodynamic relations, as we will see in this chapter. 

We have seen that problem solvinginthermodynamics frequently involves construction of hypothetical paths to find the change in a given property between two states. In applying this procedure, we often come up with a partial derivative ofone property with respect to another, holding a third constant. In this section, we will use the thermodynamic web to translate partial derivatives to forms in which experimental data are routinely reported, such as Co, Cp, jj, K, and derivatives of equations of state. 

We can now go back to the calculation for A illustrated by the hypothetical path in Figure 5.2. Recall that we can relate the differential change in internal energy to the independent properties T and v by Equation (5.5) ... 

The hypothetical path depicted in Figure 5.2 allows us to reduce Equation (5.36) to one term for each step. The first and third steps are isothermal, so the first term on the right-hand side goes to zero and we use only the second term. Conversely, step 2 is isochoric and we use only the first term. Furthermore, we constructed this path to carry out step 2 under ideal gas conditions, where heat capacity data are readily... 

SOLUTION There are many ways to get from one thermodynamic state to another using the thermodynamic web. We consider the path shown in Figure E5.3A as an alternative calculation path for Au to that presented in Figure 5.2. In this case, our hypothetical path consists of two steps isochoric heating (step 1) followed by isothermal compression (step 2). However, for the temperature change in step 1, the gas no longer behaves as an ideal gas. 

In solving these problems, it is often necessary to construct hypothetical paths to calculate the change in a desired property between two states. Similarly, the approach for developing solutions to the phase equilibria and chemical reaction equilibria problems in the second half of the text will rely on an ability to exploit property relations and form paths that allow us to use appropriate measured data. 

Consider a gas that undergoes a process from state 1 to state 2. You know the ideal gas heat capacity and an equation of state. Which of the following hypothetical paths would be most appropriate to chose to calculate Am Explain. 

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