The state postulate

Thermodynamic properties provide a powerful tool for learning about systems and making engineering calculations. Since they are independent of the calculation pathway, the clever engineer can often use data that are available in the ftterature to characterize processes or equilibrium states of interest. It turns out that once we know the value of 

If we have a system containing a pure substance, its thermodynamic state and, therefore, all its intensive thermodynamic properties can be determined from two independent intensive properties. 

We call the two intensive properties we select to constrain the state the independent variables. All other intensive properties are then dependent variables. We are free to decide what intensive properties we choose as the independent variables, as long as they are indeed independent of each other. For example, the molar volume, in [m /mol], of a pure species in the gas phase is completely specified once we know its temperature and pressure, that is, 

We write o(T, P) to indicate some general mathematical function that depends on the values of the independent variables T and P. 

On the other hand, if we wish to know the value of an extensive property of the system, we must additionally specify the size of the system. Thus, to constrain the value of an extensive property of a pure substance, we need to specify three quantities—the two intensive properties that constrain the state and a third property to indicate the size of the system. For example, if we wish to know the extensive volume of the gas, in [m ], we would need to additionally specify, for example, the total number of moles in the system  

---

Gases and superheated vapors exist at temperatures above the saturation or boiling temperature for a given pressure. Pressure and temperature are independent properties and knowledge of both, according to the state postulate, fully defines the state and all other properties. 

Fortunately, it is not normally necessary to select an internally reversible path and carry out this integration of Equation 23.46, because entropy is a property. Based on the state postulate, entropy can be determined from changes in other properties. 

Apply the state postulate and the phase rule to determine the appropriate independent properties to constrain the state of a system that contains a pure species. 

INDEPENDENT AND DEPENDENT THERMODYNAMIC PROPERTIES The State Postulate... 

The state postulate refers to the entire system. A related concept is used to determine the number of independent, intensive properties needed to constrain the properties in a given phase, which is referred to as the degrees of freedom, As we will later verify (see Example 6.17), the Gibbs phase rule says that is given by ... 

The number of phases influences which properties are independent. The determination of which two properties we can choose to constrain the system according to the state postulate depends on the number of phases that are present. First, consider a system with only one phase present (77 = l). In this case the state postulate and the Gibbs phase rule are equivalent. Equation (1.13) tells us we need two independent properties to constrain the phase and, thus, the system. Specification of any two intensive properties—such as pressure, P, and temperature, T—constrains all the other properties in the system. The... 

In this section, we explore graphical depictions of the relation between the measured variables P, v, and T. Figure 1.6 shows a PvT surface for a typical pure substance. This three-dimensional graph is constructed by plotting molar volume on the x-axis, temperature on the (/-axis, and pressure on the z-axis. The state postulate tells us that these three intensive properties are not all independent. The surface that is plotted identifies the values that all three measured properties of a given pure substance can simultaneously have. While each species has its own characteristic PvT surface, the general qualitative features shown in Figure 1.6 are common to aU species. ... 

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. 

We limit our present discussion to constant composition systems we will learn about mixtures that can change in composition in Chapter 6. Recall that the state postulate says that for systems of constant composition, values of two independent, intensive properties completely constrain the state of the system. In mathematical terms, the change in any intensive thermodynamic property of interest, z, can be written in terms of partial derivatives of the two independent intensive properties, x and y, as follows ... 

The state postulate tells us that if we specify two intensive properties for any pure species, we constrain the state of a single-phase system. For extensive properties, we must additionally specify the total number of moles. In Chapter 5, we learned how to mathematically describe any intensive thermodynamic property in terms of partial derivatives of two independent, intensive properties. Since we are now concerned with thermal and mechanical equilibrium, it makes sense to choose T and P as the independent, intensive properties. We wish to extend the formulation to mixtures with changing composition. In addition to specifying two independent properties, we must also consider the number of moles of each species in the mixture. 

Since energies never have absolute values, we need a reference state for the partial molar Gibbs energy. The reference state is indicated by a superscript o . In choosing a reference state, we must specify the appropriate number of thermodynamic properties as prescribed by the state postulate the rest of the properties of the reference state are then constrained. The reference chemical potential, juf, is the chemical potential at the reference pressure, P°, and at the same temperature as the chemical potential of interest, T. The latter constraint derives from our stipulation of constant temperature. Integrating between a reference state and the state of the system, we get ... 

The basic assumption is that the Langmuir equation applies to each layer, with the added postulate that for the first layer the heat of adsorption Q may have some special value, whereas for all succeeding layers, it is equal to Qu, the heat of condensation of the liquid adsorbate. A furfter assumption is that evaporation and condensation can occur only from or on exposed surfaces. As illustrated in Fig. XVII-9, the picture is one of portions of uncovered surface 5o, of surface covered by a single layer 5, by a double-layer 52. and so on.f The condition for equilibrium is taken to be that the amount of each type of surface reaches a steady-state value with respect to the next-deeper one. Thus for 5o... 

The relationship between tire abstract quantum-mechanical operators /4and the corresponding physical quantities A is the subject of the fourth postulate, which states ... 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

The concept of equilibrium is central in thermodynamics, for associated with the condition of internal eqmlibrium is the concept of. state. A system has an identifiable, reproducible state when 1 its propei ties, such as temperature T, pressure P, and molar volume are fixed. The concepts oi state a.ndpropeity are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see Postulates I and 3 following) is recognized much more indirectly. The number of properties for wdiich values must be specified in order to fix the state of a system depends on the nature of the system and is ultimately determined from experience. 

On a different note, after some 50 years of intensive research on high-pressure shock compression, there are still many outstanding problems that cannot be solved. For example, it is not possible to predict ab initio the time scales of the shock-transition process or the thermophysical and mechanical properties of condensed media under shock compression. For the most part, these properties must presently be evaluated experimentally for incorporation into semiempirical theories. To realize the potential of truly predictive capabilities, it will be necessary to develop first-principles theories that have robust predictive capability. This will require critical examination of the fundamental postulates and assumptions used to interpret shock-compression processes. For example, it is usually assumed that a steady state is achieved immediately after the shock-transition process. However, due to the fact that... 

The substituent effects in aromatic electrophilic substitution are dominated by resonance effects. In other systems, stereoelectronic effects or steric effects might be more important. Whatever the nature of the substituent effects, the Hammond postulate insists diat structural discussion of transition states in terms of reactants, intermediates, or products is valid only when their structures and energies are similar. 

The similarity solution for a flow field in front of a steady piston is a special case from a much larger class of similarity solutions in which certain well-defined variations in piston speed are allowed (Guirguis et al. 1983). The similarity postulate for variable piston speed solutions, however, sets stringent conditions for the gas-dynamic state of the ambient medium. These conditions are unrealistic within the scope of these guidelines, so discussion is confined to constant-velocity solutions. 

Now suppose that, from this equilibrium situation, the final state is instantaneously removed. The production of transition state species by the product state will cease. However, the production of transition state species by the reactant state is unaffected by this suppression of the final state, and, according to the third postulate of the theory, the rate of reaction is a function of the transition state concentration formed from the reactant state. This is the usual argument for the equilibrium assumption. Despite its apparent artificiality, the equilibrium assumption is generally considered to be fairly sound, with the possible exception of its application to very fast reactions. ... 

The Hammond postulate is a valuable criterion of mechanism, because it allows a reasonable transition state structure to be drawn on the basis of knowledge of the reactants and products and of energy differences between the states (i.e., AG and AG°). Throughout this chapter we have located transition states in accordance with the Hammond postulate. 

According to this very simple derivation and result, the position of the transition state along the reaction coordinate is determined solely by AG° (a thermodynamic quantity) and AG (a kinetic quantity). Of course, the potential energy profile of Fig. 5-15, upon which Eq. (5-60) is based, is very unrealistic, but, quite remarkably, it is found that the precise nature of the profile is not important to the result provided certain criteria are met, and Miller " obtained Eq. (5-60) using an arc length minimization criterion. Murdoch has analyzed Eq. (5-60) in detail. Equation (5-60) can be considered a quantitative formulation of the Hammond postulate. The transition state in Fig. 5-9 was located with the aid of Eq. (5-60). 

Obtain the partial CH and HF bond distances in eacl transition state, and compare them to the CH and HF bon( distances in propane and hydrogen fluoride, respectively Does the Hammond Postulate correctly predict whicl bond distances will be most similar Explain. 

Use of the Hammond Postulate requires that the reverse reactions both be fast. Obtain energies for the transition states leading to 1-propyl and 2-propyl radicals ipropane+Br end and propane+Br center), and draw a reaction energy diagram for each (place the diagrams on the same axes). Is use of the Hammond Postulate justified Compare the partial CH and HBr bond distances in each transition state to the corresponding distances in propane and hydrogen bromide, respectively. Does the Hammond Postulate correctly predict which bond distances will be most similar Explain. 

Obtain the energies of propene, dimethylborane, and 1-propyldimethyl borane, and calculate AH n for dimethylborane addition. Is this reaction exothermic or endothermic Use this result and the Hammond Postulate to predict whether the transition state will be more reactant like or more product like . Compare the geometry of the transition state to that of the reactants and products. Does the Hammond Postulate correctly anticipate the structure of the transition state Explain. 

Examine the structures of the two transition states (chlorine atom+methane and chlorine+methyI radical). For each, characterize the transition state as early (close to the geometry of the reactants) or as late (close to the geometry of the products) In Ught of the thermodynamics of the individual steps, are your results anticipated by the Hammond Postulate Explain. 

Would you describe the transition state for the Claisen rearrangement as early (like reactants), late (like products) or in between Given the overall thermodynamics of reaction, do you conclude that the Hammond Postulate applies Explain. 

Another factor in determining eomparative positional reactivity is the localization energy required to produce 50 or some form approaching 50 as the substrate reaehes the transition state under the influence of the nueleophile. Experimental results on azines and theoretieal considerations warrant the general postulate that the localization energy will be lower when a nitrogen atom is at the... 

An explanation of the relationship between reaction rate and intermediate stability was first advanced in 1955. Known as the Hammond postulate, the argument goes like this transition states represent energy maxima. They are high-energy activated complexes that occur transiently during the course of a reaction and immediately go on to a more stable species. Although we can t... 

---