Chapter 8 Maxwell relations and measurable properties 8.1 Maxwell relations Other thermodynamic potentials emerging from Legendre transforms allow us to switch independent variables and give rise to alternate versions of the second law at different constant conditions: entropy fundamental equation energy fundamental equation enthalpy Helmholtz free energy Gibbs free energy maximum at equilibrium minimum at equilibrium minimum at equilibrium minimum at equilibrium minimum at equilibrium Here, we discuss some of the mathematical properties of these functions, and their consequences for relationships between thermodynamic variables. These considerations will allow us to connect quantities that are difficult to measure directly like the entropy and the chemical potential to variables that we can easily access using experiments. To start, note that all of the potentials above have multiple independent variables. A general feature of well-behaved multivariate functions is that the mixed partial derivatives don t depend on the order in which they are taken, (8.1) Or, writing out the derivatives explicitly, [ ] [ ] (8.2) This fact has important consequences for potentials since their derivatives involve other thermodynamic quantities. As an example, consider the energy equation, Examine the second derivative with respect to and at constant conditions: (8.3) [ ] [ ] (8.4) 1
Substituting for the inner derivatives using the fundamental equation in (8.3), [ ] [ ] (8.5) Therefore, a relationship between the derivatives of and emerges simply from the fact that these two quantities are related by a common second derivative of a thermodynamic potential. Such equalities based on potential second derivatives are called Maxwell relations, after James Maxwell, one of the early founders of modern thermodynamics and electromagnetism. There are in fact many Maxwell relations, depending on which potential is used and which pair of independent variables is examined. Here is another example, based on the Gibbs free energy: [ ] [ ] (8.6) Substituting for the inner derivatives, (8.7) which shows a relationship between the pressure-dependence of the chemical potential and the -dependence of the system volume. In fact, the right-hand derivative is equal to. These examples illustrate a basic recipe: pick a potential and a second derivative involving two of its independent variables, and then substitute first derivative definitions to produce a Maxwell relation. Maxwell relations connect two derivatives of thermodynamic variables, and emerge due to the equivalence of potential second derivatives under a change of operation order:, where is a thermodynamic potential and and are two of its natural independent variables. Maxwell relations are quite important, for two reasons. First, they show us that derivatives of thermodynamic parameters are not completely independent. This can serve as a consistency check in both experiments and in pen-and-paper analysis. Second, they provide a method to express derivatives involving difficult-to-measure quantities in terms of ones that are readily accessible experimentally, as we will now see. 8.2 Measurable quantities How does one measure the entropy or chemical potential from experiments? These kinds of quantities are not usually directly accessible in the lab. What we can measure, typically, are mechanical quantities like pressure, bulk quantities like volume and density, and 2
thermal properties like temperature and heat flow (e.g., by slow heat exchange experiments where we can measure temperature changes in a coupled reference body). Of the thermodynamic variables that we have discussed thus far, the following are considered measureable: Measurable thermodynamic variables or temperature pressure volume number of particles or mass (related by the molecular weight) enthalpy (latent heat) of phase change There are also several readily-measured material properties that depend on derivatives of thermodynamic variables. Because these quantities measure the change in a system parameter in response to an infinitesimal perturbation, they are termed response functions: Measurable thermodynamic response functions constant volume heat capacity constant pressure heat capacity isothermal compressibility thermal expansivity / expansion coef. These response functions are defined such that they are positive in the case of normal systems, such as ideal gases and simple liquids. There are occasional exceptions in some of them; for example, the thermal expansivity of liquid water below 4 C is negative. There are additional measurable response functions that we have not listed; some of them can be expressed as combinations of the above, and some of them emerge in systems with other thermodynamic variables, such as those involving electromagnetic or interfacial work. Note that the heat capacities as presented above are extensive, while more conventionally we might see intensive versions, such as and. Maxwell relations enable us to express experimentally-inaccessible quantities in terms of the measurable ones just listed. Consider the following example derivative, (8.8) 3
This is reminiscent of the definition for, so that we can substitute: (8.9) Hence the entropy s dependence on pressure at constant temperature is related to the thermal expansivity. Many times, however, getting to measurable quantities isn t so easy. Consider the following Maxwell relation, (8.10) Here, we moved to something that appears even less measurable than before. Fortunately, there are other ways to relate thermodynamic variables using the principles of multivariate calculus. We can use the so-called triple product rule, which shows that, Rearranging, (8.11) (8.12) which finally gives an expression involving only measurable quantities, (8.13) We now turn to a more complex example that relies on additional mathematical manipulations; let us find the relationship between and. From the definitions of these quantities, we can infer a good place to start: both response functions depend on a derivative of the entropy with temperature. The difference between the two is the variable that is held constant in the derivative. We can relate a change in the constant conditions of a derivative in the following way. Construct the function, assuming constant conditions throughout. Notice that this is not a fundamental potential because is not a function of its natural variables. We can still, however, perform the construction since is a state function and and stem from different conjugate pairs. We choose as an 4
independent variable because it is the temperature derivative that is relevant to the heat capacities. We now expand in differential form as, (8.14) To start relating the constant volume conditions to those at constant pressure, we take the -derivative of this expression at constant. This is operationally equivalent to dividing by and applying constant conditions to any complete derivative that is formed, (8.15) We recognize that the LHS can be replaced using to, and the rightmost derivative connects (8.16) Finally, we use a Maxwell relation to address the final term in our expression, giving, by Maxwell relation from previous example (8.17) Simplifying everything, (8.18) As a final example, we will compute a quantity called the isentropic compressibility, defined by, (8.19) Experimentally, the isentropic compressibility measures the fractional change in volume of a system during a reversible adiabatic compression. To proceed, we use the triple product rule to remove the entropy from the constant condition, 5
[ ] (8.20) Now we use a new calculus rule, called addition of variable, that will enable us to expand the numerator and denominator by including temperature. We want to add temperature because there are no measurable properties that are derivatives of at constant and vice versa. The rule produces, (8.21) Note that the remaining derivatives all involve permutations of the same three variables. This suggests the use of the triple product rule again, [ ] [ ] (8.22) Finally, we see that we can insert the expression for the isothermal compressibility, (8.23) 8.3 General considerations for calculus manipulations Many thermodynamic calculus manipulations can be derived from a relatively simple procedure. The first step is to construct a state function involving the variables of interest. Some examples include:,,, or. Note that, if the state function is a potential, we do not necessarily need to use natural variables. We can choose any independent variables that we like so long as they form a complete set of thermodynamic information (i.e., all stem from different conjugate pairs). The second step is to write out the full differential of the state function. As an example, take, 6
(8.24) The third step is to set the differentials to zero for any terms that are constant in the problem of interest. For example, if is constant then and we have: (8.25) Note that we must keep in mind that the current equation corresponds to constant conditions. This is important as any new derivatives that are formed will acquire as a constant variable. The fourth step is to take the derivative of interest, essentially dividing by the appropriate differential. All new derivatives acquire any previously applied constant conditions. Moreover, we must specify additional constant conditions as necessary to meet one less than the number of independent variables. In our example, we can also specify to be constant when taking the derivative: (8.26) In conjunction with this procedure, one can then begin to substitute various fundamental definitions, expressions for measurable quantities, and Maxwell relations. As another example, consider the expression. When derivatives involve a thermodynamic potential, we often begin with its differential form: (8.27) Constant conditions implies, (8.28) Taking the temperature derivative at constant volume, (8.29) in which we can begin to see opportunities to substitute response functions. There are a number of common calculus manipulation techniques that are useful in thermodynamic analysis. These are briefly summarized below. 7
Rules of calculus manipulations inversion triple product rule addition of variable nonnatural derivative As we have seen earlier, there are also rules that apply specifically to thermodynamic potentials: Rules for thermodynamic potentials Maxwell relations potential transformation The combination of all of these techniques enables us to relate virtually any thermodynamic quantity to the set of measurable variables described earlier. 8
Problems prove the triple product rule 1. Fundamentals problem. Given only the ideal gas equation of state,, prove the following properties using fundamental equations and Maxwell relations. In the first two parts, you are proving that several ideal-gas properties are volume independent. a) b) and c) 2. Fundamentals problem. Express the following in terms of measurable quantities for a single component system at constant conditions: a) b) c) d) e) f) g) h) 3. Conceptual problem. Indicate whether each of the following relations is true in general: a) 9
b) c) 1. Prove that all of the response functions, and are in fact measurable. For the former two, consider the first law in conjunction with heating at different constant conditions, where can be quantified by coupling to a reference body. 3. Conceptual problem. A quasi-static adiabatic process is performed on a system at starting at state 1. Indicate the direction the system will proceed in the P-T plane (i.e., find ), in terms of measurable quantities. 4. Fundamentals problem. The Van der Waals equation of state for a gas is: where and and are positive constants. Find the change in molar internal energy,, upon isothermal expansion from molar volume to. 6. Applied problem. A Joule-Thompson process is one by which a gas is throttled (expanded) with no heat exchanged. Therefore, any change in internal energy is due to internal changes in pressure-volume work, and total enthalpy is conserved. Often during such an expansion, the temperature of the gas will change. The Joule-Thomson coefficient measures how much the temperature changes for a given pressure drop:. Here does not indicate a relationship to the chemical potential, but instead simply comes from convention. Show that the Joule-Thomson coefficient is given by. 6. Applied problem. Let the fundamental equation for an elastic band be 10
where is the length and the tension. a) An elastic band heats up when it is slowly stretched adiabatically. When it is cooled at constant tension, will the same band expand or contract? b) The same amount of heat flows into two identical elastic bands (not necessarily the same as in part a) while one is held at constant tension and the other at constant length. Which experiences the largest temperature rise? Prove this. 7. Applied problem. Liquid water exhibits a wealth of properties that are considered anomalous when compared to simple liquid models like the van der Waals fluid. One wellknown anomaly is that, at constant pressure, water exhibits a temperature of maximum density (TMD). At atmospheric pressure, this temperature is around 4 C. In general, though, we can construct a line in the plane that gives the TMD for any pressure. Consider the properties of this line: a) Along the TMD line, what must be true about the thermal expansion coefficient? Prove also that along this line. Be careful to address all of the terms in your proof. b) Show that the slope of this line is given by: c) Water s isothermal compressibility increases upon cooling. Can anything be said about the slope of the TMD line? 11