Article pubs.acs.org/jchemeduc Teaching Thermodynamics and Kinetics to Advanced General Chemistry Students and to Upper-Level Undergraduate Students Using PV Diagrams Srinivasan S. Iyengar* and Romualdo T. deSouza* Department of Chemistry, Indiana University, Bloomington, Indiana 47405, United States *S Supporting Information ABSTRACT: We describe how complex concepts in macroscopic chemistry, namely, thermodynamics and kinetics, can be taught at considerable depth both at the first-year undergraduate as well as upper levels. We begin with a careful treatment of PV diagrams, and by pictorially integrating the appropriate area in a PV diagram, we introduce work. This starting point allows us to elucidate the concept of state functions and nonstate functions. The students readily appreciate that for a given transition, the area enclosed by the PV curve (work) depends on the path taken. It is then argued that heat, within this chosen framework, is a consequence of energy conservation and the fact that work is not a state function. This leads to a visual introduction of all the components involved in the first law of thermodynamics. The PV diagrams are then used to introduce entropy as being related to the maximum possible work to be done by the system. This macroscopic description of entropy is then related to the usual microscopic view of Boltzmann. This equivalence connects the area inside the PV diagram (work from a specific kind of pathway) and the number of microstates involved in the Boltzmann expression. The connection between the macroscopic and microscopic description of entropy also illuminates the exponential dependence of the number of microstates (or probability) on an energy,knownas free energy. The Arrhenius picture of chemical kinetics then readily follows from the exponential dependence stated above, and finally, a reactive event is viewed as a statistically rare event, to further clarify the appearance of entropy in the rate expression. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Physical Chemistry, Interdisciplinary/Multidisciplinary, Curriculum, Kinetics, Mathematics/Symbolic Mathematics, Thermodynamics, Rate Law he flow of energy that often accompanies chemical This material dealing with macroscopic concepts comprises the transformations is a difficult concept to grasp for students, second half of a one-semester course for select, high-achieving first- T − teachers, and researchers.1 5 The difficulty arises from the year undergraduate students. The composition of this honors abstract nature of many energy concepts such as work, heat, course and its logistics have been previously presented,18 but for entropy, and free energy. The complexities involved in learning convenience is also provided as part of Supporting Information these topics may be gauged through the large number of (Appendix A). innovative teaching tools proposed in the literature that are It is important to juxtapose our efforts with the “organic- specific to thermodynamics. These teaching paradigms range from fi ”19 “ fi ”20 ff 6,7 rst and bio-organic- rst teaching e orts. The general innovative mnemonic tools that help visualize and internalize the philosophy of “explicitly linking what we know about chemistry relations between different thermodynamic variables, graphical − with what we do in the classroom” is something we share with ideas that illustrate these dependencies8 10 and others that gauge 11,12 Ege and co-workers as they described in ref 19. However, there the effective portrayal of entropy and free energy through ff ’ 19 ’ 13,14 15−17 are critical di erences. In Figure 2 of Ege s work, physical connections to Shannon s information theory. chemistry appears at the pinnacle of undergraduate education. Our approach is to simplify the discussion of work and state Thus, grounding in the key physical topics of thermodynamics functions, through a pictorial introduction of these concepts − and kinetics is the last area students learn before they graduate. using pressure volume phase diagrams. These ideas are 18 subsequently used as a conduit to introduce other thermody- In contrast, in this publication and a previous publication, namic quantities. For example, a discussion on reversible and we have attempted to place the physical concepts underlying irreversible processes helps connect entropy to pressure− all of chemistry and biochemistry upfront. These principles volume phase diagrams, through work. When microscopic form the underpinnings of all chemistry and biochemistry and descriptions of entropy and free energy are invoked, this allows us to introduce chemical kinetics in a coherent manner. © XXXX American Chemical Society and Division of Chemical Education, Inc. A dx.doi.org/10.1021/ed400480t | J. Chem. Educ. XXXX, XXX, XXX−XXX Journal of Chemical Education Article Figure 1. Plots in (A) and (B) show the use of PV diagrams to explain the path dependence of work. Plot (C), which is derived from (A) and (B) is used to explain the concept of state and nonstate functions. if well understood, can lead to a more rigorous treatment of PV plot. Deviations in real gas behavior are then introduced other aspects of chemistry in later years. For example, as part of through modified PV diagrams. the material presented here for honors general chemistry, we Once PV diagrams are discussed, plots similar to those in included a discussion on Michaelis−Menton kinetics following Figure 1 are used to (i) convey the concept of work, (ii) note the introduction of chemical kinetics. It was found that once that work is not a state function, and (iii) emphasize work the students became familiar with the conceptual basis performed during a closed path in a phase diagram is related to presented here, the Michaelis−Menton problem was accessible. the area enclosed within the path. Toward this goal, we first A detailed discussion on this topic is however reserved for the recall that work is force multiplied by displacement. Utilizing upper-level biochemistry class, given the time frame restric- the standard textbook example of a cylinder with a moving tions; however, we believe that connecting the problems at this piston that contains a gas, we are then able to express the PV early stage would help the students when these ideas are work done on the system as revisited in upper level. This paper is organized as follows: The pressure−volume w =− PVd ∫ (1) (PV) phase diagrams are discussed in the next section with the goal of introducing work and heat; entropy and free energy From this launching point it is possible to compare different follow through the concept of reversibility, also introduced changes made in the PV diagram. For example, the plots in using PV diagrams. We then connect these macroscopic ideas Figure 1A,B show two different pathways for the transition to the microscopic description of entropy given by Boltzmann from (P1,V1)to(P2,V2). In the path depicted in Figure 1A, the fi to arrive at a microscopic view to spontaneity. Chemical pressure is rst quickly changed from P1 to P2, holding the kinetics is then introduced through this probabilistic micro- volume constant. Then, the volume is increased suddenly from fi scopic picture arising from the discussion of entropy and free V1 to V2 while holding the pressure xed at P2. Although the energy. The case is made that a reactive process is a rare event italicized words quickly and suddenly do not have much meaning when viewed from a probabilistic or statistical angle. This course to the student at this point, they become useful when we has a prescribed textbook;21 however, there are substantial discuss reversible and irreversible processes later. Hence, the deviations as one would expect based on analysis of this paper students are asked to note this terminology for later reference. and a previous publication.18 A detailed syllabus is presented as In the second path shown in Figure 1B, the pressure is initially part of Supporting Information. We believe that this presentation maintained at P1 and the volume is increased as before from V1 fi of the material, somewhat modi ed, may also be appropriate in to V2. Next, the pressure is decreased from P1 to P2 while part to upper-level physical chemistry classes. Hence, advanced holding the volume constant at V2. Since PV work done on the material more applicable for upper-level undergraduates is also system is given by eq 1, it is clear that no work is done in the discussed in Supporting Information. vertical transitions when the volume is held constant. Moreover, it is graphically evident to the students that the USING PV DIAGRAMS TO INTRODUCE WORK, work done in each of the two paths shown in Figure 1A,B is ■ ff HEAT, AND THE CONCEPT OF STATE FUNCTIONS di erent. Due to the higher pressure in the second case, the magnitude of the work done on the system in this case is larger We begin our treatment of energy and thermodynamics using than that in the first case. This result is very significant as it PV diagrams for an ideal gas. Although many students are implies that the work done is path dependent! intimately familiar with the numerical manipulation of ideal gas Students readily recognize that reversing the direction of the laws through their high school experience, they are typically not path in the first case (Figure 1A) reverses the sign of the work. comfortable in relating the mathematical equations to curves in One is now able to combine the two paths previously presented a PV diagram. Once they understand how all ideal gas laws, can to create a closed loop, as depicted in Figure 1C. The system be derived from PV diagrams, they are in a position to elucidate begins and ends at (P1,V1) and the work done in the process is physical relationships from xy plots such as the dependence of the difference between the work done in the two half-cycles the pressure of an ideal gas on its volume under constant shown in Figure 1A,B.