In the Classroom
A Chemically Relevant Model for Teaching the Second Law of Thermodynamics Bryce E. Williamson* Department of Chemistry, University of Canterbury, Private Bag 4800, Christchurch, New Zealand;
[email protected] Tetsuo Morikawa Department of Chemistry, Joetsu University of Education, Joetsu 943-8512, Japan
Introduction The thermodynamic quantities entropy and Gibbs energy play extremely important roles in chemistry. However, the majority of students find them to be among the most demanding of the concepts that they encounter in their university chemistry education. The connection between thermodynamics and spontaneity is usually introduced at the first-year university level, typically by reference to model processes such as the shuffling of cards or the free expansion of an ideal gas into a number of interconnected chambers, which are easily conceptualized by students. In the latter example, the fact that initial and final states have the same energy also demonstrates the fallacy of the naive belief that spontaneous change is driven by minimization of the internal energy of a system. At this point entropy is introduced as a basis for developing an authentic criterion for spontaneity. Despite wellrecognized limitations (1, 2), it is usually described in terms of randomness or disorder, with qualitative arguments that demonstrate that high-entropy states are very much more common than low-entropy states (3). Stochastic processes consequently and inevitably lead to a net increase of the entropy of the universe, a result that is formally stated as the second law of thermodynamics. Various approaches invoking concepts such as “disorder”, “freedom”, dispersal of energy, and the number of accessible microstates of a system can then be employed to rationalize the spontaneous direction of change for many chemical and physical transformations, including those for which energy is exchanged between the system and surroundings. Even having developed a solid grasp of the first-year treatment, many students struggle with the second law and its consequences at higher levels where entropy and Gibbs energy are more rigorously defined. In particular, they find it difficult to connect the theoretical formulations for idealized systems with the real chemical processes to which the thermodynamics is to be applied. This problem is exacerbated by the use of model systems that are outside the students’ experience and of little relevance to chemistry. Some of the most acute problems stem from a weak comprehension of the notions of path dependence and reversibility. To illuminate these concepts, physical chemistry textbooks conventionally appeal to hypothetical heat engines operating around Carnot cycles (4 ). Although such models might be appropriate in physics and engineering, they are of little direct relevance to chemistry, and most chemistry students find them very challenging. It would be better to replace (or at least supplement) them with models that involve
inherently chemical processes with which the students have prior familiarity. This article presents a chemically relevant model, based on an old research paper by Hans Jahn (5), that exemplifies many aspects of the second law of thermodynamics. The model illustrates concepts such as reversibility, path dependence, and extrapolation in terms of electrochemistry and calorimetry. These are important experimental methods for obtaining thermochemical data and will be familiar to most students who have completed first-year university chemistry. The Conventional Approach In chemical thermodynamics, the universe is divided into a system and its surroundings. The chemical process, including all reactants and products, is confined to the system but can influence the surroundings via exchange of energy. A firstyear university treatment of the second law is succinctly summarized by the Clausius inequality, which, for a process at temperature T, can be written ∆Suniv = ∆S – q/T ≥ 0
(1)
The net entropy change of the universe is ∆Suniv . ∆S is the entropy change of the system and ᎑q/T is the entropy change induced in the surroundings by transferred heat, q, defined to be negative if it is lost from the system.1 A corollary of eq 1, for an isothermal process at constant pressure (conditions that are generally regarded as being most relevant to chemists), is ∆G ≡ ∆H – T∆S ≤ 0
(2)
where ∆H and ∆G are, respectively, the enthalpy and Gibbs energy changes of the system. At first-year level, mention is generally made, though often in passing and without substantiation, that ∆G represents the maximum amount of energy “free” to do useful work under conditions of constant pressure and temperature (3). More precisely, it is the maximum disposable energy that we are free to use in any way we wish, provided we can devise an appropriate mechanism. This result is used to relate the electromotive force of a galvanic electrochemical cell to thermodynamic properties, but the principles underpinning this connection are not normally presented. At more advanced levels, entropy change is formally defined in terms of a reversible process (4). Under isothermal conditions, ∆S = qrev /T
(3)
where qrev refers to the heat change that occurs when the process is conducted by following a reversible path. This definition
JChemEd.chem.wisc.edu • Vol. 79 No. 3 March 2002 • Journal of Chemical Education
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In the Classroom
B R
∆E 1/R
0
T ∆S = q rev
we q rev A
∆G = we,rev we,rev
anode
q
∆H = q + we
cathode
Figure 1. A hypothetical apparatus for measuring the enthalpy, entropy, and Gibbs energy changes for an electrochemical process. A and B are ideal, isothermal constant-pressure calorimeters, which, respectively, accommodate an ideal electrochemical cell and a resistor. When the switch is closed, the work (we) required to shift electrons from the anode to the cathode through the resistor is dissipated as heat in calorimeter B. The remainder of the enthalpy change of the process is dissipated as heat (q) in calorimeter A.
Figure 2. Plots of the idealized energy changes as a function of 1/R, due to a spontaneous process occurring in the apparatus shown in Figure 1. The heat (q) is indicated by diamonds and the nonexpansion work (we) by circles. ∆G and T∆S are defined by the reversible (R = ∞) limits we,rev and qrev, determined by extrapolation. The state function ∆ H is the sum of q and w e and is independent of R. In this case, ∆S is negative and the process is necessarily exothermic.
makes S a state function, as can be demonstrated by using a Carnot cycle (4). Furthermore, for a constant-pressure process it allows a formal derivation of the result that the maximum extra work (in excess of any work required to maintain constant pressure) that can done by the process is achieved by following a reversible path and is equal to ∆G:
current passing through the resistor. The calorimeters and the components of the external circuit do not constitute part of the system, which is composed of the cell only. The cell is immersed in ice-water to maintain isothermal conditions.2 Rather than a change of temperature, energy changes in calorimeter A are manifest as changes of the quantities of ice and water.3 When the switch is closed, a spontaneous chemical reaction within the cell will generate or absorb heat, q, resulting in the melting of ice or freezing of water in calorimeter A. A small amount of expansion work will be required to maintain constant pressure but, more importantly to this experiment, work will also be done by the system to shift electrons across the resistor from the anode to the cathode. The latter constitutes the extra work, we, which is actually dissipated as heat by the resistor in calorimeter B. No assumption is made that the process is thermodynamically reversible. Instead, the required physical quantities under reversible conditions will be determined by extrapolation. A series of experiments is conducted with different values of R. The switch is closed until the cell reaction is complete to a predefined extent. The enthalpy change is given by
∆G = we,max = we,rev
(4)
Effective teaching of physical sciences should convey not only the theoretical concepts, but also how the associated quantities are physically manifest in a particular context and how measurements are made of them. Although most physical chemistry students can follow the algebra leading to eq 4, many find it difficult to relate to physical reality. Without appropriate illustration and guidance, the second law can become a blur of obscure symbols dealt with by rote learning and “hand-cranking” computation. Anyone who accepts these precepts must conclude that the Carnot cycle is, at best, of marginal utility in the teaching of chemical thermodynamics. It seems to us that the subject is long overdue for a chemical relevant alternative. A Chemically Relevant Thought Experiment The following thought experiment uses an ideal galvanic electrochemical cell, free from junction potentials. The electrodes of the cell are connected, using wires that are perfect electrical conductors and thermal insulators, through an electrical circuit comprising a switch and a resistor of resistance R. The cell and resistor are placed in separate calorimeters, A and B, respectively (Fig. 1), which are perfect thermal insulators. The only energy transfer that can occur between calorimeters is that associated (according to Joule’s law) with the electrical 340
∆H = q + we
(5)
The dependence of q and we on R is represented in Figure 2. In the short-circuit limit (R = 0) all of the energy change occurs in calorimeter A; we = 0 and q = ∆H. As R is increased, the current will be reduced and the reaction time will be longer. At the same time, we becomes negative (the system is doing work) but this effect is exactly compensated by an increase in q (the cell dissipates less, or absorbs more, heat) so that ∆H is unchanged.4 This is a reflection of the fact that enthalpy is a state function.
Journal of Chemical Education • Vol. 79 No. 3 March 2002 • JChemEd.chem.wisc.edu
In the Classroom A
Table 1. Examples of Redox Reactions (6)
∆E q rev
Cu2+(aq) + Zn(s) → Cu(s) + Zn2+(aq)
2
q
3A Cu2+(aq) + Cd(s) → Cu(s) + Cd2+(aq)
T ∆S
1/R
0
∆H °298K / ∆G °298K / T∆S °298K / kJ mol ᎑1 kJ mol ᎑1 kJ mol ᎑1
Cell Reactiona
Fig.
3B
∆H
᎑
᎑219
᎑213
᎑6.3
᎑141
᎑143
2.3
13
᎑73
᎑
MnO4 (aq) + 8H (aq) + 5Cl (aq) → Mn2+(aq) + 4H2O(ᐉ) + 5⁄2Cl2(g)
aAll
+
86
solutes are at unit activity.
∆G we
Hence the absolute value of ∆G (remembering ∆G < 0 for a spontaneous process) represents the maximum amount of energy that can (in principle) be employed to do useful work. The thermodynamics of all possible spontaneous electrochemical processes conducted with the apparatus in Figure 1 can be represented using diagrams of the type in Figure 2. If ∆G is normalized, such diagrams differ only by displacement of the curve for q along the energy axis (compare Figs. 2 and 3). These representations also provide a useful graphical tool for summarizing some of the general consequences of the second law. For example, if, as in Figure 2, ∆S is negative, the system must lose heat (q < 0) sufficiently to increases the entropy of the surroundings (by, say, melting ice) by at least the same amount. Consequently, q must be negative for all paths (all values of R in the context of this experiment) and the process is necessarily exothermic. If ∆S is positive (Fig. 3), then the system can absorb heat (q > 0) to the extent that the decrease in entropy induced in the surrounding matches the increase for the system. The process may then be either exothermic (Fig. 3A) or endothermic (Fig. 3B). Examples for each of these cases are listed in Table 1.
we,rev
B ∆E q rev
T ∆S
q
1/R ∆H
0
∆G
we,rev
Figure 3. Idealized energy-change plots for two processes for which ∆S is positive. Depending on the magnitude of T∆S, the process can be (A) exothermic or (B) endothermic.
In the limit as R tends to infinity, the system is never far from equilibrium (the rate of the process becomes infinitesimal) and q and we will asymptotically approach their reversible (infinite-R) values, qrev and we,rev. Although these quantities cannot be measured directly, they can be obtained by extrapolating to 1/R = 0 (Fig. 2). From the definition of entropy in eq 3, the result for heat can be expressed as q ≤ qrev = T∆S
(6)
So T∆S is the least negative exchange of heat that can be achieved for the spontaneous process at temperature T. In other words, it represents that part of the energy change that must necessarily be transferred between the surroundings and the system in order to accommodate the entropy change of the latter. This energy is unavoidably thermal and can never be employed to do useful work. From eqs 5, 6, and 2 one deduces for the nonexpansion work, we ≥ we,rev = ∆H – T∆S = ∆G
(7)
Work, Gibbs Energy, and Electromotive Force An immediate advantage of the model described above is that it leads simply and directly to an illustration of the relationship between Gibbs energy and the electromotive force of an electrochemical cell. The current through the external circuit is associated with an electrical potential (a voltage), Ᏹ, across the resistor. The work required to shift charge Q across this potential is we = ᏱQ
(8)
In the thought experiment, Q is determined by the predefined extent to which the reaction proceeds; hence the absolute magnitude of ᏱQ is clearly maximized by using the largest possible Ᏹ. Following the discussion around eqs 6 and 7, the extra work done by the cell is maximized as R tends to infinity. At the same time, the voltage across the resistor tends to the open-circuit electromotive force (emf), Ᏹcell, of the cell. So, from eq 4, one has ∆G = ᏱcellQ
(9)
The molar change in Gibbs energy is therefore given by ∆G = ᎑nF Ᏹcell
(10)
where n is the number of electrons transferred per unit of reaction (defined by the reaction stoichiometry) and the Faraday constant, F, is the absolute charge of one mole of electrons.
JChemEd.chem.wisc.edu • Vol. 79 No. 3 March 2002 • Journal of Chemical Education
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In the Classroom
Summary
Notes
The important points of this note can be summarized as follows for a chemical reaction under isothermal, constantpressure conditions. A system and its surroundings can exchange energy as heat and work, both of which are path dependent. ∆H is the net change of energy apart from a small amount of work required to maintain constant pressure—it comprises the sum of heat and nonexpansion work (eq 5). T∆S represents the part of the energy change that must be thermal for the process to be spontaneous at temperature T (eq 6)—it can never be used to do useful work. The maximum amount of energy that can be employed to do useful (nonexpansion) work is achieved under reversible conditions (eq 7) and is equal to ∆G (eq 4). Reversible paths are unachievable for real processes, but the thermodynamic parameters pertaining to such paths can be determined by extrapolation to asymptotic limits. All of these points are contained in conventional textbook treatments of chemical thermodynamics. However, the chemically relevant model presented here should give students a better appreciation of the significance of the thermodynamic concepts and quantities in a chemical context. Diagrams of the type shown in Figure 2 and 3 elegantly summarize the relationships between the physical quantities pertaining to the second law and clearly illustrate the path dependence of heat and work as the process is varied between its reversible and extreme irreversible limits.
1. We adopt the convention that all changes refer to the system. Thus, q and w are negative if they represent loss of energy from the system to the surroundings. 2. In variations of this experiment, calorimeter A could be a conventional constant-pressure or constant-volume calorimeter. The major advantage of the “ice-water” apparatus is that it permits one to avoid the complications of discussing the determination of isothermal properties from changes of temperature. 3. This amounts to an entropy change of the surroundings, since the entropy of the water is greater than that of the ice. 4. If the resistor in calorimeter B were also immersed in ice-water, the total amount of ice melted (or water frozen) in both calorimeters would be independent of the path.
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Literature Cited 1. Styer, D. F. Am. J. Phys. 2000, 68, 1090–1096. 2. Lambert F. L. J. Chem. Educ. 2002, 79, 187–192. 3. Chang, R. Chemistry, 5th ed.; McGraw Hill: Boston, 1998; pp 726–727. 4. Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University Press: Oxford, 1998; pp 100–105. 5. Jahn, H. Z. physik. Chem. 1895, 18, 399–425. 6. CRC Handbook of Chemistry and Physics, 71st ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1990.
Journal of Chemical Education • Vol. 79 No. 3 March 2002 • JChemEd.chem.wisc.edu
