Reclaiming the Central Role of Equations of State in Thermodynamics

An equation of state expresses the characteristic relationship between state variables for a particular thermodynamic system. Knowledge of the equations of state for a given system can be used to derive all its thermodynamic properties. Despite their central role in thermodynamics, the topic of equations of state is poorly addressed in traditional physical chemistry textbooks, and in most cases is typically associated with the properties of gases. This approach certainly minimizes the importance of the subject and hinders students from building a comprehensive understanding of classical thermodynamics. The central goal of this paper is to present an alternative approach to the derivation, analysis, and discussion of equations of state in physical chemistry courses. The strategy relies on the use of response coefficients (partial derivatives involving the system’s thermodynamic variables) and the systematic manipulation of simple thermodynamic relationships. The methodology can be used to derive the equation of state of a wide variety of systems and to analyze the similarities and differences in their thermodynamic behavior. INTRODUCTION Equations of state are key elements in the study of the thermodynamic behavior of physical systems. They describe the relationships that exist between relevant extensive and intensive parameters for any given system, and thus are very useful tools for prediction. Moreover, all the thermodynamic information of a system can be derived from the knowledge of a limited number of equations of state, which interrelate the system’s state variables (1). However, despite their predictive efficacy and thermodynamic relevance, equations of state tend to receive little attention in traditional physical chemistry courses. In fact, most conventional textbooks in this area introduce the concept of equation of state in a very restricted way. An equation of state is normally defined as the relationship between temperature (T), pressure (P), and volume (V) in a given system, and in many situations it is further limited to the case of gases or liquids (2-4). Fewer authors approach the subject in a more general fashion (5-7). Although chemical engineering textbooks and courses may discuss equations of state in further detail, their main emphasis is on the behavior of fluids. Moreover, a review of many of the papers published in this journal on the topic of equations of state illustrates the pervasiveness of this narrow definition (8-11). This approach to the study of equations of state in thermodynamics is problematic not only for being limited in scope but also because it may mislead students. The traditional association between the topics of equations of state and properties of gases leads students to believe either that only the behavior of gases can be described by these types of relationships or that equations such as PV=nRT can be applied to any type of system (12). Additionally, students are not given the opportunity to develop any real understanding of how equations of state for systems as varied as a rubber band or a paramagnetic solid may be systematically derived from experimental measurements of their thermodynamic properties. The central goal of this paper is to present an alternative approach to the derivation, analysis, and discussion of equations of state in our physical chemistry courses. The strategy relies on the use of “response coefficients” that can be measured experimentally (13-14) and on the systematic manipulation of simple thermodynamic relationships. These response coefficients are simple partial derivatives involving the thermodynamic state variables of the system. The use of response coefficients such as the compressibility of a fluid, κ=1/V(∂V/∂P)T, or the magnetic susceptibility of a paramagnetic solid, χT=(∂M/∂H)T, to derive an equation of state highlights the importance of experimental work geared towards the measurement of these types of physical properties. It illustrates how equations of state in classical thermodynamics are derived from experimental results and not from theoretical models. This methodology also makes explicit the intrinsic similarities among the thermodynamic behavior of diverse systems. In the following section we describe the proposed strategy and illustrate how it can be used to derive basic equations of state for a variety of systems. GENERAL FORMALISM Physical systems tend to reach, when isolated or unperturbed, a state in which no further change is perceivable. This state is characterized by the specific values of a small number of quantities, identified as the state variables of the system. The number of independent variables needed to describe this state is not known a priori; thus, one must rely on experience to answer this question. What parameters are selected Journal of Chemical Education, Accepted 11/04 1 to accomplish the task depends on the nature of the system and how readily measurable the parameters are. Through experience summarized in the zeroth law of thermodynamics, we know that for thermodynamic systems it is possible to measure the absolute temperature T. From additional experiments for different systems we find that one can express the absolute temperature as a function of selected independent variables: T=T(X1, X2,....;Y1, Y2,....), (1) where Xi and Yi represent, respectively, the relevant extensive and intensive thermodynamic variables of the system. (5,6). For a simple fluid, for example, Eq. (1) can be expressed as T=T(V,P) for a fixed amount of substance, and equilibrium states may be specified by any two of the three variables P, V, and T. In a fluid mixture, the equation of state will also include a functional dependence on the concentration of the various components. For other types of systems such as an elastic band, the equation of state may define the functional relationship between the band’s temperature, length L, and applied tension τ (T=T(L,τ)). For a dielectric solid, the equation of state interrelates the solid’s temperature, polarization PE, and applied electric field E (T=T(PE ,E)). The structure of the equation of state for a given system may be derived from experimental measurements. For example, one can measure the volume of a certain amount of gas at several temperatures and pressures and analyze the data to find the function V=V(T,P) that best represents the experimental behavior. This procedure can be systematized for different types of systems by using the experimental information to derive what we will call the system’s response coefficients. These quantities contain information about the system’s response to changes in the state variables. The proposed systematic determination of a system’s equation of state is best accomplished by rewriting Eq. (1) selecting the extensive parameter X1 as the dependent variable: X1=X1(T; X2,...Xn; Y1,Y2,...Ym). (2) Any infinitesimal perturbation in the state of the system due to changes in the independent variables will then result in the change i