The Thermodynamic Web

Apply the thermodynamic web to relate measured, fundamental, and derived thermodynamic properties. In doing so, apply the fundamental property relations. Maxwell relations, the chain rule, derivative inversion, the cyclic relation, and Equations (5.22), (5.23), and (5.24). Use Figure 5.3 to rewrite partial derivatives with T,P,s, and v in more convenient forms. 

In this section, we present three other mathematical relations that will be of use in helping us surf the thermodynamic web. The first relationship is the chain rule, which can be written in general as follows ... 

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. 

Figure 5.3 presents a way to navigate the thermodynamic web when partial derivatives with T,P,s, and v are encountered. It provides 12 permutations of partial derivatives between these properties. Derivative inversion can also be applied to form 12 additional... 

Figure 5.3 Roadmap through the thermodynamic web. Partial derivatives of T, P, s, and v are related to each other and to reported properties c , Cp, j8, and K.

Using the paths in the diagram, we can rewrite any derivative in Figure 5.3. For example, we can follow the diagram to represent as - dP/dv)s as — Cp/vKc . See if you can justify this relation (and the others indicated in Figure 5.3). This exercise will pay dividends when you encounter these forms in solving problems with the thermodynamic web. 

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. 

We can obtain an expression for the change in enthalpy in terms of the independent properties T and P by applying the thermodynamic web in a manner similar to that used for entropy and internal energy. We first write the differential expression in the form of Equation (5.4) using the independent properties T and P. 

First-law—Open-System—Calculation Using the Thermodynamic Web... 

If we integrate Equation (E5.4L), we get a result identical to Equation (E5.4F), so the rest of the problem is equivalent to the part above where we used T and v as independent properties. We come up with the same result applying the thermodynamic web to each path the form h = h T, v) yields an equivalent result to h = h T, P). However, the first choice of independent properties made the math easier. This result is not surprising in light of the discussion after Equation (5.33) that is, T and v are the convenient independent properties when we have a pressure-explicit equation of state. 

As the name Thermodynamic Web implies, many approaches can be used to develop the useful relations between measured properties and the fundamental and derived properties that we need to solve problems. As we learned in Chapter 1, pressure and temperature are the properties that drive systems toward mechanical and thermal equilibrium, respectively. Because we will focus on equilibrium systems in the remainder of the text (phase equilibrium in Chapters 6-8 and chemical reaction equilibrium in Chapter 9), it is instructive to recast the thermodynamic web, exclusively in terms of independent properties, T and P. 

We now need to come up with an expression for the enthalpy departure function so that we can solve Equation (5.47). Since enthalpy departure at a given state is related to the intermolecular forces involved, we will need to use the PvT relation developed in Chapter 4 and then apply the relationships of the thermodynamic web to come up with an expression for the enthalpy departure function. In the development that follows, we will use the generalized compressibility charts and tables discussed in Section 4.4 to develop values for the generalized enthalpy departure function based on corresponding... 

Using the thermodynamic web, we can determine each difference on the right-hand side of Equation (5.53). At constant temperature, the entropy can be written in terms of the independent property P. According to Equation (5.32),... 

We can use the thermodynamic web to develop an expression for /tjt in terms of PvT property relations and heat capacities. We begin with Equation (5.37) ... 

Based on the definition in Problem 5.55, use the thermodynamic web to come up with an expression and a value for [Vsound] in water at 20°C. Use the steam tables for thermodynamic property data of liquid water. 

Apply the fundamental property relation for Gibbs energy and other tools of the thermodynamic web to predict how the pressure of a pure species in phase equilibrium changes with temperature and how other properties change in relation to one another. Write the Clapeyron equation and use it to relate Tand Pfor a pure species in phase equilibrium. Derive the Clausius-Clapeyron equation for vapor-liquid mixtures, and state the assumptions used. Relate the Clausius-Clapeyron equation to the Antoine equation. 

As we saw in Chapter 5, the thermodynamic web provides a useful vehicle for relating derived thermodynamic properties to measured properties. Applying the fundamental property relation for g [Equation (5.9)] to each phase, we get ... 

Partial molar properties can be related to each other, further extending the thermodynamic web. For example, consider the total solution enthalpy ... 

In analogy to the case for pure species we saw in Section 6.2, we can determine how a mixture in equilibrium responds to changes in measured variables. We can use the thermodynamic web to relate the change in chemical potential and the criteria for chemical equilibrium between phases with changes in pressure, temperature, and mole fraction. 

Apply the thermodynamic web to show that the partial derivative of this function with respect to temperature at constant pressure is given by ... 

---